COMP90048
Declarative Programming Subject Notes for Semester 1, 2020
search bst :: Tree k v −> k −> Maybe v search bst Leaf = Nothing
search bst (Node k v l r) sk =
if sk == k Just v else if sk
then
< k then search bst l sk
else
search bst r sk
School of Computing and Information Systems
The University of Melbourne
A language that doesn’t affect the way you think about programming, is not worth knowing.
— Alan Perlis
Section 0: Subject Introduction
Welcome to COMP90048 Declarative Programming
Lecturer: Peter Schachte and Michelle Blom
Contact information is available from the LMS.
Peter will mostly teach the functional programming part of the subject and Michelle will mostly teach the logic programming part. There will be two one-hour lectures per week.You are welcome to ask questions during or after a lecture.
There will be eleven one-hour workshops (labs), starting in week 2.
You should have already been allocated a workshop. Please check your personal timetable after the lec- ture.
• PleaseturnmobilephonesOFFduringlectures and workshops.
• Please do not use your laptop for anything dis- tracting to others.
NOTE That includes playing games, checking Face- book, displaying still or moving pictures (even if done by screensavers), and anything that generates noise, including loud keyclicks.
– section 0 slide 1 – Grok
We use Grok to provide added self-paced instruc- tional material, exercises, and self-assessment for both Haskell and Prolog.
You can access Grok by following the link from the subject LMS page.
If you are unable to access Grok or find that it is not working correctly, please email
Grok University Support ~~~~
from your university email account and explain the problem.
If you have questions regarding the Grok lessons or exercises, please post a message to the subject LMS discussion forum.
– section 0 slide 2 –
Workshops
The workshops will reinforce the material from lec- tures, partly by asking you to apply it to small scale programming tasks.
To get the most out of each workshop, you should read and attempt the exercises before your work- shop.
This semester, workshops will be available through Grok, so they can be undertaken even if you are not present in Australia. Sample solutions for each set of workshop exercises will also be available through Grok.
Most programming questions have more than one correct answer; your answer may be correct even if it differs from the sample solution.
NOTE If your laptop can access the building’s wire- less network, you will be able to log onto the MSE servers and use their installed versions of this sub- ject’s languages, Haskell and Prolog. If your lap- top cannot access the building’s wireless network,
then you will be able to test your Haskell or Pro- log code if you install the implementations of those languages on your machine yourself. For both lan- guages this is typically fast and simple.
– section 0 slide 3 –
Resources
The lecture notes contain copies of the slides pre- sented in lectures, plus some additional material.
All subject materials (lecture notes, workshop ex- ercises, project specifications etc) will be available online through the LMS.
The recommended text is
• Bryan O’Sullivan, John Goerzen and Don Stewart: Real world Haskell. O’Reilly Media, 2009. ISBN 978-0-596-51498-3. Available on- line at http://book.realworldhaskell.org/read.
Other recommended resources are listed on the LMS.
– section 0 slide 4 –
Assessment
The subject has the following assessment compo- nents:
5% short Haskell project, is due in Week 4
15% larger Haskell project, due in Week 6 or 7 5% short Prolog project, due in Week 9
15% larger Prolog project, due in Week 11 or 12 60% two-hour written final examination
To pass the subject (get 50%), you must pass both the project component and the exam component.
The exam is closed book, and will be held during the usual examination period after the end of the semester. Practice versions of the final exam are available on the LMS.
– section 0 slide 5 –
Disruptions
This semester we are transitioning to a new LMS. This has turned out to be more disruptive than expected in a number of ways.
On top of that, this semester almost half of the students in this class are unable to enter Australia to attend classes, and we do not know when they will arrive.
We will do all we can to accommodate these disrup- tions, but we ask your patience as we try to make things run as smoothly as we can.
– section 0 slide 6 – Academic Integrity
All assessment for this subject is individual; what you submit for assessment must be your work and your work alone.
It is important to distinguish project work (which is assessed) from tutorials and other unassessed ex- ercises.
We are well aware that there are many online sources of material for subjects like this one; you are encouraged to learn from any online sources, and from other students, but do not submit for assessment anything that is not your work alone.
Do not provide or show your project work to any other student.
Do not store your project work in a public Github or other repository.
We use sophisticated software to find code that is similar to other submissions this year or in past years. Students who submit another person’s work as their own or provide their work for another stu- dent to submit in whole or in part will be subject to disciplinary action.
– section 0 slide 7 –
How to succeed
Declarative programming is substantially different from imperative programming.
Even after you can understand declarative code, it can take a while before you can master writing your own.
If you have been writing imperative code all your programming life, you will probably try to write im- perative code even in a declarative language. This often does not work, and when it does work, it usu- ally does not work well.
Writing declarative code requires a different mind- set, which takes a while to acquire.
This is why attending the workshops, and practic- ing, practicing and practicing some more are essen- tial for passing the subject.
– section 0 slide 8 –
Sources of help During contact hours:
• Ask me during or after a lecture (not before). • Ask the demonstrator in your workshop.
Outside contact hours:
• The LMS discussion board (preferred: every- one can see the discussion)
• Email (if not of interest to everyone)
• Attend my consultation hours (see LMS for
schedule)
• Email to schedule an appointment
Announcements will be made both in lectures and on the LMS.
Please do not pose a question both on the LMS and in private email; if it is on the LMS, I will see it.
– section 0 slide 9 –
Do you want to represent this class?
Would you like to represent this class on the de- partment’s student-staff liaison committee?
The job is to collect the concerns of your classmates and present them to me and to the committee. I will post your name and email address on the LMS so classmates can contact you.
The bribe is that you get pizza, and something to put on your resum ́e, and the knowledge you’ve done something to help both your classmates and the university, and possibly make the subject better for yourself.
If you would like to volunteer, please email me. – section 0 slide 10 –
Objectives
On completion of this subject, students should be able to:
• apply declarative programming techniques;
• write medium size programs in a declarative language;
• write programs in which different components use different languages;
• select appropriate languages for each compo-
nent task in a pro ject.
These objectives are not all of equal weight; we will spend almost all of our time on the first two objec- tives.
– section 0 slide 11 – Content
• introduction to functional programming and Haskell
• declarative programming techniques
• introduction to logic programming and Prolog
• tools for declarative programming, such as de- buggers
• interfacing to imperative language code
• introduction to constraint programming
This subject will teach you Haskell and Prolog, with an emphasis on Haskell.
– section 0 slide 12 –
Why Declarative Programming
Declarative programming languages are quite dif- ferent from imperative and object oriented lan- guages.
• They give you a different perspective: a focus on what is to be done, rather than how.
• They work at a higher level of abstraction.
• They make it easier to use some powerful pro-
gramming techniques.
• Their clean semantics means you can do things with declarative programs that you can’t do with conventional programs.
The ultimate objective of this subject is to widen your horizons and thus to make you better pro- grammers, and not just when using declarative pro- gramming languages.
– section 0 slide 13 – Imperative vs functional
Imperative languages are based on commands, in the form of instructions and statements.
• Commands are executed. • Commands have an effect.
Assignment statements update state, and later code may depend on this update. Example: x := 3 + 4 ; …
Functional languages are based on expressions. • Expressions are evaluated.
• Expressions have no effect.
– section 0 slide 14 –
Side effects
Code is said to have a side effect if, in addition to producing a value, it also modifies some state or has an observable interaction with calling functions or the outside world. For example, a function might
• modify a global or a static variable,
• modify one of its arguments,
• raise an exception (e.g. divide by zero),
• write data to a display, file or network,
• read data from a keyboard, mouse, file or net- work, or
• call other side-effecting functions.
NOTE Reading from a file is a side effect because it moves the current position in the file being read, so that the next read from that file will get something else.
– section 0 slide 15 –
An example: destructive update
In imperative languages, the natural way to insert a new entry into a table is to modify the table in place: a side-effect. This effectively destroys the old table.
In declarative languages, you would instead create a new version of the table, but the old version (with- out the new entry) would still be there.
The price is that the language implementation has to work harder to recover memory and to ensure efficiency.
The benefit is that you don’t need to worry what other code will be affected by the change. It also allows you to keep previous versions, for purposes of comparison, or for implementing undo.
The immutability of data structures also makes par- allel programming much easier. Some people think that programming the dozens of cores that CPUs will have in future is the killer application of declar- ative programming languages.
– section 0 slide 16 –
• •
•
Guarantees
If you pass a pointer to a data structure to a function, can you guarantee that the function does not update the data structure?
If not, you will need to make a copy of the data structure, and pass a pointer to that.
You add a new field to a structure. Can you guarantee that every piece of code that handles the structure has been updated to handle the new field?
If not, you will need many more test cases, and will need to find and fix more bugs.
Can you guarantee that this function does not read or write global variables? Can you guar- antee that this function does no I/O?
If the answer to either question is “no”, you will have much more work to do during testing and debugging, and parallelising the program will be a lot harder.
– section 0 slide 17 –
Some uses of declarative languages
In a US Navy study in which several teams wrote code for the same task in several lan- guages, declarative languages like Haskell were much more productive than imperative lan- guages.
Mission Critical used Mercury to build an in- surance application in one third the time and cost of the next best quote (which used Java).
Ericsson, one of the largest manufacturers of phone network switches, uses Erlang in some of their switches.
•
• •
• The statistical machine learning algorithms be- hind Bing’s advertising system are written in F#.
• Facebook used Haskell to build the system they use to fight spam. Haskell allowed them to in- crease power and performance over their pre- vious system.
NOTE Erlang, F# and Lisp are of course declara- tive languages.
For a whole bunch of essays about programming, in- cluding some about the use of Lisp in Yahoo! Store, see paulgraham.com.
– section 0 slide 18 –
The Blub paradox
Consider Blub, a hypothetical average program- ming language right in the middle of the power con- tinuum.
When a Blub programmer looks down the power continuum, he knows he is looking down. Lan- guages below Blub are obviously less powerful, be- cause they are missing some features he is used to.
But when a Blub programmer looks up the power continuum, he does not realize he is looking up. What he sees are merely weird languages. He thinks they are about equivalent in power to Blub, but with some extra hairy stuff. Blub is good enough for him, since he thinks in Blub.
When we switch to the point of view of a program- mer using a language higher up the power contin- uum, however, we find that she in turn looks down upon Blub, because it is missing some things she is used to.
Therefore understanding the differences in power between languages requires understanding the most powerful ones.
NOTE This slide is itself paraphrased from one of Paul Graham’s essays. (The full quotation is too big to fit on one slide.)
The least abstract and therefore least powerful lan- guage is machine code. One step above that is as- sembler, and one step above that are the lowest level imperative languages, like C and Fortran. Ev- eryone agrees on that. Most people (but not all) would also agree that modern object-oriented lan- guages like Java and C#, scripting languages like awk and Perl and multi-paradigm languages like Python and Ruby are more abstract and more pow- erful than C and Fortran, but there is no general agreement on their relative position on the contin- uum. However, almost everyone who knows declar- ative programming languages agrees that they are more abstract and more powerful than Java, C#, awk, Perl, Python and Ruby.
A large part of that extra power is the ability to offer many more guarantees.
– section 0 slide 19 –
Section 1: Introduction to Functional Programming
Functional programming
The basis of functional programming is equational
reasoning. This is a grand name for a simple idea: • if two expressions have equal values, then one
can be replaced by the other.
You can use equational reasoning to rewrite a com- plex expression to be simpler and simpler, until it is as simple as possible. Suppose x = 2 and y = 4, and you start with the expression x + (3 * y):
The notation [] means the empty list, while x:xs means a nonempty list whose head (first element) is represented by the variable x, and whose tail (all the remaining elements) is represented by the vari- able xs.
The notation [“a”, “b”] is syntactic sugar for “a”:”b”:[]. As in most languages, “a” represents the string that consists of a single character, the first character of the alphabet.
– section 1 slide 2 –
Functions
A function definition consists of equations, each of which establishes an equality between the left and right hand sides of the equal sign.
len [] = 0
len (x:xs) = 1 + len xs
Each equation typically expects the input argu- ments to conform to a given pattern; [] and (x:xs) are two patterns.
The set of patterns should be exhaustive: at least one pattern should apply for any possible call.
It is good programming style to ensure that the set of patterns is also exclusive, which means that at most one pattern should apply for any possible call.
If the set of patterns is both exhaustive and exclu- sive, then exactly one pattern will apply for any possible call.
NOTE ghc, the most popular Haskell compiler, has options you can specify if you want some help following these rules of programming style. If given the option -fwarn-incomplete-patterns, ghc will give you a warning if the pat- terns are not exhaustive; if given the option -fwarn-overlapping-patterns, ghc will give you a warning if the patterns are not exclusive.
step 0:
step 1:
step 2:
step 3: 2+12 step 4: 14
x+(3 *y) 2+(3 *y) 2+(3 *4)
– section 1 slide 1 –
Lists
Of course, programs want to manipulate more com- plex data than just simple numbers.
Like most functional programming languages, Haskell allows programmers to define their own types, using a much more expressive type system than the type system of e.g. C.
Nevertheless, the most frequently used type in Haskell programs is probably the builtin list type.
If the set of patterns for a function is not exhaus- tive, and no equation applies for a given function call, then the function call will generate an excep- tion, which will typically cause the program to be aborted.
– section 1 slide 3 – Aside: syntax
• In most languages, a function call looks like f(fa1, fa2, fa3).
• In Haskell, it looks like f fa1 fa2 fa3.
If the second argument is not fa2 but the function g applied to the single argument ga1, then in Haskell you would need to write
f fa1 (g ga1) fa3
since Haskell would interpret f fa1 g ga1 fa3 as
a call to f with four arguments.
In Haskell, there are no parentheses around the whole argument list of a function call, but paren- theses may be needed around individual arguments. This applies on the left as well the right hand sides of equations.
This is why the recursive call is len xs and not len(xs), and why the left hand side of the second equation is len (x:xs) instead of len x:xs.
– section 1 slide 4 –
More syntax issues
Comments start with two minus signs and continue to the end of the line.
The names of functions and variables are sequences of letters, numbers and/or underscores that must start with a lower case letter.
Suppose line1 starts in column n, and the follow- ing nonblank line, line2, starts in column m. The offside rule says that
• if m > n, then line2 is a continuation of the construct on line1;
• ifm=n,thenline2isthestartofanewcon- struct at the same level as line1;
• if m < n, then line2 is either the continuation of something else that line1 is part of, or a new item at the same level as something else that line1 is part of.
This means that the structure of the code as shown by indentation must match the structure of the code.
NOTE Actually, it is also ok for function names to consist wholly of graphic characters like +, but only builtin functions should have such names.
If your source code includes tabs as well as spaces, two lines can look like they have the same level of indentation even if they do not. It is therefore best to ensure that Haskell source files do not contain tabs.
If you use vim as your editor, you can ensure this by putting
-- vim: ts=4 sw=4 expandtab syntax=haskell
at the top of the source file. The expandtab tells vim to expand all tabs into several spaces, while the ts=4 and sw=4 tell vim to set up tab stops every four columns. The syntax=haskell specifies the set of rules vim should use for syntax highlighting.
The following function definition is wrong: since the second line is indented further than the first, Haskell considers it to be a continuation of the first equation, rather than a separate equation.
len [] = 0
len (x:xs) = 1 + len xs
The following definition is acceptable to the offside rule, though it is not an example of good style:
len [] = 0
len (x:xs) = 1+
len xs
step 0: len ["a", "b"] -- ("a":("b":[])) step 1: 1 + len ["b"] -- ("b":[]) step2: 1+1+len[]
step3: 1+1+0
step4: 1+1 step 5: 2
NOTE In general, if there is more than one applica- ble equation, the Haskell implementation picks the first one.
You can think of builtin functions as being implic- itly defined by a very long list of equations; on a 32 bit machine, you would need 232 ∗ 232 = 264 equations. In practice, such functions are of course implemented using the arithmetic instructions of the platform.
– section 1 slide 7 –
Expression evaluation
To evaluate an expression, the Haskell runtime sys- tem conceptually executes a loop, each iteration of which consists of these steps:
• looks for a function call in the current expres- sion,
• searches the list of equations defining the func- tion from the top downwards, looking for a matching equation,
• sets the values of the variables in the match- ing pattern to the corresponding parts of the actual arguments, and
• replaces the left hand side of the equation with the right hand side.
The loop stops when the current expression con- tains no function calls, not even calls to such builtin “functions” as addition.
The actual Haskell implementation is more sophis- ticated than this loop, but the effect it achieves is the same.
– section 1 slide 5 –
Recursion
The definition of a function to compute the length of a list, like many Haskell functions, reflect the structure of the data: a list is either empty, or has a head and a tail.
The first equation for len handles the empty list case.
len [] = 0
This is called the base case.
The second equation handles nonempty lists. This is called the recursive case, since it contains a re- cursive call.
len (x:xs) = 1 + len xs
If you want to be a good programmer in a declar- ative language, you have to get comfortable with recursion, because most of the things you need to do involve recursion.
– section 1 slide 6 – Using a function
len []
len (x:xs) = 1 + len xs
Given a function definition like this, the Haskell implementation can use it to replace calls to the function with the right hand side of an applicable equation.
= 0
– section 1 slide 8 –
Order of evaluation
The first step in each iteration of the loop, “look for a function call in the current expression”, can find more than one function call. Which one should we select?
len [] = 0
len (x:xs) = 1 + len xs
evaluation order A:
evaluation order B:
step 0: len ["a"] + len [] step 1: len ["a"] + 0 step2: 1+len[]+0 step3: 1+0+0
step4: 1+0
step 5: 1
NOTE In this example, evaluation order A chooses the leftmost call, while evaluation order B chooses the rightmost call.
– section 1 slide 9 –
Church-Rosser theorem
In 1936, Alonzo Church and J. Barkley Rosser proved a famous theorem, which says that for the rewriting system known as the lambda calculus, re- gardless of the order in which the original term’s
subterms are rewritten, the final result is always the same.
This theorem also holds for Haskell and for several other functional programming languages (though not for all).
This is not that surprising, since most modern func- tional languages are based on one variant or another of the lambda calculus.
We will ignore the order of evaluation of Haskell expressions for now, since in most cases it does not matter. We will come back to the topic later.
The Church-Rosser theorem is not applicable to im- perative languages.
NOTE Each rewriting step replaces the left hand side of an equation with the right hand side.
– section 1 slide 10 – Order of evaluation: efficiency
all_pos [] = True
all_pos (x:xs) = x > 0 && all_pos xs
evaluation order A:
0: all_pos [-1, 2] 1:-1>0&&all_pos[2] 2: False && all_pos [2] 3: False
evaluation order B:
0: all_pos [-1, 2]
1: -1 > 0 && all_pos [2] 2:-1>0&&2>0&&all_pos[] 3: -1 > 0 && 2 > 0 && True
4: -1 > 0 && True && True
5: -1 > 0 && True
6: False && True
7: False
step 0: step 1: step2: step 3: step4: step 5: 1
len [“a”] + len [] 1+len[]+len[] 1+0+len[] 1+len[]
1+0
NOTE The definition of conjunction for Booleans does not need to know the value of second operand if the value of the first operand is False:
False & _ = False True &b=b
Note that all pos, like len, has one equation for empty lists and one equation for nonempty lists. Functions that operate on data with the similar structures often have similar structures themselves.
– section 1 slide 11 –
Imperative vs declarative languages
In the presence of side effects, a program’s behavior depends on history; that is, the order of evaluation matters.
Because understanding an effectful program re- quires thinking about all possible histories, side ef- fects often make a program harder to understand.
When developing larger programs or working in teams, managing side-effects is critical and difficult; Haskell guarantees the absence of side-effects.
What really distinguishes pure declarative lan- guages from imperative languages is that they do not allow side effects.
There is only one benign exception to that: they do allow programs to generate exceptions.
We will ignore exceptions from now on, since in the programs we deal with, they have only one effect: they abort the program.
NOTE This is OK. The only thing that depends on the order of evaluation is which of several ex- ceptions that a program can raise will actually be raised. Unless you are writing the exception han- dler, you still don’t have to understand all possible histories.
– section 1 slide 12 –
Referential transparency
The absence of side effects allows pure functional languages to achieve referential transparency, which means that an expression can be replaced with its value. This requires that the expression has no side effects and is pure, i.e. always returns the same re- sults on the same input.
By contrast, in imperative languages such as C, functions in general are not pure and are thus not functions in a mathematical sense: two identical calls may return different results.
Impure functional languages such as Lisp are called impure precisely because they do permit side effects like assignments, and thus their programs are not referentially transparent.
NOTE Impure functional languages share charac- teristics both with imperative languages and with pure functional languages, so they are effectively somewhere between them on the spectrum of pro- gramming languages.
In the rest of the subject, we will use “functional languages” as a shorthand to mean “pure functional languages”, except in contexts where we are specif- ically talking about impure functional languages.
Even in imperative languages, some functions are pure. This is usually true e.g. of implementations of actual mathematical functions, such as logarithm, sine, cosine, etc, but also of many others.
– section 1 slide 13 –
Single assignment
One consequence of the absence of side effects is that assignment means something different in a functional language than in an imperative lan- guage.
• In conventional, imperative languages, even object-oriented ones (including C, Java, and Python), each variable has a current value (a garbage value if not yet initialized), and assignment statements can change the current value of a variable.
• In functional languages, variables are single as- signment, and there are no assignment state- ments. You can define a variable’s value, but you cannot redefine it. Once a variable has a value, it has that value until the end of its lifetime.
– section 1 slide 14 –
Giving variables values
Haskell programs can give a variable a value in one of two ways.
The explicit way is to use a let clause: let pi = 3.14159 in …
This defines pi to be the given value in the expres- sion represented by the dots. It does not define pi anywhere else.
The implicit way is to put the variable in a pattern on the left hand side of an equation:
len (x:xs) = 1 + len xs
If len is called with a nonempty list, Haskell will
bind x to its head and xs to its tail.
NOTE Actually, there are other ways, both explicit
and implicit, but these are enough for now. – section 1 slide 15 –
Section 2: Builtin Haskell types
The Haskell type system
Haskell has a strong, safe and static type system.
The strong part means that the system has no loop- holes; one cannot tell Haskell to e.g. consider an integer to be a pointer, as one can in C with (char *) 42.
The safe part means that a running program is guaranteed never to crash due to a type error. (A C program that dereferenced the above pointer would almost certainly crash.)
The static part means that types are checked when the program is compiled, not when the program is run.
This is partly what makes the safe part possible; Haskell will not even start to run a program with a type error.
NOTE There is an “out” from the static nature of the type system, for use in cases where this is warranted Haskell also supports a dynamic type, and operations on values of this type are checked for type correctness only at runtime.
Note also that different people mean different things when they talk about e.g. the strength or safety of a type system, so these are not the only definitions you will see in the computer science lit- erature. However, we will use these definitions in this subject.
– section 2 slide 1 –
Basic Haskell types
Haskell has the usual basic types. These include:
• The Boolean type is called Bool. It has two values: True and False.
• The native integer type is called Int. Values of this type are 32 or 64 bits in size, depending on the platform. Haskell also has a type for integers of unbounded size: Integer.
• The usual floating-point type is Double. (Float is also available, but its use is discour- aged.)
• The character type is called Char.
There are also others, e.g. integer types with 8, 16,
32 and 64 bits regardless of platform. There are more complex types as well.
– section 2 slide 2 – The types of lists
In Haskell, list is not a type; it is a type constructor.
Given any type t, it constructs a type for lists whose elements are all of type t. This type is writ- ten as [t], and it is pronounced as “list of t”.
You can have lists of any type. For example,
• [Bool] is the type of lists of Booleans,
• [Int] is the type of lists of native integers,
• [[Int]] is the type of lists of lists of native integers.
These are similar to LinkedList
Haskell considers strings to be lists of characters, whose type is [Char]; String is a synonym for [Char].
The names of types and type constructors should be identifiers starting with an upper case letter; the list type constructor is an exception.
NOTE In fact, you can have lists of lists of lists of lists [[[[Int]]]], lists of lists of lists of lists of lists [[[[[Int]]]]], and so on. The only effective limit is the programmer’s ability to do something useful with the values of the type.
– section 2 slide 3 – ghci
The usual implementation of Haskell is ghc, the Glasgow Haskell Compiler. It also comes with an interpreter, ghci.
The prelude is Haskell’s standard library.
ghci uses its name as the prompt to remind users
that they can call its functions. NOTE Once you invoke ghci,
• you type an expression on a line;
• it typechecks the expression;
• it evaluates the expression (if it is type cor- rect);
• it prints the resulting value.
$ ghci
…
Prelude> let x = 2 Prelude> let y = 4 Prelude> x + (3 * y) 14
You can also load Haskell code into ghci with :load filename.hs. The suffix .hs, the standard suffix for Haskell source files, can be omitted.
– section 2 slide 4 – Types and ghci
You can ask ghci to tell you the type of an ex- pression by prefacing that expression with :t. The command :set +t tells ghci to print the type as well as the value of every expression it evaluates.
The notation x::y says that expression x is of type y. In this case, it says “abc” is of type [Char].
it is ghci’s name for the value of the expression just evaluated.
– section 2 slide 5 – Function types
You can also ask ghci about the types of functions. Consider this function, which checks whether a list is empty:
isEmpty [] = True isEmpty (_:_) = False
( is a special pattern that matches anything.) If you ask ghci about its type, you get
A function type lists the types of all the arguments and the result, all separated by arrows. We’ll see what the a means a bit later.
NOTE This function is already defined in standard Haskell; it is called null. The len function is also already defined, under the name length.
– section 2 slide 6 –
Function types
Programmers should declare the type of each func- tion. The syntax for this is similar to the notation printed by ghci: the function name, a double colon, and the type.
module Emptiness where
isEmpty :: [t] -> Bool isEmpty [] = True isEmpty _ = False
Declaring the type of functions is required only by good programming style. The Haskell implementa- tion will infer the types of functions if not declared.
Haskell also infers the types of all the local vari- ables.
Later in the subject, we will briefly introduce the algorithm Haskell uses for type inference.
NOTE A Haskell source file should contain one Haskell module. We will cover the Haskell module system later.
– section 2 slide 7 – Function type declarations
> :t isEmpty
isEmpty :: [a] -> Bool
Prelude> :t “abc” “abc” :: [Char] Prelude> :set +t Prelude> “abc” “abc”
…
it :: [Char]
With type declarations, Haskell will report an error and refuse to compile the file if the declared type of a function is incompatible with its definition.
It’s also an error if a call to the function is incom- patible with its declared type.
Without declarations, Haskell will report an error if the types in any call to any function are incom- patible with its definition. Haskell will never allow code to be run with a type error.
Type declarations improve Haskell’s error mes- sages, and make function definitions much easier to understand.
– section 2 slide 8 – Number types
Haskell has several numeric types, including Int, Integer, Float, and Double. A plain integer con- stant belongs to all of them. So what does Haskell say when asked what the type of e.g. 3 is?
In these messages, a and p are type variables; they are variables that stand for types, not values.
The notation Num p means “the type p is a member of type class Num”. Num is the class of numeric types, including the four types above.
The notation 3 :: Num p => p means that “if p is a numeric type, then 3 is a value of that type”.
NOTE Similarly, the notation [1, 2] :: Num a => [a] means that “if a is a numeric type, then [1, 2] is a list of values of that type”.
– section 2 slide 9 –
Number type flexibility
The usual arithmetic operations, such as addition, work for any numeric type:
The notation a -> a -> a denotes a function that takes two arguments and returns a result, all of which have to be of the same type (since they are denoted by the same type variable, a), which in this case must be a member of the Num type class.
This flexibility is nice, but it does result in confus- ing error messages:
NOTE The error message is trying to say that Bool, the type of True, is not a member or instance of Num type class. If it were a numeric type, then the list could a list of elements of that type, since the integer constant is a member of any numeric type, and thus the types of the two elements would be the same (which must hold for all Haskell lists).
The fix suggested by the error message is to have the programmer add a declaration to the effect that Bool is a numeric type. Since Booleans are not numbers, this fix would be worse than useless, but since ghc has no idea about what each type actually means, it does not know that.
Programmers new to Haskell should probably han- dle type errors by simply going to the location spe- ficied in the error message (which has been removed from this example to make it fit on the slide), and looking around for the error, ignoring the rest of the error message.
Prelude> :t (+)
(+) :: (Num a) => a -> a -> a
Prelude> [1, True]
No instance for (Num Bool)
arising from the literal ‘1’ …
Prelude> :t 3
3 :: Num p => p Prelude> :t [1, 2]
[1, 2] :: Num a => [a]
Normally, + is a binary infix operator, but you can tell Haskell you want to use it as an ordinary func- tion name by wrapping it in parentheses. (You can do the same with any other operator.) This means that(+) 1 2meansthesameas1 + 2.
– section 2 slide 10 – if-then-else
— Definition A iota n =
if n == 0 then [] else iota (n-1) ++ [n]
Definition A uses an if-then-else. If-then-else in Haskell differs from if-then-elses in imperative lan- guages in that
• the else arm is not optional, and
• the then and else arms are expressions, not
statements.
NOTE The expressions representing the then and else arms must be of the same type, since the result of the if-then-else will be one of them. It will be the expression in the then arm if the condition is true, and it will be the expression in the else arm if the condition is false.
In Haskell, = separates the left and right hand sides of equations, while == represents a test for equality.
A language called APL in the 1960s had a whole bunch of builtin functions working with numbers and vectors and matrices, and many of these were named after letters of the greek alphabet. This function is named after the iota function of APL, which also returned the first n integers when in- voked with n as its argument.
– section 2 slide 11 – Guards
— Definition B iota n
|n==0 =[]
|n >0 =iota(n-1)++[n]
Definition B uses guards to specify cases. Note the first line does not end with an “=”; each guard line specifies a case and the value for that case, much as in definition A.
Note that the second guard specifies n > 0. What should happen if you do iota (-3)? What do you expect to happen? What about for definition A?
– section 2 slide 12 –
Structured definitions
Some Haskell equations do not fit on one line, and even the ones that do fit are often better split across several. Guards are only one example of this.
— Definition C iota n =
if n == 0 then
[] else
iota (n-1) ++ [n]
The offside rule says that
• the keywords then and else, if they start a line, must be at the same level of indentation as the corresponding if, and
• if the then and else expressions are on their own lines, these must be more indented than those keywords.
– section 2 slide 13 – Parametric polymorphism
Here is a version of the code of len complete with type declaration:
len :: [t] -> Int
len [] = 0
len (_:xs) = 1 + len xs
This function, like many others in Haskell, is poly- morphic. The phrase “poly morph” means “many shapes” or “many forms”. In this context, it means that len can process lists of type t regardless of what type t is, i.e. regardless of what the form of the elements is.
The reason why len works regardless of the type of the list elements is that it does not do anything with the list elements.
This version of len shows this in the second pat- tern: the underscore is a placeholder for a value you want to ignore.
NOTE This is called parametric polymorphism be- cause the type variable t is effectively a type pa- rameter.
Since the underscore matches all values in its posi- tion, it is often called a wild card.
– section 2 slide 14 –
Section 3: Defining Haskell types
Type definitions
Like most languages, Haskell allows programmers to define their own types. The simplest type defi- nitions define types that are similar to enumerated types in C:
data Gender = Female | Male data Role = Staff | Student
This defines two new types. The type called Gender has the two values Female and Male, while the type called Role has the two values Staff and Student.
Both types are also considered arity-0 type con- structors; given zero argument types, they each construct a type.
The four values are also called data constructors. Given zero arguments, they each construct a value (a piece of data).
The names of type constructors and data construc- tors must be identifiers starting with upper-case let- ters.
NOTE ”Arity” means the number of arguments. A function of arity 0 takes 0 arguments, a function of arity 1 takes 1 argument, a function of arity 2 takes 2 arguments, and so on. Similarly for type constructors. A type constructor of arity 0 con- structs a type from 0 other types, a type construc- tor of arity 1 constructs a type from 1 other type, a type constructor of arity 2 constructs a type from 2 other types, and so on.
– section 3 slide 1 –
Using Booleans
You do not have to use such types. If you wish, you can use the standard Boolean type instead, like this:
show1 :: Bool -> Bool -> String
— intended usage: show1 isFemale isStaff show1 True True = “female staff”
show1 True False = “female student” show1 False True = “male staff”
show1 False False = “male student”
You can use such a function like this:
– section 3 slide 2 –
Using defined types vs using Booleans
The problem with using Booleans is that of these two calls to show1, only one matches the program- mer’s intention, but since both are type correct (both supply two Boolean arguments), Haskell can- not catch errors that switch the arguments.
show2 :: Gender -> Role -> String
With show2, Haskell can and will detect and re- port any accidental switch. This makes the pro- gram safer and the programmer more productive.
In general, you should use separate types for sepa- rate semantic distinctions. You can use this tech- nique in any language that supports enumerated types.
– section 3 slide 3 –
Representing cards
Here is one way to represent standard western play- ing cards:
data Suit = Club | Diamond | Heart | Spade data Rank
= R2 | R3 | R4 | R5 | R6 | R7 | R8
| R9 | R10 | Jack | Queen | King | Ace
data Card = Card Suit Rank
The types Suit and Rank would be enumerated types in C, while the type Card would be a struc- ture type.
On the right hand side of the definition of the type Card, Card is the name of the data constructor, while Suit and Rank are the types of its two argu- ments.
NOTE In general, each alternative starts with the name of the constructor, which is followed by the types of all the arguments (if there are any).
– section 3 slide 4 – Creating structures
data Card = Card Suit Rank
In this definition, Card is not just the name of the type (from its first appearance), but (from its sec- ond appearance) also the name of the data con- structor which constructs the “structure” from its arguments.
In languages like C, creating a structure and filling it in requires a call to malloc or its equivalent, a check of its return value, and an assignment to each field of the structure. This typically takes several lines of code.
> let isFemale = True
> let isStaff = False
> show1 isFemale isStaff
> show1 isFemale isStaff > show1 isStaff isFemale
In Haskell, you can construct a structure just by writing down the name of the data constructor, fol- lowed by its arguments, like this: Card Club Ace. This typically takes only part of one line of code.
In practice, this seemingly small difference has a significant impact, because it removes much clutter (details irrelevant to the main objective).
NOTE This is the reason for the name “data con- structor”.
– section 3 slide 5 –
Printing values
Many programs have code whose job it is to print out the values of a given type in a way that is mean- ingful to the programmer. Such functions are par- ticularly useful during various forms of debugging.
The Haskell approach is use a function that returns a string. However, writing such functions by hand can be tedious, because each data constructor re- quires its own case:
showrank :: Rank -> String showrank R2 = “R2” showrank R3 = “R3”
…
– section 3 slide 6 –
Show
The Haskell prelude has a standard string conver- sion function called show. Just as the arithmetic functions are applicable to all types that are mem- bers of the type class Num, this function is applica- ble to all types that are members of the type class Show.
You can tell Haskell that the show function for val- ues of the type Rank is showrank:
instance Show Rank where show = showrank
This of course requires defining showrank. If you don’t want to do that, you can get Haskell to define the show function for a type by adding deriving Show to the type’s definition, like this:
data Rank =
R2 | R3 | R4 | R5 | R6 | R7 | R8 | R9 | R10 | Jack | Queen | King | Ace deriving Show
NOTE The offside rule applies to type definitions as well as function definitions, so e.g.
data Rank =
R2 | R3 | R4 | R5 | R6 | R7 | R8 | R9 | R10 | Jack | Queen | King | Ace
deriving Show
would not be valid Haskell: since deriving is part of the definition of the type Rank, it must be in- dented more than the line that starts the type def- inition.
– section 3 slide 7 –
Eq and Ord
Another operation even more important than string conversion is comparison for equality.
To be able to use Haskell’s == comparison operation for a type, it must be in the Eq type class. This can also be done automatically by putting deriving Eq at the end of a type definition.
To compare values of a type for order (using <, <=, etc.), the type must be in the Ord type class, which can also be done by putting deriving Ord at the end of a type definition. To be in Ord, the type must also be in Eq.
To derive multiple type classes, parenthesise them:
data Suit = Club | Diamond | Heart | Spade deriving (Show, Eq, Ord)
– section 3 slide 8 – Disjunction and conjunction
data Suit = Club | Diamond | Heart | Spade data Card = Card Suit Rank
A value of type Suit is either a Club or a Diamond or a Heart or a Spade. This disjunction of values corresponds to an enumerated type.
A value of type Card contains a value of type Suit and a value of type Rank. This conjunction of val- ues corresponds to a structure type.
In most imperative languages, a type can represent either a disjunction or a conjunction, but not both at once.
Haskell and related languages do not have this lim- itation.
– section 3 slide 9 – Discriminated union types
Haskell has discriminated union types, which can include both disjunction and conjunction at once.
Since disjunction and conjunction are operations in Boolean algebra, type systems that allows them to be combined in this way are often called algebraic type systems, and their types algebraic types.
data JokerColor = Red | Black data JCard =
NormalCard Suit Rank | JokerCard JokerColor
A value of type JCard is constructed
• either using the NormalCard constructor, in which case it contains a value of type Suit and a value of type Rank,
• or using the JokerCard constructor, in which case it contains a value of type JokerColor.
– section 3 slide 10 – Discriminated vs undiscriminated unions
In C, you could try to represent JCard like this:
struct normalcard_struct { ... }; struct jokercard_struct { ... }; union card_union {
};
struct normalcard_struct struct jokercard_struct
normal;
joker;
but you wouldn’t know which field of the union is applicable in any given case. In Haskell, you do (the data constructor tells you), which is why Haskell’s unions are said to be discriminated.
Note that unlike C’s union types, C’s enumera- tion types and structure types are special cases of Haskell’s discriminated union types.
Discriminated union types allow programmers to define types that describe exactly what they mean.
NOTE C’s unions are undiscriminated.
A Haskell discriminated union type in which none of the data constructors have arguments corresponds to a C enumeration type. A Haskell discriminated union type with only one data constructor corre- sponds to a C structure type.
– section 3 slide 11 – Maybe
In languages like C, if you have a value of type *T for some type T, or in languages like Java, if you have a value of some non-primitive type, can this value be null?
If not, the value represents a value of type T. If yes, the value may represent a value of type T, or it may represent nothing. The problem is, often the reader of the code has no idea whether it can be null.
And even if the value must not be null, there’s no guarantee it won’t be. This can lead to segfaults or NullPointerExceptions.
In Haskell, if a value is optional, you indicate this by using the maybe type defined in the prelude:
data Maybe t = Nothing | Just t
For any type t, a value of type Maybe t is either Nothing, or Just x, where x is a value of type t. This is a polymorphic type, like [t].
– section 3 slide 12 –
Section 4: Using Haskell Types
Representing expressions in C
typedef enum {
EXPR_NUM, EXPR_VAR, EXPR_BINOP, EXPR_UNOP
} ExprKind;
typedef struct expr_struct *Expr; struct expr_struct
{
ExprKind kind;
int value;
char *name;
Binop binop;
Unop unop;
Expr subexpr1;
Expr subexpr2;
/* if EXPR_NUM */ /* if EXPR_VAR */ /* if EXPR_BINOP */ /* if EXPR_UNOP */ /* if EXPR_BINOP
or EXPR_UNOP */ /* if EXPR_BINOP */
};
– section 4 slide 1 – Representing expressions in Java
public abstract class Expr { ... abstract methods ...
}
public class NumExpr extends Expr { int value;
... implementation of abstract methods ... }
public class VarExpr extends Expr { String name;
... implementation of abstract methods ... }
– section 4 slide 2 – Representing expressions in Java (2)
public class BinExpr extends Expr { Binop binop;
– section 4 slide 4 –
Comparing representions: errors
By far the most important difference is that the C representation is quite error-prone.
• You can access a field when that field is not meaningful, e.g. you can access the subexpr2 field instead of the subexpr1 field when kind is EXPR UNOP.
• You can forget to initialize some of the fields, e.g. you can forget to assign to the name field when setting kind to EXPR VAR.
• You can forget to process some of the alter- natives, e.g. when switching on the kind field, you may handle only three of the four enum values.
The first mistake is literally impossible to make with Haskell, and would be caught by the Java compiler. The second is guaranteed to be caught by Haskell, but not Java. The third will be caught by Java, and by Haskell if you ask ghc to be on the lookout for it.
NOTE You can do this by invoking ghc with the op- tion -fwarn-incomplete-patterns, or by setting the same option inside ghci with
– section 4 slide 5 –
Comparing representions: memory
The C representation requires more memory: seven words for every expression, whereas the Java and Haskell representations needs a maximum
Expr arg1;
Expr arg2;
... implementation of abstract
public class UnExpr extends Expr { Unop unop;
Expr arg;
... implementation of abstract
methods ...
methods ...
}
}
data Expr
= Number Int
| Variable String
| Binop Binopr Expr Expr | Unop Unopr Expr
data Binopr = Plus | Minus | Times | Divide data Unopr = Negate
As you can see, this is a much more direct definition of the set of values that the programmer wants to represent.
It is also much shorter, and entirely free of notes that are meaningless to the compiler and under- stood only by humans.
– section 4 slide 3 – Representing expressions in Haskell
> :set -fwarn-incomplete-patterns
of four (one for the kind, and three for argu- ments/members).
Using unions can make the C representation more compact, but only at the expense of more complex- ity, and therefore a higher probability of program- mer error.
Even with unions, the C representation needs four words for all kinds of expressions. The Java and Haskell representations need only two for numbers and variables, and three for expressions built with unary operators.
This is an example where a Java or Haskell program can actually be more efficient than a C program.
– section 4 slide 6 –
Comparing representions: maintenance Adding a new kind of expression requires:
Java: Adding a new class and implementing all the methods for it
C: Adding a new alternative to the enum and adding the needed members to the type, and adding code for it to all functions handling that type
Haskell Adding a new alternative, with argu- ments, to the type, and adding code for it to all functions handling that type
Adding a new operation for expressions requires:
Java: Adding a new method to the abstract Expr class, and implementing it for each class
C: Writing one new function Haskell Writing one new function
– section 4 slide 7 –
Switching on alternatives
You do not have to have separate equations for each possible shape of the arguments. You can test the value of a variable (which may or may not be the value of an argument) in the body of an equation, like this:
is_static :: Expr -> Bool is_static expr =
case expr of Number _
Variable _
Unop _ expr1 -> is_static expr1 Binop _ expr1 expr2 ->
is_static expr1 && is_static expr2
This function figures out whether the value of an expression can be known statically, i.e. without hav- ing to know the values of variables.
NOTE As with equations, Haskell matches the value being switched on against the given patterns in order, from the top down. If you want, you can use an underscore as a wildcard pattern that matches any value, like in this example:
is_atomic :: Expr -> Bool is_atomic expr =
case expr of
Unop _ _ -> False Binop _ _ _ -> False _ -> True
– section 4 slide 8 – Missing alternatives
If you specify the option -fwarn-incomplete- patterns, ghc and ghci will warn about any miss- ing alternatives, both in case expressions and in sets of equations.
-> True
-> False
This option is particularly useful during program maintenance. When you add a new alternative to an existing type, all the switches on values of that type instantly become incorrect. To fix them, a programmer must add a case to each such switch to handle the new alternative.
If you always compile the program with this option, the compiler will tell you all the switches in the program that must be modified.
Without such help, programmers must look for such switches themselves, and they may not find them all.
– section 4 slide 9 –
The consequences of missing alternatives
If a Haskell program finds a missing alternative at runtime, it will throw an exception, which (unless caught and handled) will abort the program.
Without a default case, a C program would simply go on and silently compute an incorrect result. If a default case is provided, it is likely to just print an error message and abort the program. C pro- grammers thus have to do more work than Haskell programmers just to get up to the level of safety offered by Haskell.
If an abstract method is used in Java, this gives the same safety as Haskell. However, if overriding is used alone, forgetting to write a method for a subclass will just inherit the (probably wrong) be- haviour of the superclass.
NOTE Switches in C usually need default clauses to compensate for the absence of type safety. Con- sider our expression representation example. In theory, a switch on expr->kind should need only four cases: EXPR NUM, EXPR VAR, EXPR BINOP, and EXPR UNOP. However, some other part of the pro- gram could have violated type safety by assigning e.g. (ExprKind) 42 to expr->kind. You need a default case to catch and report such errors.
In Haskell, such bugs cannot happen, since the com- piler will not let the programmer violate type safety.
– section 4 slide 10 –
Binary search trees
Here is one possible representation of binary search trees in C:
typedef struct bst_struct *BST; struct bst_struct {
char *key;
int value;
BST left;
BST right;
};
Here it is in Haskell:
data Tree
= Leaf
| Node String Int Tree Tree
The Haskell version has two alternatives, one of which has no associated data. The C version uses a null pointer to represent this alternative.
– section 4 slide 11 – Counting nodes in a BST
countnodes :: Tree -> Int countnodes Leaf = 0 countnodes (Node _ _ l r) =
1 + (countnodes l) + (countnodes r)
int countnodes(BST tree) {
if (tree == NULL) { return 0;
}else{
return 1 +
countnodes(tree->left) + countnodes(tree->right); {
} }
NOTE Since all we are doing is counting nodes, the values of the keys and values in nodes do not matter.
In most cases, using one-character variable names is not a good idea, since such names are usually not readable. However, for anyone who knows what binary search trees are, it should be trivial to figure out that l and r refer to the left and right subtrees. Similarly for k and v for keys and values.
– section 4 slide 12 –
Pattern matching vs pointer dereferencing
The left-hand-side of the second equation in the Haskell definition naturally gives names to each of the fields of the node that actually need names (be- cause they are used in the right hand side).
These variables do not have to be declared, and Haskell infers their types.
The C version refers to these fields using syntax that dereferences the pointer tree and accesses one of the fields of the structure it points to.
The C code is longer, and using Haskell-like names for the fields would make it longer still:
BST l = tree->left;
BST r = tree->right;
…
– section 4 slide 13 – Searching a BST in C (iteration)
search_bst(BST tree, char *key, int *value_ptr)
while (tree != NULL) { int cmp_result;
cmp_result =
strcmp(key, tree->key);
if (cmp_result == 0) { *value_ptr = tree->value; return TRUE;
} else if (cmp_result < 0) { tree = tree->left;
} else {
tree = tree->right;
} }
return FALSE;
}
NOTE C allows functions to assign to their formal parameters (tree in this case), since it considers these to be ordinary local variables that just happen to be initialized by the caller.
– section 4 slide 14 – Searching a BST in Haskell
search_bst :: Tree -> String -> Maybe Int search_bst Leaf _ = Nothing
search_bst (Node k v l r) sk
| sk == k = Just v
|sk< k =search_bstlsk | otherwise = search_bst r sk
• If the search succeeds, this function returns Just v, where v is the searched-for value.
• If the search fails, it returns Nothing.
• We could have used Haskell’s if-then-else for this, but guards make the code look much nicer and easier to read.
int
NOTE When the tree being searched is the empty tree, the value of the key being searched for doesn’t matter.
– section 4 slide 15 –
Data structure and code structure
The Haskell definitions of countnodes and search bst have similar structures:
• an equation handling the case where the tree is empty (a Leaf), and
• an equation handling the case where the tree is nonempty (a Node).
The type we are processing has two alternatives, so these two functions have two equations: one for each alternative.
This is quite a common occurrence:
• a function whose input is a data structure will need to process all or a selected part of that data structure, and
• what the function needs to do often depends on the shape of the data, so the structure of the code often mirrors the structure of the data.
NOTE If a function doesn’t need to process any part of a data structure, it shouldn’t be passed that data structure as an argument.
– section 4 slide 16 –
Section 5: Adapting to Declarative Programming
Writing code
Consider a C function with two loops, and some other code around and between the loops:
... somefunc(...) {
straight line code A loop 1
straight line code B loop 2
straight line code C
How can you get the same effect in Haskell? – section 5 slide 1 –
The functional equivalent
loop1func base case loop1func recursive case
loop2func base case loop2func recursive case
somefunc =
let ... = ... in
let r1 = loop1func ... in let ... = ... in
let r2 = loop2func ... in ...
}
The only effect of the absence of iteration con- structs is that instead of writing a loop inside somefunc, the Haskell programmer needs to write an auxiliary recursive function, usually outside somefunc.
NOTE In fact, Haskell allows one function defini- tion to contain another, and if he or she wished, the programmer could put the definition of e.g. loop1func inside the definition of somefunc.
– section 5 slide 2 – Example: C
int f(int *a, int size) {
int i;
int target;
int first_gt_target;
after_skip = skip_init_le_zero list
in
case after_skip of
(x:xs) ->
let target = 2 * x in let
first_gt_target = find_gt xs target
in
3 * first_gt_target
skip_init_le_zero :: [Int] -> [Int] skip_init_le_zero [] = [] skip_init_le_zero (x:xs) =
if x <= 0 then skip_init_le_zero xs
else (x:xs)
i = 0;
while (i
i++;
}
target =
i++;
while (i
i++; }
first_gt_target = a[i];
return 3 * first_gt_target; }
– section 5 slide 3 – Example: Haskell version
f :: [Int] -> Int f list =
let
(x:xs) target = if x <= target then
< size && a[i] <= 0) {
2 * a[i];
< size && a[i] <= target) {
find_gt xs target else
x
NOTE This version has the steps of f directly one
find_gt
find_gt
:: [Int] -> Int -> Int
after another.
– section 5 slide 4 –
Recursion vs iteration
Functional languages do not have language con- structs for iteration. What imperative language programs do with iteration, functional language programs do with recursion.
For a programmer who has known nothing but im- perative languages, the absence of iteration can seem like a crippling limitation.
In fact, it is not a limitation at all. Any loop can be implemented with recursion, but some recursions are difficult to implement with iteration.
There are several viewpoints to consider:
• How does this affect the process of writing code?
• How does this affect the reliability of the re- sulting code?
• How does this affect the productivity of the programmers?
• How does this affect the efficiency of the re- sulting code?
NOTE There are a few languages called functional languages, such as Lisp, that do have constructs for iteration. However, these languages are hy- brids, with some functional programming features and some imperative programming features, and their constructs for iteration belong to the second category.
– section 5 slide 5 –
C version vs Haskell versions
The Haskell versions use lists instead of arrays, since in Haskell, lists are the natural representa- tion.
With -fwarn-incomplete-patterns, Haskell will warn you that
• there may not be a strictly positive number in the list;
• there may not be a number greater than the target in the list,
and that these situations need to be handled.
The C compiler cannot generate such warnings. If the Haskell code operated on an array, the Haskell compiler couldn’t either.
The Haskell versions give meaningful names to the jobs done by the loops.
NOTE Haskell supports arrays, but their use re- quires concepts we have not covered yet.
– section 5 slide 6 –
Reliability
The names of the auxiliary functions should remind readers of their tasks.
These functions should be documented like other functions. The documentation should give the meaning of the arguments, and describe the re- lationship between the arguments and the return value.
This description should allow readers to construct a correctness argument for the function.
The imperative language equivalent of these func- tion descriptions are loop invariants, but they are as rare as hen’s teeth in real-world programs.
The act of writing down the information needed for a correctness argument gives programmers a chance to notice situations where the (implicit or explicit) correctness argument doesn’t hold water.
The fact that such writing down occurs much more often with functional programs is one factor that tends to make them more reliable.
NOTE As we saw on the previous slide, the Haskell compiler can help spot bugs as well.
– section 5 slide 7 –
Productivity
Picking a meaningful name for each auxiliary func- tion and writing down its documentation takes time.
This cost imposed on the original author of the code is repaid manyfold when
• other members of the team read the code, and find it easier to read and understand,
• the original author reads the code much later, and finds it easier to read and understand.
Properly documented functions, whether created as auxiliary functions or not, can be reused. Separat- ing the code of a loop out into a function allows the code of that function to be reused, requiring less code to be written overall.
In fact, modern functional languages come with large libraries of prewritten useful functions.
NOTE Programmers who do not document their code often find later themselves in the position of reading a part of the program they are working on, finding that they do not understand it, ask “who wrote this unreadable mess?”, and then finding out they they wrote it.
Just because you understand your code today does not mean that you will understand it a few months or few years from now. Using meaningful names, documenting the meaning of each data structure, the purpose of each function, the reasons for each design decision, and in general following the rules of good programming style will help your future self as well as your teammates in both the present and the future.
– section 5 slide 8 – Efficiency
The recursive version of e.g. search bst will allo- cate one stack frame for each node of the tree it traverses, while the iterative version will just allo- cate one stack frame period.
The recursive version will therefore be less efficient, since it needs to allocate, fill in and then later deal- locate more stack frames.
The recursive version will also need more stack
space. This should not be a problem for
search bst, but the recursive versions of some other functions can run out of stack space.
However, compilers for declarative languages put huge emphasis on the optimization of recursive code. In many cases, they can take a recursive al- gorithm in their source language (e.g. Haskell), and generate iterative code in their target language.
– section 5 slide 9 –
Efficiency in general
Overall, programs in declarative languages are typ- ically slower than they would be if written in C. De- pending on which declarative language and which language implementation you are talking about, and on what the program does, the slowdown can range from a few percent to huge integer factors, such as 10% to a factor of a 100.
However, popular languages like Python and Javascript typically also yield significantly slower programs than C. In fact, their programs will typ- ically be significantly slower than corresponding Haskell programs.
In general, the higher the level of a programming language (the more it does for the programmer), the slower its programs will be on average. The price of C’s speed is the need to handle all the details yourself.
The right point on the productivity vs efficiency tradeoff continuum depends on the project (and component of the project).
– section 5 slide 10 –
Declarative thinking
Some problems are difficult to approach impera- tively, and are much easier to think about declara- tively.
The imperative approach is procedural: we devise a way to solve the problem step by step. As an af- terthought we may think about grouping the steps into chunks (methods, procedures, functions, etc.)
The declarative approach breaks down the problem into chunks (functions), assembling the results of the chunks to construct the result.
You must be careful in imperative languages, be- cause the chunks may not compose due to side- effects. The chunks always compose in purely declarative languages.
– section 5 slide 11 –
Recursive thinking
One especially useful approach is recursive think- ing: use the function you are defining as one of the chunks.
To take this approach:
1. Determine how to produce the result for the whole problem from the results for the parts of the problem (recursive case);
2. Determine the solution for the smallest part of the input (base case).
Keep in mind the specification of the problem, but it also helps to think of concrete examples.
For lists, (1) usually means generating the result for the whole list from the list head and the result for the tail; (2) usually means the result for the empty list.
This works perfectly well in most imperative lan- guages, if you’re careful to ensure your function composes. But it takes practice to think this way.
– section 5 slide 12 –
Immutable data structures
In declarative languages, data structures are im- mutable: once created, they cannot be changed. So what do you do if you do need to update a data structure?
You create another version of the data structure, one which has the change you want to make, and use that version from then on.
However, if you want to, you can hang onto the old version as well. You will definitely want to do so if some part of the system still needs the old version (in which case imperative code must also make a modified copy).
The old version can also be used
• because both old and new are needed, as in
sublists
• to implement undo
• to gather statistics, e.g. about how the size of
a data structure changes over time – section 5 slide 13 –
Updating a BST
insert_bst :: Tree -> String -> Int -> Tree insert_bst Leaf ik iv = Node ik iv Leaf Leaf insert_bst (Node k v l r) ik iv
| ik == k = Node ik iv l r
| ik < k = Node k v (insert_bst l ik iv) r | otherwise = Node k v l (insert_bst r ik iv)
Note that all of the code of this function is con- cerned with the job at hand; there is no code con- cerned with memory management.
In Haskell, as in Java, memory management is au- tomatic. Any unreachable cells of memory are re- covered by the garbage collector.
– section 5 slide 14 –
Section 6: Polymorphism
Polymorphic types
Our definition of the tree type so far was this:
data Tree =
Leaf |
Node String Int Tree Tree
This type assumes that the keys are strings and the values are integers. However, the functions we have written to handle trees (countnodes and search bst do not really care about the types of the keys and values.
We could also define trees like this:
data Tree k v = Leaf
| Node k v (Tree k v) (Tree k v)
In this case, k and v are type variables, variables standing in for the types of keys and values, and Tree is a type constructor, which constructs a new type from two other types.
– section 6 slide 1 –
Using polymorphic types: countnodes
With the old, monomorphic definition of Tree, the type declaration or signature of countnodes was:
countnodes :: Tree -> Int
With the new, polymorphic definition of Tree, it will be
countnodes :: Tree k v -> Int
Regardless of the types of the keys and values in the tree, countnodes will count the number of nodes in it.
The exact same code works in these cases.
NOTE In C++, you can use templates to arrange to get the same piece of code to work on values of different types, but the C++ compiler will generate different object code for each type. This can make executables significantly bigger than they need to be.
– section 6 slide 2 –
Using polymorphic types: search bst
countnodes does not touch keys or values, but search bst does perform some operations on keys. Replacing
search_bst :: Tree -> String -> Maybe Int
with
search_bst :: Tree k v -> k -> Maybe v
will not work; it will yield an error message.
The reason is that search bst contains these two tests:
• a comparison for equality: sk == k, and • a comparison for order: sk < k.
– section 6 slide 3 –
Comparing values for equality and order
Some types cannot be compared for equality. For example, two functions should be considered equal
if for all sets of input argument values, they com- pute the same result. Unfortunately, it has been proven that testing whether two functions are equal is undecidable. This means that building an algo- rithm that is guaranteed to decide in finite time whether two functions are equal is impossible.
Some types that can be compared for equality can- not be compared for order. Consider a set of in- tegers. It is obvious that {1, 5} is not equal to {2, 4}, but using the standard method of set com- parison (set inclusion), they are otherwise incom- parable; neither can be said to be greater than the other.
– section 6 slide 4 – Eq and Ord
In Haskell,
• comparison for equality can only be done on values of types that belong to the type class Eq, while
• comparison for order can only be done on val- ues of types that belong to the type class Ord.
Membership of Ord implies membership of Eq, but not vice versa.
The declaration of search bst should be this: search_bst ::
Ord k => Tree k v -> k -> Maybe v
The construct Ord k => is a type class constraint; it says search bst requires whatever type k stands for to be in Ord. This guarantees its membership of Eq as well.
– section 6 slide 5 –
Data.Map
The polymorphic Tree type described above is de- fined in the standard library with the name Map, in the module Data.Map. You can import it with the declaration:
import Data.Map as Map
The key functions defined for Maps include:
insert
Map.lookup
(!) size
:: Ordk=>k->a->Mapka -> Mapka
:: Ordk=>k->Mapka
-> Maybe a
:: Ordk=>Mapka->k->a :: Map k a -> Int
. . . and many, many more functions; see the docu- mentation.
NOTE
(!) is an infix operator, and m ! k throws an ex-
ception if key k is not present in map m. – section 6 slide 6 –
Deriving membership automatically
data Suit = Club | Diamond | Heart | Spade deriving (Show, Eq, Ord)
data Card = Card Suit Rank deriving (Show, Eq, Ord)
The automatically created comparison function takes the order of data constructors from the or- der in the declaration itself: a constructor listed earlier is less than a constructor listed later (e.g. Club < Diamond).
If the two values being compared have the same top level data constructor, the automatically created comparison function compares their arguments in turn, from left to right. This means the argu- ment types must also be instances of Ord. If the
corresponding arguments are not equal, the com- parison stops (e.g. Card Club Ace < Card Spade Jack); if the corresponding argument are equal, it goes on to the next argument, if there is one (e.g. Card Spade Ace > Card Spade Jack). This is called lexicographic ordering.
– section 6 slide 7 – Recursive vs nonrecursive types
data Tree
= Leaf
| Node String Int Tree Tree data Card = Card Suit Rank
Tree is a recursive type because some of its data constructors have arguments of type Tree.
Card is a non-recursive type because none of its data constructors have an arguments of type Card.
A recursive type needs a nonrecursive alternative, because without one, all values of the type would have infinite size.
– section 6 slide 8 – Mutually recursive types
Some types are recursive but not directly recursive.
data BoolExpr
= BoolConst Bool
| BoolOp BoolOp BoolExpr BoolExpr | CompOp CompOp IntExpr IntExpr
data IntExpr
= IntConst Int
| IntOp IntOp IntExpr IntExpr
| IntIfThenElse BoolExpr IntExpr IntExpr
In a mutually recursive set of types, it is enough for one of the types to have a nonrecursive alternative.
These types represent Boolean- and integer-valued expressions in a program. They must be mutu- ally recursive because comparison of integers re- turns a Boolean and integer-valued conditionals use a Boolean.
NOTE Provided that at least one of the recursive alternatives is only mutually recursive, not directly recursive. That way, a value built using that alter- native can be of finite size.
– section 6 slide 9 –
Structural induction
Code that follows the shape of a nonrecursive type tends to be simple. Code that follows the shape of a directly or mutually recursive type tends to be more interesting.
Consider a recursive type with one nonrecursive data constructor (like Leaf in Tree) and one recur- sive data constructor (like Node in Tree). A func- tion that follows the structure of this type will typ- ically have
• anequationforthenonrecursivedataconstruc- tor, and
• an equation for the recursive data constructor.
Typically, recursive calls will occur only in the sec- ond equation, and the switched-on argument in the recursive call will be strictly smaller than the cor- responding argument in the left hand side of the equation.
– section 6 slide 10 –
Proof by induction
You can view the function definition’s structure as the outline of a correctness argument.
The argument is a proof by induction on n, the number of data constructors of the switched-on type in the switched-on argument.
• Base case: If n = 1, then the applicable equa- tion is the base case. If the first equation is correct, then the function correctly handles the case where n = 1.
• Induction step: Assume the induction hypoth- esis: the function correctly handles all cases where n ≤ k. This hypothesis implies that all the recursive calls are correct. If the second equation is correct, then the function correctly handles all cases where n ≤ k + 1.
The base case and the induction step together im- ply that the function correctly handles all inputs.
– section 6 slide 11 –
Formality
If you want, you can use these kinds of arguments to formally prove the correctness of functions, and of entire functional programs.
This typically requires a formal specification of the expected relationship between each function’s ar- guments and its result.
Typical software development projects do not do formal proofs of correctness, regardless of what kind of language their code is written in.
However, pro jects using functional languages do tend to use informal correctness arguments slightly more often.
The support for this provided by the original pro- grammer usually consists of nothing more than a natural language description of the criterion of cor- rectness of each function. Readers who want a cor- rectness argument can then construct it for them- selves from this and the structure of the code.
– section 6 slide 12 –
Structural induction for more complex types
If a type has nr nonrecursive data constructors and r recursive data constructors, what happens when nr > 1 or r > 1, like BoolExpr and IntExpr?
You can do structural induction on such types as well.
Such functions will typically have nr nonrecursive equations and r recursive equations, but not al- ways. Sometimes you need more than one equation to handle a constructor, and sometimes one equa- tion can handle more than one constructor. For example, sometimes all base cases need the same treatment.
Picking the right representation of the data is im- portant in every program, but when the structure of the code follows the structure of the data, it is particularly important.
NOTE If all the base cases can be handled the same way, then the function can start with r equations for the recursive constructors, followed by a single equation using a wildcard pattern that handles all the nonrecursive constructors.
– section 6 slide 13 –
Let clauses and where clauses
assoc_list_to_bst ((hk, hv):kvs) = let t0 = assoc_list_to_bst kvs in insert_bst t0 hk hv
A let clause let name = expr in mainexpr intro- duces a name for a value to be used in the main expression.
assoc_list_to_bst ((hk, hv):kvs) = insert_bst t0 hk hv
where t0 = assoc_list_to_bst kvs
A where clause mainexpr where name = expr has the same meaning, but has the definition of the name after the main expression.
Which one you want to use depends on where you want to put the emphasis.
But you can only use where clauses at the top level of a function, while you can use a let for any ex- pression.
NOTE In theory, you can also mix the two, like this:
Section 7: Higher order functions
First vs higher order
First order values are data.
Second order values are functions whose arguments and results are first order values.
Third order values are functions whose arguments and results are first or second order values.
In general, nth order values are functions whose arguments and results are values of any order from first up to n−1.
Values that belong to an order higher than first are higher order values.
Java 8, released mid-2014, supports higher order programming. C also supports it, if you work at it. Higher order programming is a central aspect of Haskell, often allowing Haskell programmers to avoid writing recursive functions.
– section 7 slide 1 –
A higher order function in C
IntList filter(Bool (*f)(int), IntList list) {
IntList filtered_tail, new_list;
if (list == NULL) { return NULL;
} else { filtered_tail =
let name1 =
in mainexpr
where name2
expr1
= expr2
However, you should not do this, since it is defi- nitely bad programming style.
– section 6 slide 14 –
Defining multiple names
You can define multiple names with a single let or where clause:
let name1 = expr1 name2 = expr2
in mainexpr
or
mainexpr
where
name1 = expr1
name2 = expr2
The scope of each name includes the right hand sides of the definitions of the following names, as well as the main expression, unless one of the later definitions defines the same name, in which case the original definition is shadowed and not visible from then on.
– section 6 slide 15 –
filter(f, list->tail); if ((*f)(list->head)) {
new_list = checked_malloc( sizeof(*new_list));
new_list->head = list->head; new_list->tail = filtered_tail; return new_list;
} else {
return filtered_tail;
}
NOTE This code assumes type definitions like these:
typedef struct intlist_struct *IntList;
struct intlist_struct { int head;
IntList tail;
};
typedef int Bool;
Unfortunately, although the last typedef allows pro- grammers to write Bool instead of int to tell read- ers of the code that something (in this case, the return value of the function) is meant to be used to represent only a TRUE/FALSE distinction, the C compiler will not report as errors any arithmetic operations on booleans, any mixing of booleans and integers, or in general any unintended use of a boolean as an integer or vice versa. Since booleans and integers are distinct builtin types in Haskell, any such errors in Haskell programs will be caught by the Haskell implementation.
– section 7 slide 2 –
A higher order function in Haskell
Haskell’s syntax for passing a function as an argu- ment is much simpler than C’s syntax. All you need to do is wrap the type of the higher order argument in parentheses to tell Haskell it is one argument.
filter :: (a -> Bool) -> [a] -> [a] filter _ [] = []
filter f (x:xs) =
if f x then x:fxs else fxs where fxs = filter f xs
Even though it is significantly shorter, this function is actually more general than the C version, since it is polymorphic, and thus works for lists with any type of element.
filter is defined in the Haskell prelude. – section 7 slide 3 –
Using higher order functions You can call filter like this:
… filter is_even [1, 2, 3, 4] …
… filter is_pos [0, -1, 1, -2, 2] …
… filter is_long [“a”, “abc”, “abcde”] …
given definitions like this:
is_even :: Int -> Bool is_even x =
} }
if (mod x True
else False
is_pos :: Int is_pos x = if
2) == 0 then
-> Bool
x > 0 then True else False
is_long :: String -> Bool is_long x =
if length x > 3 then True
else False
NOTE length is a function defined in the Haskell prelude. As its name implies, it returns the length
of a list. Remember that in Haskell, a string is a list of characters.
The mod function is similar to the % operator in C: in this case, it returns the remainder after dividing x by 2.
– section 7 slide 4 –
Backquote
Modulo is a built-in infix operator in many lan- guages. For example, in C or Java, 5 modulo 2 would be written 5 % 2.
Haskell uses mod for the modulo operation, but Haskell allows you to make any function an infix operator by surrounding the function name with backquotes (backticks, written ‘).
So a friendlier way to write the is even function would be:
is_even :: Int -> Bool is_even x = x ‘mod‘ 2 == 0
Operators written with backquotes have high prece- dence and associate to the left.
It’s also possible to explicitly declare non- alphanumeric operators, and specify their associa- tivity and fixity, but this feature should be used sparingly.
– section 7 slide 5 –
Anonymous functions
In some cases, the only thing you need a function for is to pass as an argument to a higher order function like filter. In such cases, readers may find it more convenient if the call contained the definition of the function, not its name.
In Haskell, anonymous functions are defined by lambda expressions, and you use them like this.
… filter (\x -> x ‘mod‘ 2 == 0) [1, 2, 3, 4] …
… filter (\s -> length s > 3) [“a”, “abc”, “abcde”] …
This notation is based on the lambda calculus, the basis of functional programming.
In the lambda calculus, each argument is preceded by a lambda, and the argument list is followed by a dot and the expression that is the function body. For example, the function that adds together its two arguments is written as λa.λb.a + b.
NOTE These calls are equivalent to the calls
… filter is_even [1, 2, 3, 4] …
… filter is_long [“a”, “abc”, “abcde”] …
given our earlier definitions of is even and is long.
– section 7 slide 6 – Map
(Not to be confused with Data.Map.)
map is one of the most frequently used Haskell func- tions. (It is defined in the Haskell prelude.) Given a function and a list, map applies the function to every member of the list.
map :: (a -> b) -> [a] -> [b] map _ [] = []
map f (x:xs) = (f x):(map f xs)
Many things that an imperative programmer would do with a loop, a functional programmer would do with a call to map. An example:
get_names :: [Customer] -> [String] get_names customers =
map customer_name customers
This assumes that customer name is a function whose type is Customer -> String.
– section 7 slide 7 – Partial application
Given a function with n arguments, partially apply- ing that function means giving it its first k argu- ments, where k < n.
The result of the partial application is a closure that records the identity of the function and the values of those k arguments.
This closure behaves as a function with n − k ar- guments. A call of the closure leads to a call of the original function with both sets of arguments.
is_longer :: Int -> String -> Bool is_longer limit x = length x > limit
…
filter (is_longer 4)
[“ab”, “abcd”, “abcdef”] …
In this case, the function is longer takes two ar- guments. The expression is longer 4 partially applies this function, and creates a closure which records 4 as the value of the first argument.
NOTE There is no way to partially apply a function (as opposed to an operator, see below) by supplying it with k arguments if those not are not the first k arguments.
Sometimes, programmers define small, sometimes anonymous helper functions which simply call an- other function with a different order of arguments, the point being to bring the k arguments you want to supply to that function to the start of the argu- ment list of the helper function.
– section 7 slide 8 –
Calling a closure: an example
filter f (x:xs) =
if f x then x:fxs else fxs where fxs = filter f xs
…
filter (is_longer 4)
[“ab”, “abcd”, “abcdef”]
In this case, the code of filter will call is longer
three times:
• is longer 4 “ab”
• is longer 4 “abcd”
• is longer 4 “abcdef”
Each of these calls comes from the higher order call f x in filter. In this case f represents the clo- sure is longer 4. In each case, the first argument comes from the closure, with the second being the value of x.
– section 7 slide 9 –
Operators and sections
If you enclose an infix operator in parentheses, you can partially apply it by enclosing its left or right operand with it; this is called a section.
You can use section notation to partially apply ei- ther of its arguments.
…
Prelude> map (*3) [1, 2, 3] [3,6,9]
Prelude> map (5 ‘mod‘) [3, 4, 5, 6, 7] [2,1,0,5,5]
Prelude> map (‘mod‘ 3) [3, 4, 5, 6, 7] [0,1,2,0,1]
– section 7 slide 10 – Types for partial application
In most languages, the type of a function with n arguments would be something like:
f :: (at1, at2, … atn) -> rt
where at1, at2 etc are the argument types, (at1, at2, … atn) is the type of a tuple containing all the arguments, and rt is the result type.
To allow the function to be partially applied by supplying the first argument, you need a function with a different type:
f :: at1 -> ((at2, … atn) -> rt)
This function takes a single value of type at1, and returns as its result another function, which is of type (at2, … atn) -> rt.
– section 7 slide 11 –
Currying
You can keep transforming the function type until every single argument is supplied separately:
f :: at1 -> (at2 -> (at3 -> … (atn -> rt)))
The transformation from a function type in which all arguments are supplied together to a function
type in which the arguments are supplied one by one is called currying.
In Haskell, all function types are curried. This is why the syntax for function types is what it is. The arrow that makes function types is right associative, so the second declaration below just shows explic- itly the parenthesization implicit in the first:
is_longer :: Int -> String -> Bool is_longer :: Int -> (String -> Bool)
NOTE Currying and the Haskell programming lan- guage are both named after the same person, the English mathematician Haskell Brooks Curry. He did a lot to develop the lambda calculus, the mathe- matical foundation of functional programming, and a result, he is a popular guy in functional pro- gramming circles. In fact, there are two functional programming languages named for him: besides Haskell, there is another one named Curry.
– section 7 slide 12 –
Functions with all their arguments
Given a function with curried argument types, you can supply the function its first argument, then its second, then its third, and so on. What happens when you have supplied them all?
is_longer 3 “abcd”
There are two things you can get:
• a closure that contains all the function’s argu- ments, or
• the result of the evaluation of the function.
In C and in most other languages, these would be very different, but in Haskell, as we will see later, they are equivalent.
– section 7 slide 13 –
Composing functions
Any function that makes a higher order function call or creates a closure (e.g. by partially applying another function) is a second order function. This means that both filter and its callers are second order functions.
filter has a piece of data as an argument (the list to filter) as well as a function (the filtering func- tion). Some functions do not take any piece of data as arguments; all their arguments are functions.
The builtin operator ‘.’ composes two functions. The expression f . g represents a function which first calls g, and then invokes f on the result:
(f . g) x = f (g x)
If the type of x is represented by the type variable a,thenthetypeofgmustbea -> bforsomeb, andthetypeoffmustbeb -> cforsomec. The type of . itself is therefore (b -> c) -> (a -> b) -> (a -> c).
NOTE There is nothing preventing some or all of those type variables actually standing for the same type.
– section 7 slide 14 –
Composing functions: some examples
Suppose you already have a function that sorts a list and a function that returns the head of a list, if it has one. You can then compute the minimum of the list like this:
minimum = head . sort
If you also have a function that reverses a list, you can also compute the maximum with very little ex- tra code:
maximum = head . reverse . sort
This shows that functions created by composition, such as reverse . sort, can themselves be part of further compositions.
This style of programming is sometimes called point-free style, though value-free style would be a better description, since its distinguishing charac- teristic is the absence of variables representing (first order) values.
NOTE The . operator is right associative, so head . reverse . sort parenthesizes as head . (reverse . sort), not as (head . reverse) . sort, even though the two parenthesizations in fact yield functions that compute the same answers for all possible argument values.
Given the above definition, maximum xs is equiv- alent to (head . reverse . sort) xs, which in turn is equivalent to head (reverse (sort xs)).
Code written in point-free style can contain vari- ables representing functions; the definition of . is an example.
Code written in point free style is usually very short, and it can be very elegant. However, while elegance is nice, it is not the most important char- acteristic that programmers should strive for. If most readers cannot understand a piece of code, its elegance to the few readers that do understand it is of little concern, and unfortunately, a large frac- tion of programmers find code written in point-free style hard to understand.
– section 7 slide 15 –
Composition as sequence
Function composition is one way to express a se- quence of operations. Consider the function com- position step3f . step2f . step1f.
1. You start with the input, x.
2. You compute step1f x.
3. You compute step2f (step1f x).
4. You compute step3f (step2f (step1f x)).
This idea is the basis of monads, which is the mech- anism Haskell uses to do input/output.
– section 7 slide 16 –
Section 8: Functional design patterns
Higher order programming
Higher order programming is widely used by func- tional programmers. Its advantages include
• code reuse,
• a higher level of abstraction, and
• a set of canned solutions to frequently encoun- tered problems.
In programs written by programmers who do not use higher order programming, you frequently find pieces of code that have the same structure but slot different pieces of code into that structure.
Such code typically qualifies as an instance of the copy-and-paste programming antipattern, a pattern that programmers should strive to avoid.
NOTE This is an antipattern because the many dif- ferent copies of the code structure violate a variant of the principle of single point of control. If you find a bug in one of the copies, that bug may be present in other copies as well, but you don’t get help in finding out where those copies are. With higher order code, fixing the code of the higher or- der function itself fixes that bug in all calls to it.
– section 8 slide 1 – Folds
We have already seen the functions map and filter, which operate on and transform lists.
The other class of popular higher order functions on lists are the reduction operations, which reduce a list to a single value.
The usual reduction operations are folds. There are three main folds: left, right and balanced.
left ((((I ⊙X1)⊙X2)…)⊙Xn)
right (X1 ⊙(X2 ⊙(…(Xn ⊙I)))) balanced ((X1 ⊙X2)⊙(X3 ⊙X4))⊙…
Here ⊙ denotes a binary function, the folding op- eration, and I denotes the identity element of that operation. (The balanced fold also needs the iden- tity element in case the list is empty.)
– section 8 slide 2 – Foldl
foldl :: (v -> e -> v) -> v -> [e] -> v foldl _ base [] = base
foldl f base (x:xs) =
let newbase = f base x in foldl f newbase xs
suml :: Num a => [a] -> a suml = foldl (+) 0
productl :: Num a => [a] -> a productl = foldl (*) 1
concatl :: [[a]] -> [a] concatl = foldl (++) []
– section 8 slide 3 – Foldr
foldr :: (e -> v -> v) -> v -> [e] -> v foldr _ base [] = base
foldr f base (x:xs) =
let fxs = foldr f base xs in f x fxs
sumr = foldr (+) 0 productr = foldr (*) 1 concatr = foldr (++) []
You can define sum, product and concatenation in terms of both foldl and foldr because addition and multiplication on integers, and list append, are all associative operations.
NOTE The declarations of sumr, productr and concatr differ from the declarations of suml, productl and concatl only in the function name.
Addition and multiplication are not associative on floating point numbers, because of their limited pre- cision. For the sake of simplicity in the discussion, suppose the limit is four decimal digits in the frac- tion, and suppose you want to sum up the list of numbers [0.25, 0.25,0.25, 0.25, 1000]. If you do the additions from left to right, the first addition adds 0.25 and 0.25 giving 0.5, the second adds 0.5 and 0.25 giving 0.75, the third adds 0.75 and 0.25 giving 1.0, and the fourth adds 1 and 1000, yielding 1001. The final addition of the identity element 0 does not change this result. However, if you do the additions right to left, the result is different. This is because adding 0.25 and 1000 cannot yield 1000.25, since that has too many digits. Instead, 1000.25 must be rounded to the nearest number with four decimal digits, which will be 1000. The next three additions of 0.25 and the final addition of 0 will still leave the overall result at 1000.
While concatl and concatr are guaranteed to gen- erate the same result, concatr is much more effi- cient than concatl. We will discuss this later.
– section 8 slide 4 – Balanced fold
balanced_fold :: (e -> e -> e) -> e -> [e] -> e
balanced_fold _ b [] = b balanced_fold _ _ (x:[]) = x balanced_fold f b l@(_:_:_) =
let
len = length l
that compute
foldl1 o [X1,X2,…,Xn]=((X1 ⊙X2)…⊙Xn)
foldr1 o[X1,X2,…,Xn]=(X1⊙(X2⊙…Xn)) maximum = foldr1 max
minimum = foldr1 min
You could equally well use foldl1 for these.
– section 8 slide 6 –
Folds are really powerful
You can compute the length of a list by summing 1 for each element, instead of the element itself. So if we can define a function that takes anything and returns 1, together with (+) we can use fold to define length.
const :: a -> b -> a const a b = a
length = foldr ((+) . const 1) 0 You can map over a list with foldr:
map f = foldr ((:) . f) []
– section 8 slide 7 – Fold can reverse a list
If we had a “backwards” (:) operation, call it snoc, then foldl could reverse a list:
reverse [X1, X2, . . . Xn] =[]snocX1snocX2 …snocXn
in
(half1, half2) =
splitAt (len ‘div‘ 2) l
value1 = balanced_fold f b half1 value2 = balanced_fold f b half2
f value1 value2
splitAt n l returns a pair of the first n elements of l and the rest of l. It is defined in the standard Prelude.
NOTE This code does some wasted work. Each call to balanced fold computes the length of its list, but for recursive calls, the caller already knows the length. (It is div len 2 for half1 and len – (div len 2) for half2.) Also, balanced fold does not make recursive calls unless the length of the list is at least two. This means that the length of both half1 and half2 will be at least one, which means that the test for zero length lists can succeed only for the top level call; for all the recursive calls, that test is wasted work. It is of course possible to write a balanced fold that does not have either of these performance problems.
– section 8 slide 5 – More folds
The Haskell prelude defines sum, product, and concat.
For maximum and minimum, there is no identity el- ement, and it is an error if the list is empty. For such cases, the Haskell Prelude defines:
foldl1 :: (a -> a -> a) -> [a] -> a foldr1 :: (a -> a -> a) -> [a] -> a
snoc :: [a] -> a -> [a] snoc tl hd = hd:tl
reverse = foldl snoc []
But the Haskell Prelude defines a function to flip the arguments of a binary function:
flip :: (a -> b -> c) -> b -> a -> c flip f x y = f y x
reverse = foldl (flip (:)) []
– section 8 slide 8 –
Foldable
But what about types other than lists? Can we fold over them?
Actually, we can:
In fact we can fold over any type in the type class
Foldable
We can declare our own types to be instances of Foldable by defining foldr for our type; then many standard functions, such length, sum, etc. will work on that type, too.
– section 8 slide 9 – List comprehensions
Haskell has special syntax for one class of higher order operations. These two implementations of quicksort do the same thing, with the first using conventional higher order code, and the second us- ing list comprehensions:
qs1 [] = []
qs1
(x:xs) = qs1 littles ++ [x] ++
= filter (>=x) xs
qs2 littles ++ [x] ++
qs2 bigs
qs2 [] = [] qs2 (x:xs) =
qs1 bigs
littles = filter (
– section 8 slide 10 –
List comprehensions
List comprehensions can be used for things other
than filtering a single list.
In general, a list comprehension consists of
• a template (an expression, which is often just a variable)
• one or more generators (each of the form var <- list),
• zero or more tests (boolean expressions),
• zero or more let expressions defining local vari-
ables.
Some more examples:
Prelude> sum (Just 7) 7
Prelude> sum Nothing 0
Prelude> :t foldr
foldr :: Foldable t => (a -> b -> b)
-> b -> t a -> b
columns = “abcdefgh” rows = “12345678” chess_squares = [[c, r]
| c <- columns, r <- rows]
pairs = [(a, b)
| a <- [1, 2, 3], b <- [1, 2, 3]]
nums = [10*a+b
| a <- [1, 2, 3], b <- [1, 2, 3]]
– section 8 slide 11 –
Traversing HTML documents Types to represent HTML documents:
type HTML = [HTML_element] data HTML_element
= HTML_text String
| HTML_font Font_tag HTML | HTML_p HTML
data Font_tag = Font_tag (Maybe Int) (Maybe String)
(Maybe Font_color)
data Font_color
= Colour_name String | Hex Int
| RGB Int Int Int
– section 8 slide 12 – Collecting font sizes
font_sizes_in_html ::
HTML -> Set Int -> Set Int
font_sizes_in_html elements sizes =
foldr font_sizes_in_elt sizes elements
font_sizes_in_elt ::
HTML_element -> Set Int -> Set Int
font_sizes_in_elt (HTML_text _) sizes =
sizes font_sizes_in_elt
(HTML_font font_tag html) sizes = let
Font_tag maybe_size _ _ = font_tag newsizes = case maybe_size of
Nothing ->
sizes
Just fontsize -> Data.Set.insert
fontsize sizes
in
font_sizes_in_html html newsizes
font_sizes_in_elt (HTML_p html) sizes = font_sizes_in_html html sizes
NOTE Normally, font sizes in html would be invoked with the empty set as the second argu- ment, which would mean that the returned set is the set of font sizes appearing in the given HTML page description. The second argument of font sizes in html thus plays the role of an ac- cumulator.
– section 8 slide 13 – Collecting font names
font_names_in_html ::
HTML -> Set String -> Set String
font_names_in_html elements names =
foldr font_names_in_elt names elements
font_names_in_elt ::
HTML_element -> Set String -> Set String
font_names_in_elt (HTML_text _) names = names font_names_in_elt
(HTML_font font_tag html) names = let
Font_tag _ maybe_name _ = font_tag newnames = case maybe_name of
Nothing ->
names
Just fontname ->
Data.Set.insert fontname names
in
font_names_in_html html newnames
font_names_in_elt (HTML_p html) names = font_names_in_html html names
– section 8 slide 14 – Collecting any font information
font_stuff_in_html ::
(Font_tag -> a -> a) -> HTML -> a -> a
font_stuff_in_html f elements stuff = foldr (font_stuff_in_elt f) stuff
elements
font_stuff_in_elt :: (Font_tag -> a -> a) -> HTML_element -> a -> a
font_stuff_in_elt f
(HTML_text _) stuff = stuff
font_stuff_in_elt f
(HTML_font font_tag html) stuff = let newstuff = f font_tag stuff in font_stuff_in_html f html newstuff
font_stuff_in_elt f (HTML_p html) stuff = font_stuff_in_html f html stuff
– section 8 slide 15 – Collecting font sizes again
font_sizes_in_html’ ::
HTML -> Set Int -> Set Int
font_sizes_in_html’ html sizes = font_stuff_in_html accumulate_font_sizes
html sizes
accumulate_font_sizes font_tag sizes =
let Font_tag maybe_size _ _ = font_tag in case maybe_size of
Nothing -> sizes
Just fontsize -> Data.Set.insert
fontsize sizes
Using the higher order version avoids duplicating the code that traverses the data structure. The benefit you get from this scales linearly with the complexity of the data structure being traversed.
– section 8 slide 16 – Comparison to the visitor pattern
The function font stuff in html does a job that is very similar to the job that the visitor design pat- tern would do in an object-oriented language like Java: they both traverse a data structure, invok- ing a function at one or more selected points in the code. However, there are also differences.
• In the Haskell version, the type of the higher order function makes it clear whether the code executed at the selected points just gathers in- formation, or whether it modifies the traversed data structure. In Java, the invoked code is imperative, so it can do either.
• The Java version needs an accept method in every one of the classes that correspond to Haskell types in the data structure (in this case, HTML and HTML element).
• In the Haskell version, the functions that im- plement the traversal can be (and typically are) next to each other. In Java, the corre- sponding methods have to be dispersed to the classes to which they belong.
– section 8 slide 17 – Libraries vs frameworks
The way typical libraries work in any language (in- cluding C and Java as well as Haskell) is that code written by the programmer calls functions in the library.
In some cases, the library function is a higher or- der function, and thus it can call back a function supplied to it by the programmer.
Application frameworks are libraries but they are not typical libraries, because they are intended to be the top layer of a program.
When a program uses a framework, the framework is in control, and it calls functions written by the programmer when circumstances call for it.
For example, a framework for web servers would handle all communication with remote clients. It would itself implement the event loop that waits for the next query to arrive, and would invoke user code only to generate the response to each query.
– section 8 slide 18 –
Frameworks: libraries vs application generators Frameworks in Haskell can be done like this, with
framework simply being a library function:
main = framework plugin1 plugin2 plugin3
plugin1 = … plugin2 = … plugin3 = …
This approach could also be used in other lan- guages, since even C and Java support callbacks, though sometimes clumsily.
Unfortunately, many frameworks instead just gen- erate code (in C, C#, Java, …) that the program- mer is then expected to modify. This approach throws abstraction out the window, and is much more error-prone.
NOTE Application generators expose to the pro- grammer all the details of their implementation. This allows the programmer to modify those de- tails if needed, but the next invocation of the ap- plication generator will destroy those modifications, which means the application generator can be used only once. If a new version of the application gen- erator comes out that fixes some problems with the old version, the programmer has no good options: there is no easy way to integrate the new version’s bug fixes with his or her own earlier modifications.
However, bugs in which programmers modify the wrong part of the generated code or forget to mod- ify a part they should have modified can be ex- pected to be made considerably more frequently.
– section 8 slide 19 –
Section 9: Exploiting the type system
Representation of C programs in gcc
The gcc compiler has one main data type to repre- sent the code being compiled. The node type is a giant union which has different fields for different kinds of entities. A node can represent, amongst other things,
• a data type,
• a variable,
• an expression or • a statement.
Every link to another part of a program (such as the operands of an operator) is a pointer to a tree node of this can-represent-everything type.
When Stallman chose this design in the 1980s, he was a Lisp programmer. Lisp does not have a static type system, so the Blub paradox applies here in reverse: even C has a better static type system than Lisp. It’s up to the programmer to design types to exploit the type system.
NOTE The giant union is definitely very big: in gcc 4.4.1, it has 40 alternatives, many more than the four listed above.
Like Python, Lisp does have a type system, but it operates only at runtime.
– section 9 slide 1 – Representation of if-then-elses
To represent if-then-else expressions such as C’s ternary operator
(x > y) ? x : y
the relevant union field is a structure that has an array of operands, which should have exactly three elements (the condition, the then part and the else part). All three should be expressions.
This representation is subject to two main kinds of error.
• The array of operands could have the wrong number of operands.
• Any operand in the array could point to the wrong kind of tree node.
gcc has extensive infrastructure designed to detect these kinds of errors, but this infrastructure itself has three problems:
• it makes the source code harder to read and write;
• if enabled, it slows down gcc by about 5 to 15%; and
• it detects violations only at runtime.
NOTE When trying to compiler some unusual C programs, usually those generated by a program rather than a programmer, you may gcc to abort with a message talking about an ”internal compiler error”. The usual reason for such aborts is that the compiler expected to find a node of a particu- lar kind in a given position in the tree, but found a node of a different kind. In other words, the prob- lem is the failure of a runtime type check. A better representation that took advantage of the type sys- tem would allow such bugs to be caught at compile time.
– section 9 slide 2 –
Exploiting the type system
A well designed representation using algebraic types is not vulnerable to either kind of error, and is not subject any of those three kinds of problems.
data Expr
= Const Const
| Var String
| Binop Binop Expr Expr | Unop Unop Expr
| Call String [Expr]
| ITE Expr Expr Expr
data Binop = Add | Sub | … data Unop = Uminus | … data Const
= IntConst Int
| FloatConst Double | …
NOTE You can do much better than gcc’s design even in C. A native C programmer would have chosen to have represent these four very different kinds of entities using four different main types, to- gether with more types representing their compo- nents. This representation could start from some- thing like the following.
union expr_union { Const
Var
struct unop_expr_struct struct binop_expr_struct struct call_expr_struct struct ite_expr_struct
};
typedef enum { ADD, SUB, …
} Binop;
typedef enum { UMINUS, …
} Unop;
struct binop_expr_struct { Binop *binop;
Expr binop_arg_1; Stmt binop_arg_2;
};
struct unop_expr_struct { Unop *unop;
Expr unop_arg_1;
};
struct call_expr_struct { char *funcname; Exprs func_args;
};
struct ite_expr_struct { Expr *cond;
Stmt then_part;
Stmt else_part;
};
struct exprs_struct { Expr *head; Exprs *tail;
};
*expr_const;
*expr_var;
*expr_unop;
*expr_binop;
*expr_call;
*expr_ite;
typedef enum {
EXPR_CONST, EXPR_VAR, EXPR_BINOP, EXPR_UNOP, EXPR_CALL, EXPR_ITE
} ExprKind;
typedef struct expr_struct typedef union expr_union typedef struct exprs_struct typedef struct const_struct typedef struct var_struct
struct expr_struct {
Expr;
ExprUnion;
Exprs;
Const;
Var;
};
There is a paper titled ”Statically typed trees in
ExprKind
ExprUnion
expr_kind;
*expr_union;
GCC” by Sidwell and Weinberg that describes the drawbacks of gcc’s current program representa- tion and proposes a roadmap for switching to a more type-safe representation, something like the representation shown above. Unfortunately, that roadmap has not been implemented, and probably never will be, partly because its implementation in- terfere considerably with the usual development of gcc.
– section 9 slide 3 – Generic lists in C
typedef struct generic_list List; struct generic_list {
void *head;
List *tail;
};
…
List *int_list;
List *p;
int item;
for (p = int_list; p != NULL; p = p->tail) { item = (int) p->head;
… do something with item …
}
NOTE Despite the name of the variable holding the list being int list, C’s type system cannot guarantee that the list elements are in fact integers.
– section 9 slide 4 –
Type system expressiveness
Programmers who choose to use generic lists in a C program need only one list type and therefore one set of functions operating on lists.
The downside is that every loop over lists needs to cast the list element to the right type, and this cast is a frequent source of bugs.
The other alternative in a C program is to define and use a separate list type for every different type of item that the program wants to put into a list. This is type safe, but requires repeated duplication of the functions that operate on lists. Any bugs in those those functions must be fixed in each copy.
Haskell has a very expressive type system that is increasingly being copied by other languages. Some OO/procedural languages now support generics. A few such languages (Rust, Swift, Java 8) support option types, like Haskell’s Maybe. No well-known such languages support full algebraic types.
NOTE However, even in Haskell, some program- ming practices help the compiler to catch problems early (at compile time), and some do not.
– section 9 slide 5 –
Units
One typical bug type in programs that manipulate physical measurements is unit confusion, such as adding 2 meters and 3 feet, and thinking the result is 5 meters. Mars Climate Orbiter was lost because of such a bug.
Such bugs can be prevented by wrapping the num- ber representing the length in a data constructor giving its unit.
data Length = Meters Double
meters_to_length :: Double -> Length meters_to_length m = Meters m
feet_to_length :: Double -> Length feet_to_length f = Meters (f * 0.3048)
add_lengths :: Length -> Length -> Length
add_lengths (Meters a) (Meters b) = Meters (a+b)
NOTE Making Length an abstract type, and ex- porting only type-safe operations to the rest of the program, improves safety even further.
Wrapping a data constructor around a number looks like adding overhead, since normally, each data constructor requires a cell of its own on the heap for itself and its arguments. However, types that have exactly one data constructor with ex- actly one argument can be implemented as if the data constructor were not there, eliminating the overhead. The Mercury language uses this repre- sentation scheme for types like this, and in some circumstances GHC can avoid the indirection.
For Mars Climate Orbiter, the Jet Propulsion Lab- oratory expected a contractor to provide thruster control data expressed in newtons, the SI unit of force, but the contractor provided the data in pounds-force, the usual unit of force in the old En- glish system of measurement. Since one pound of force is 4.45 newtons, the thruster calculations were off by a factor of 4.45. The error was not caught because the data files sent from the contractor to JPL contained only numbers and not units, and because other kinds of tests that could have caught the error were eliminated in an effort to save money. The result was that Mars Climate Orbiter dived too steeply into the Martian atmosphere and burned up.
– section 9 slide 6 –
Different uses of one unit
Sometimes, you want to prevent confusion even be- tween two kinds of quantities measured in the same units.
For example, many operating systems represent time as the number of seconds elapsed since a fixed epoch. For Unix, the epoch is 0:00am on 1 Jan 1970.
data Duration = Seconds Int
data Time = SecondsSinceEpoch Int
add_durations ::
Duration -> Duration -> Duration
add_durations (Seconds a) (Seconds b) = Seconds (a+b)
add_duration_to_time ::
Time -> Duration -> Time
add_duration_to_time
(SecondsSinceEpoch sse) (Seconds t) =
SecondsSinceEpoch (sse + t)
NOTE It makes sense to add together two durations or a time and a duration, but it does not make sense to add two times.
– section 9 slide 7 –
Different units in one type
Sometimes, you cannot apply a fixed conversion rate between different units. In such applications, each operation may need to do conversion on de- mand at whatever rate is applicable at the time of its execution.
data Money
= USD_dollars Double | AUD_dollars Double | GBP_pounds Double
For financial applications, using Doubles would not be a good idea, since accounting rules that precede the use of computers specify rounding methods (e.g. for interest calculations) that binary floating point numbers do not satisfy.
One workaround is to use fixed-point numbers, such as integers in which 1 represents not one dollar, but one one-thousandth of one cent.
NOTE Those accounting rules specified rounding algorithms for human accountants working with
decimal numbers, not for automatic computers working with binary numbers. The difference is sig- nificant, since floating-point numbers cannot even represent exactly such simple but important frac- tions as one tenth and one percent.
– section 9 slide 8 –
Mapping over a Maybe Suppose we have a type
type Marks = Map String [Int]
giving a list of all the marks for each student.
(A type declaration like this declares that Marks is an alias for Map String [Int], just the way String is an alias for [Char].)
We want to write a function
studentTotal :: Marks -> String -> Maybe Int
that returns Just the total mark for the specified student, or Nothing if the specified student is un- known. Nothing means something different from Just 0.
– section 9 slide 9 –
Mapping over a Maybe
This definition will work, but it’s a bit verbose:
studentTotal marks student =
case Map.lookup student marks of
Nothing -> Nothing
Just ms -> Just (sum ms)
We’d like to do something like this:
studentTotal marks student =
sum (Map.lookup student marks)
but it’s a type error:
Couldn’t match expected type ‘Maybe Int’ with actual type ‘Int’
– section 9 slide 10 – The Functor class
IfwethinkofaMaybe aasa“box”thatholdsana (or maybe not), we want to apply a function inside the box, leaving the box in place but replacing its content with the result of the function.
That’s actually what the map function does: it ap- plies a function to the contents of a list, returning a list of the results.
What we want is to apply a function a -> b to the contents of a Maybe a, returning a Maybe b. We want to map over a Maybe.
The Functor type class is the class of all types that can be “mapped” over. This includes [a], but also Maybe a.
– section 9 slide 11 – The Functor class
You can map over a Functor type with the fmap function:
fmap :: Functor f => (a -> b) -> f a -> f b
This gives us a much more succinct definition:
studentTotal marks student =
fmap sum (Map.lookup student marks)
Functor is defined in the standard prelude, so you can use fmap over Maybes (and lists) without im- porting anything.
The standard prelude also defines a function <$> as an infix operator alias for fmap, so the following code will also work:
sum <$> Map.lookup student marks
– section 9 slide 12 –
Beyond Functors
Suppose our students work in pairs, with either teammate submitting the pair’s work. We want to write a function
pairTotal :: Marks -> String -> String -> Maybe Int
to return the total of the assessments of two stu- dents, or Nothing if either or both of the students are not enrolled.
This code works, but is disappointingly verbose:
pairTotal marks student1 student2 = case studentTotal marks student1 of
Nothing -> Nothing Just t1 ->
case studentTotal marks student2 of Nothing -> Nothing
Just t2 -> Just (t1 + t2)
ifwetakeftobeMaybe,atobeIntandbtobe Int -> Int, then we see that
fmap (+) (studentTotal marks student1) returns a value of type Maybe (Int -> Int).
So all we need is a way to extract a function from inside a functor and fmap that over another functor. We want to apply one functor to another.
– section 9 slide 14 –
Applicative functors
Enter applicative functors. These are functors that can contain functions, which can be applied to other functors, defined by the Applicative class.
The most important function of the Applicative class is
(<*>) :: f (a -> b) -> f a -> f b
which does exactly what we need.
Happily, Maybe is in the Applicative class, so the following definition works:
pairTotal marks student1 student2 = let mark = studentTotal marks in (+) <$> mark student1
<*> mark student2
– section 9 slide 15 –
Applicative
The second function defined for every Applicative
type is
pure :: a -> f a
– section 9 slide 13 – Putting functions in Functors
Functor works nicely for unary functions, but not for greater arities. If we try to use fmap on a binary function and a Maybe, we wind up with a function in a Maybe.
Remembering the type of fmap,
fmap :: Functor f => (a -> b) -> f a -> f b
which just inserts a value into the applicative func- tor. For the Maybe class, pure = Just. For lists, pure = (:[]) (creating a singleton list).
For example, if we wanted to subtract a student’s score from 100, we could do:
(-) <$> pure 100 <*> studentTotal marks student
but a simpler way to do the same thing is:
(100-) <$> studentTotal marks student
In fact, every Applicative must also be a Functor,
just as every Ord type must be Eq. – section 9 slide 16 –
Lists are Applicative <*> gets even more interesting for lists:
Section 10: Introduction to Logic Programming
Logic programming
Functional programming languages are based on the lambda calculus of Alonzo Church and the con- cept of a function: a thing that maps inputs to outputs.
Logic programming languages are based on the predicate calculus of Gottlob Frege and the concept of a relation, which captures a relationship among a number of individuals, and the predicate that re- lates them.
…> (++) <$> [“do”,”something”,”good”] <*> pure “!”
[“do!”,”something!”,”good!”]
…> (++) <$> [“a”,”b”,”c”] <*> [“1″,”2”] [“a1″,”a2″,”b1″,”b2″,”c1″,”c2”]
You can think of <*> as being like a Cartesian prod- uct, hence the “*”.
– section 9 slide 17 –
A function is a special kind of relation that can only be used in one direction (inputs to outputs), and can only have one result. Relations in general do not have these limitations. In this sense, logic programming generalises functional programming.
While the first functional programming language was Lisp, implemented by John McCarthy’s group at MIT in 1958, the first logic programming lan- guage was Prolog, implemented by Alain Colmer- auer’s group at Marseille in 1971.
NOTE
Since the early 1980s, the University of Melbourne has been one of the world’s top centers for research in logic programming.
Lee Naish designed and implemented MU-Prolog, and led the development of its successor, NU- Prolog.
Zoltan Somogyi led the development of Mercury,
and was one of its main implementors.
The name “Prolog” was chosen by Philippe Roussel as an abbreviation for “programmation en logique”, which is French for “programming in logic”.
MU-Prolog and NU-Prolog are two closely-related dialects of Prolog. There are many others, since most centers of logic programming research in the 1980s implemented their own versions of the lan- guage.
The other main centers of logic programming re- search are in Leuven, Belgium; Uppsala, Sweden; Madrid, Spain; and Las Cruces, New Mexico, USA.
– section 10 slide 1 – Relations
A relation specifies a relationship; for example, a family relationship. In Prolog syntax,
parent(queen_elizabeth, prince_charles).
specifies (a small part of the) parenthood relation, which relates parents to their children. This says that Queen Elizabeth is a parent of Prince Charles.
The name of a relation is called a predicate. Predi- cates have no directionality: it makes just as much sense to ask of whom is Queen Elizabeth a parent as to ask who is Prince Charles’s parent. There is also no promise that there is a unique answer to either of these questions.
– section 10 slide 2 –
Facts A statement such as:
parent(queen_elizabeth, prince_charles).
is called a fact. It may take many facts to define a relation:
% (A small part of) the British Royal family parent(queen_elizabeth, prince_charles). parent(prince_philip, prince_charles). parent(prince_charles, prince_william). parent(prince_charles, prince_harry). parent(princess_diana, prince_william). parent(princess_diana, prince_harry).
.
Text between a percent sign (%) and end-of-line is treated as a comment.
– section 10 slide 3 –
Using Prolog
Most Prolog systems have an environment similar to GHCi. A file containing facts like this should be written in a file whose name begins with a lower- case letter and contains only letters, digits, and un- derscores, and ends with “.pl”.
A source file can be loaded into Prolog by typing its filename (without the .pl extension) between square brackets at the Prolog prompt (?-). Prolog prints a message to say the file was compiled, and true to indicate it was successful (user input looks
):
Some Prolog GUI environments provide other, more convenient, ways to load code, such as menu items or drag-and-drop.
like this
NOTE
?- [royals].
% royals compiled 0.00 sec, 8 clauses true.
?-
– section 10 slide 4 –
Queries
Once your code is loaded, you can use or test it by issuing queries at the Prolog prompt. A Prolog query looks just like a fact. When written in a source file and loaded into Prolog, it is treated as a true statement. At the Prolog prompt, it is treated as a query, asking if the statement is true.
?- parent(prince_charles, prince_william). true .
?- parent(prince_william, prince_charles). false.
– section 10 slide 5 – Variables
Each predicate argument may be a variable, which in Prolog begins with a capital letter or underscore and follows with letters, digits, and underscores. A query containing a variable asks if there exists a value for that variable that makes that query true, and prints the value that makes it true.
If there is more than one answer to the query, Pro- log prints them one at a time, pausing to see if more solutions are wanted. Typing semicolon asks for more solutions; just hitting enter (return) fin- ishes without more solutions.
This query asks: of whom Prince Charles is a par- ent?
– section 10 slide 6 –
Multiple modes
The same parenthood relation can be used just as easily to ask who is a parent of Prince Charles or even who is a parent of whom. Each of these is a different mode, based on which arguments are bound (inputs; non-variables) and which are un- bound (outputs; variables).
?- parent(X, prince_charles). X = queen_elizabeth ;
X = prince_philip.
?- parent(X, Y).
X = queen_elizabeth, Y = prince_charles ; X = prince_philip,
Y = prince_charles ;
.
– section 10 slide 7 –
Compound queries
Queries may use multiple predicate applications (called goals in Prolog and atoms in predicate logic). The simplest way to combine multiple goals is to separate them with a comma. This asks Prolog for all bindings for the variables that satisfy both (or all) of the goals. The comma can be read as “and”. In relational algebra, this is called an inner join (but do not worry if you do not know what that is).
?- parent(queen_elizabeth, X), | parent(X, Y).
X = prince_charles,
Y = prince_william ;
X = prince_charles, Y = prince_harry.
?- parent(prince_charles, X). X = prince_william ;
X = prince_harry.
– section 10 slide 8 – Rules
Predicates can be defined using rules as well as facts. A rule has the form
Head :- Body,
where Head has the form of a fact and Body has the form of a (possibly compound) query. The :- is read “if”, and the clause means that the Head is true if the Body is. For example
– section 10 slide 10 – Equality
Equality in Prolog, written “=” and used as an in- fix operator, can be used both to bind variables and to check for equality. Like Haskell, Prolog is a single-assignment language: once bound, a variable cannot be reassigned.
?- X = 7. X = 7.
?- a = b.
false.
?- X = 7, X = a.
false.
?- X = 7, Y = 8, X = Y.
false.
– section 10 slide 11 –
Disjunction
Goals can be combined with disjunction (or) as well as conjunction (and). Disjunction is written “;” and used as an infix operator. Conjunction (“,”) has higher precedence (binds tighter) than disjunction, but paren- theses can be used to achieve the desired precedence.
Who are the children of Queen Elizabeth or Princess Diana?
– section 10 slide 12 – Negation
grandparent(X,Z) :- parent(X, Y), parent(Y, Z).
means “X is grandparent of Z if X is parent of Y and Y is parent of Z.”
Rules and facts are the two different kinds of clauses. A predicate can be defined with any num- ber of clauses of either or both kinds, intermixed in any order.
– section 10 slide 9 –
Recursion
Rules can be recursive. Like Haskell, Prolog has no looping constructs, so recursion is widely used. Prolog does not have as well-developed a library of higher-order operations as Haskell, so recursion is used more in Prolog than in Haskell.
A person’s ancestors are their parents and the an- cestors of their parents.
ancestor(Anc, Desc) :- parent(Anc, Desc).
ancestor(Anc, Desc) :- parent(Parent, Desc),
ancestor(Anc, Parent).
?- parent(queen_elizabeth, X)
| ; parent(princess_diana, X). X = prince_charles ;
X = prince_william ;
X = prince_harry.
Negation in Prolog is written “\+” and used as a pre- fix operator. Negation has higher (tighter) precedence than both conjunction and disjunction. Be sure to leave a space between the \+ and an open parenthesis.
Who are the parents of Prince William other than Prince Charles?
Disequality in Prolog is written as an infix “\=”. So X \= Yisthesameas\+ X = Y.
– section 10 slide 13 –
The Closed World Assumption
Prolog assumes that all true things can be derived from the program. This is called the closed world assump- tion. Of course, this is not true for our parent relation (that would require tens of billions of clauses!).
but Princess Anne is a daughter of Queen Elizabeth. Our program simply does not know about her.
So use negation with great care on predicates that are not complete, such as parent.
– section 10 slide 14 –
Negation as failure
Prolog executes \+ G by first trying to prove G. If this fails, then \+ G succeeds; if it succeeds, then \+ G fails. This is called negation as failure.
In Prolog, failing goals can never bind variables, so any variable bindings made in solving G are thrown away when \+ G fails. Therefore, \+ G cannot solve for any variables, and goals such as these cannot work properly.
Is there anyone of whom Queen Elizabeth is not a par- ent?
Is there anyone who is not Queen Elizabeth?
– section 10 slide 15 –
Execution Order
The solution to this problem is simple: ensure all vari- ables in a negated goal are bound before the goal is executed.
Prolog executes goals in a query (and the body of a clause) from first to last, so put the goals that will bind the variables in a negation before the negation (or \=).
In this case, we can generate all people who are either parents or children, and ask whether any of them is different from Queen Elizabeth.
– section 10 slide 16 –
Datalog
The fragment of Prolog discussed so far, which omits data structures, is called Datalog. It is a generalisa- tion of what is provided by relational databases. Many modern databases now provide Datalog features or use Datalog implementation techniques.
?- parent(X, prince_william), | \+ X = prince_charles.
X = princess_diana.
?- \+ parent(queen_elizabeth, X). false.
?- X \= queen_elizabeth.
false.
?- parent(X, prince_william), | X \= prince_charles.
X = princess_diana.
?- \+ parent(queen_elizabeth, princess_anne). true.
?- (parent(X,_) ; parent(_,X)), | X \= queen_elizabeth.
X = prince_philip ;
.
capital(australia, canberra). capital(france, paris).
.
continent(australia, australia). continent(france, europe).
.
population(australia, 22_680_000). population(france, 65_700_000).
Section 11: Beyond Datalog
Terms
In Prolog, all data structures are called terms. A term can be atomic or compound, or it can be a variable. Datalog has only atomic terms and variables.
Atomic terms include integers and floating point num- bers, written as you would expect, and atoms.
An atom begins with a lower case letter and follows with letters, digits and underscores, for example a, queen elizabeth, or banana.
An atom can also be written beginning and ending with a single quote, and have any intervening characters. The usual character escapes can be used, for example \n for newline, \t for tab, and \’ for a single quote. For example: ’Queen Elizabeth’ or ’Hello, World!\n’.
– section 11 slide 1 – Compound Terms
In the syntax of Prolog, each compound term is a func- tor (sometimes called function symbol ) followed by zero or more arguments; if there are any arguments, they are shown in parentheses, separated by commas. Functors are Prolog’s equivalent of data constructors, and have the same syntax as atoms.
For example, the small tree that in Haskell syntax would be written as
Node Leaf 1 (Node Leaf 2 Leaf)
would be written in Prolog syntax as the term
node(leaf, 1, node(leaf, 2, leaf))
Because Prolog is dynamically typed, each argument of a term can be any term, and there is no need to declare types.
.
– section 10 slide 17 –
Datalog Queries
What is the capital of France?
?- capital(france, Capital). Capital = paris.
What are capitals of European countries?
?- continent(Country, europe), | capital(Country, Capital). Country = france,
Capital = paris.
What European countries have populations > 50,000,000?
?- continent(Country, europe), | population(Country, Pop),
| Pop > 50_000_000.
Country = france,
Pop = 65700000.
– section 10 slide 18 –
Prolog has special syntax for some functors, such as infix notation.
– section 11 slide 2 –
Variables
A variable is also a term. It denotes a single unknown term.
A variable name begins with an upper case letter or underscore, followed by any number of letters, digits, and underscores.
A single underscore is special: it specifies a different variable each time it appears, much like in Haskell pattern matching.
Like Haskell, Prolog is a single-assignment language: a variable can only be bound (assigned) once.
Because the arguments of a compound term can be any terms, and variables are terms, variables can appear in terms.
For example f(A,A) denotes a term whose functor is f and whose two arguments can be anything, as long as they are the same; f( , ) denotes a term whose functor is f and has any two arguments.
– section 11 slide 3 –
List syntax
Like Haskell, Prolog has a special syntax for lists. Both denote the empty list by [].
Both denote the list with the three elements 1, 2 and 3 by [1, 2, 3].
While Haskell uses x:xs to denote a list whose head is x and whose tail is xs, the Prolog syntax is [X | Xs] (not to be confused with list comprehensions, which Prolog lacks).
The Prolog syntax for what Haskell would represent with x1:x2:xs is [X1, X2 | Xs].
– section 11 slide 4 – Ground vs nonground terms
A term is a ground term if it contains no variables, and it is a nonground term if it contains at least one variable.
3 and f(a, b) are ground terms.
Since Name and f(a, X) each contain at least one vari-
able, they are nonground terms.
– section 11 slide 5 –
Substitutions
A substitution is a mapping from variables to terms.
Applying a substitution to a term means consistently replacing all occurrences of each variable in the map with the term it is mapped to.
Note that a substitution only replaces variables, never atomic or compound terms.
For example, applying the substitution {X1 → leaf, X2 → 1, X3 → leaf} to the term node(X1,X2,X3) yields the term node(leaf,1,leaf).
Since you can get node(leaf,1,leaf) from node(X1,X2,X3) by applying a substitution to it, node(leaf,1,leaf) is an instance of node(X1,X2,X3).
Any ground Prolog term has only one instance, while a nonground Prolog terms has an infinite number of instances.
– section 11 slide 6 – Unification
The term that results from applying a substitution θ to a term t is denoted tθ.
A term u is therefore an instance of term t if there is some substitution θ such that u = tθ.
A substitution θ unifies two terms t and u if tθ = uθ.
Consider the terms f(X, b) and f(a, Y).
Applying a substitution {X → a} to those two terms yields f(a, b) and f(a, Y), which are not syntactically identical, so this substitution is not a unifier.
On the other hand, applying the substitution {X → a, Y → b} to those terms yields f(a, b) in both cases, so this substitution is a unifier.
– section 11 slide 7 –
Recognising proper lists
A proper list is either empty ([]) or not ([X|Y]), in which case, the tail of the list must be a proper list. We can define a predicate to recognise these.
proper_list([]). proper_list([Head|Tail]) :-
proper_list(Tail).
– section 11 slide 8 –
Detour: singleton variables
The variable Head appears only once in this clause:
proper_list([Head|Tail]) :- proper_list(Tail).
This often indicates a typo in the source code. For example, if Tail were spelled Tial in one place, this would be easy to miss. But Prolog’s singleton warning would alert us to the problem.
– section 11 slide 9 –
Detour: singleton variables
In this case, there is no problem; to avoid the warn- ing, we should begin the variable name Head with an underscore, or just name the variable .
proper_list([]). proper_list([_Head|Tail]) :-
proper_list(Tail).
General programming advice: always fix compiler warn- ings (if possible). Some warnings may indicate a real problem, and you will not see them if they’re lost in a sea of unimportant warnings. It is easier to fix a prob- lem when the compiler points it out than when you have to find it yourself.
– section 11 slide 10 –
Append
Appending two lists is a common operation in Prolog. This is a built in predicate in most Prolog systems, but could easily be implemented as:
append([], C, C). append([A|B], C, [A|BC]) :-
append(B, C, BC).
?- [list].
% list compiled 0.00 sec, 1 clauses true.
?- [list].
Warning: list.pl:3:
Singleton variables: [Head] % list compiled 0.00 sec, 1 clauses true.
Warning: list.pl:3:
Singleton variables: [Head]
?- append([a,b,c],[d,e],List). List = [a, b, c, d, e].
This is similar to ++ in Haskell.
– section 11 slide 11 –
append is like proper list
Compare the code for proper list to the code for
append:
proper_list([]). proper_list([Head|Tail]) :-
proper_list(Tail).
append([], C, C). append([A|B], C, [A|BC]) :-
append(B, C, BC).
This is common: code for a predicate that handles a term often follows the structure of that term (as we saw in Haskell).
While the proper list predicate is not very useful it- self, it was worth designing, as it gives a hint at the structure of other code that traverses lists. Since types are not declared in Prolog, predicates like proper list can serve to indicate the notional type.
– section 11 slide 12 –
Appending backwards
Unlike ++ in Haskell, append in Prolog can work in other modes:
?- append([1,2,3], Rest, [1,2,3,4,5]). Rest = [4, 5].
?- append(Front, [3,4], [1,2,3,4]). Front = [1, 2] ;
false.
?- append(Front,Back,[a,b,c]). Front = [],
Back = [a, b, c] ;
Front = [a],
Back = [b, c] ;
Front = [a, b],
Back = [c] ;
Front = [a, b, c], Back = [] ;
false.
– section 11 slide 13 – Length
The length/2 built-in predicate relates a list to its length:
The …termsarehowPrologprintsoutunboundvari- ables. The number reflects when the variable was cre- ated; because these variables are all printed differently, we can tell they are all distinct variables.
[ 2956, 2962, 2968] is a list of three distinct un- bound variables, and each unbound variable can be any term, so this can be any three-element list, as specified by the query.
– section 11 slide 14 – Putting them together
How would we implement take in Prolog? take(N,List,Front) should hold if Front is the first N
elements of List. So length(Front,N) should hold. Also, append(Front, , List) should hold. Then:
take(N, List, Front) :- length(Front,N),
append(Front, _, List).
Prolog coding hint: sometimes it is easier to write code if you think about checking if the result is correct rather than computing it. That is, think declaratively.
?- length([a,b,c], Len). Len = 3.
?- length(List, 3).
List = [_2956, _2962, _2968].
Then you need to think about whether your code will work the ways you want it to. We will return to that.
– section 11 slide 15 –
Member Here is list membership, two ways:
member1(Elt, List) :- append(_,[Elt|_], List).
member2(Elt, [Elt|_]).
member2(Elt, [_|Rest]) :- member2(Elt, Rest).
These behave the same, but the second is a bit more efficient because the first builds and ignores the list of elements before Elt in List, and the second does not.
Note the recursive version does not exactly match the structure of our earlier proper list predicate. This is because Elt is never a member of the empty list, so we do not need a clause for []. In Prolog, we do not need to specify when a predicate should fail; only when it should succeed. We also have two cases to consider when the list is non-empty (like Haskell in this respect).
– section 11 slide 16 –
Arithmetic
In Prolog, terms like 6 * 7 are just data structures, and
= does not evaluate them, it just unifies them.
The built-in predicate is/2 (an infix operator) evalu-
ates expressions.
– section 11 slide 17 –
Arithmetic modes Use is/2 to evaluate expression
square(N, N2) :- N2 is N * N.
Unfortunately, square only works when the first argu- ment is bound. This is because is/2 only works if its second argument is ground.
Later we shall see how to write code to do arithmetic in different modes.
– section 11 slide 18 –
Arithmetic
Prolog provides the usual arithmetic operators, includ- ing:
?- square(5, X).
X = 25.
?- square(X, 25).
ERROR: is/2: Arguments are not sufficiently instantiated
?- 25 is X * X.
ERROR: is/2: Arguments are not sufficiently instantiated
+-* / // mod
– integer float
add, subtract, multiply
division (may return a float) integer division (rounds toward 0) modulo (result has same sign as
second argument) unary minus (negation)
coersions (not operators)
More arithmetic predicates (infix operators; both argu- ments must be ground expressions):
?- X = 6 * 7.
X = 6*7.
?- X is 6 * 7.
X = 42.
< =<
> >= =:= =\=
less, less or equal (note!) greater, greater or equal
equal, not equal (only numbers)
– section 11 slide 19 –
Section 12: Understanding and Debugging Prolog code
List Reverse
To reverse a list, put the first element of the list at the end of the reverse of the tail of the list.
rev1([], []). rev1([A|BC], CBA) :-
rev1(BC, CB),
append(CB, [A], CBA).
reverse/2 is an SWI Prolog built-in, so we use a dif- ferent name to avoid conflict.
NOTE This version of reverse/2 is called naive re- verse, because it has quadratic complexity. We’ll see a more efficient version a little later.
– section 12 slide 1 –
List Reverse
The mode of a Prolog goal says which arguments are bound (inputs) and which are unbound (outputs) when the predicate is called.
rev1/2 works as intended when the first argument is ground and the second is free, but not for the opposite mode.
Prolog hangs at this point. We will use the Prolog de- bugger to understand why. For now, hit control-C and then ’a’ to abort.
– section 12 slide 2 –
The Prolog Debugger
To understand the debugger, you will need to under- stand the Byrd box model. Think of goal execution as a box with a port for each way to enter and exit.
A conventional language has only one way to enter and one way to exit; Prolog has two of each.
The four debugger ports are:
call exit fail redo
call initial entry fail final failure
exit successful completion redo backtrack into goal
Turn on debugger with trace, and off with nodebug, at the Prolog prompt.
– section 12 slide 3 –
Using the Debugger
The debugger prints the current port, execution depth, and goal (with the current variable bindings) at each step.
“lists:” in front of append is a module name.
?- trace, rev1([a,b], Y).
Call: (7) rev1([a, b], _12717) ? creep
Call: (8) rev1([b], _12834) ? creep
Call: (9) rev1([], _12834) ? creep
Exit: (9) rev1([], []) ? creep
Call: (9) lists:append([], [b], _12838) ? creep Exit: (9) lists:append([], [b], [b]) ? creep Exit: (8) rev1([b], [b]) ? creep
Call: (8) lists:append([b], [a], _12717) ? creep Exit: (8) lists:append([b], [a], [b, a]) ? creep Exit: (7) rev1([a, b], [b, a]) ? creep
Y = [b, a].
?- rev1([a,b,c], Y). Y = [c, b, a].
?- rev1(X, [c,b,a]). X = [a, b, c] ;
– section 12 slide 4 – Reverse backward
Now try the “backwards” mode of rev1/2. We shall use a smaller test case to keep it manageable.
– section 12 slide 5 –
Reverse backward, continued
after showing the first solution, Prolog goes on for- ever like this:
rev1([], []). rev1([A|BC], CBA) :-
rev1(BC, CB), append(CB, [A], CBA).
The problem is that the goal rev1(X,[a]), re- solves to the goal rev1(BC, CB), append(CB, [A], [a]). The call rev1(BC, CB) produces an infinite backtracking sequence of solutions {BC → [],CB → []}, {BC → [Z],CB → [Z]}, {BC → [Y,Z], CB → [Z,Y]}, . . .. For each of these solu- tions, we call append(CB, [A], [a]).
append([], [A], [a]) succeeds, with {A → [a]}. However, append([Z], [A], [a]) fails, as does this goal for all following solutions for CB. This is an infinite backtracking loop.
– section 12 slide 7 –
Infinite backtracking loop
We could fix this problem by executing the body goals in the other order:
rev2([], []). rev2([A|BC], CBA) :-
append(CB, [A], CBA), rev2(BC, CB).
But this definition does not work in the forward direction:
NOTE
Prolog uses a fixed left-to-right execution order, so nei- ther order works for both directions.
?- trace, rev1(X, [a]).
Call: (7) rev1(_11553, [a]) ? creep
Call: (8) rev1(_11661, _11671) ? creep
Exit: (8) rev1([], []) ? creep
Call: (8) lists:append([], [_11660], [a]) ? creep Exit: (8) lists:append([], [a], [a]) ? creep Exit: (7) rev1([a], [a]) ? creep
X = [a] ;
Redo: (8) rev1(_11661, _11671) ? creep
Call: (9) rev1(_11664, _11674) ? creep
Exit: (9) rev1([], []) ? creep
Call: (9) lists:append([], [_11663], _11678) ? creep Exit: (9) lists:append([], [_11663], [_11663]) ? creep Exit: (8) rev1([_11663], [_11663]) ? creep
Call: (8) lists:append([_11663], [_11660], [a]) ? creep
Fail: (8) lists:append([_11663], [_11660], [a]) ? creep
Redo: (9) rev1(_11664, _11674) ? creep
Call: (10) rev1(_11667, _11677) ? creep
Exit: (10) rev1([], []) ? creep
Call: (10) lists:append([], [_11666], _11681) ? creep Exit: (10) lists:append([], [_11666], [_11666]) ? creep
Exit: (9) Call: (9) Exit: (9)
Exit: (8) Call: (8) Fail: (8)
.
rev1([_11666], [_11666]) ? creep lists:append([_11666], [_11663], _11684) ? creep lists:append([_11666], [_11663], [_11666, _11663]) ? creep
rev1([_11663, _11666], [_11666, _11663]) ? creep lists:append([_11666, _11663], [_11660], [a]) lists:append([_11666, _11663], [_11660], [a])
– section 12 slide 6 – Infinite backtracking loop
?- rev2(X, [a,b]). X = [b, a] ; false.
?- rev2([a,b], Y).
Y = [b, a] ;
^CAction (h for help) ? abort % Execution Aborted
– section 12 slide 8 –
Working in both directions
The solution is to ensure that when rev1 is called, the first argument is always bound to a list. We do this by observing that the length of a list must always be the same as that of its reverse. When samelength/2 succeeds, both arguments are bound to lists of the same fixed length.
rev3(ABC, CBA) :- samelength(ABC, CBA),
rev1(ABC, CBA).
samelength([], []). samelength([_|Xs], [_|Ys]) :-
samelength(Xs, Ys).
– section 12 slide 9 –
Documenting Prolog Code
Your code files should have two levels of documen- tation – file level documentation and predicate level comments. Each file should start with comments that outline: the purpose of the file; its author; the date at which the code was written; and a brief summary of what the code does and any underlying rationale.
Comments should be provided above all significant and non-trivial predicates in a consistent format. These comments should identify: the meaning of each argument; what the predicate does; and the modes in which the predicate is designed to operate.
An excellent resource on coding standards in Prolog is the paper “Coding guidelines for Prolog” by Covington et al. (2011).
– section 12 slide 10 –
Predicate level documentation
The following is an example of predicate level documen- tation from Covington et al. (2011). This predicate removes duplicates from a list.
%% remove duplicates(+List, -Result)
%
% Removes the duplicates in List, giving Result.
% Elements are considered to match if they can be
% unified with each other; thus, a partly
% uninstantiated element may become further
% instantiated during testing. If several elements % match, the last of them is preserved.
Predicate arguments are prefaced with a: + to indicate that the argument is an input and must be instantiated to a term that is not an unbound variable; – if the ar- gument is an output and may be an unbound variable; or a ? to indicate that the argument can be either an input or an output.
– section 12 slide 11 –
More on the Debugger Some useful debugger commands:
h display debugger help
c creep to the next port (also space, enter)
s skip over goal; go straight to exit or fail port
r back to initial call port of goal, undoing all bindings done since starting it;
a abort whole debugging session
+ set spypoint (like breakpoint) on this pred – remove spypoint from this predicate
l leap to the next spypoint
b pause this debugging session and enter a “break level,” giving a new Prolog prompt; end of file reenters debugger
?- rev3(X, [a,b]). X = [b, a].
?- rev3([a,b], Y). Y = [b, a].
– section 12 slide 12 –
More on the Debugger
Built-in predicates for controlling the debugger:
spy(Predspec) Place a spypoint on Predspec, which can be a name/arity pair, or just a predicate name.
nospy(Predspec) RemovethespypointfromPredspec. trace Turn on the debugger
debug Turn on the debugger and leap to first spypoint nodebug Turn off the debugger
A “Predspec” is a predicate name or name/arity – section 12 slide 13 –
Using the debugger
Note the r (retry) debugger command restarts a goal from the beginning, “time travelling” back to the time when starting to execute that goal.
The s (skip) command skips forward in time, over the whole execution of a goal, to its exit or fail port.
This leads to a quick way of tracking down most bugs:
1. When you arrive at a call or redo port: skip.
2. If you come to an exit port with the correct results
(or a correct fail port): creep.
3. If you come to an incorrect exit or fail port: retry,
then creep.
Eventually you will find a clause that has the right input and wrong output (or wrong failure); this is the bug. This will not help find infinite recursion, though.
– section 12 slide 14 –
Spypoints
For larger computations, it may take some time to get to the part of the computation where the bug lies. Usually, you will have a good idea, or at least a few good guesses, which predicates you suspect of being buggy (usually the predicates you have edited most recently). In cases of infinite recursion you may suspect certain predicates of being involved in the loop.
In these cases, spypoints will be helpful. Like a break- point in most debuggers, when Prolog reaches any port of a predicate with a spypoint set, Prolog stops and shows the port. The l (leap) command tells Prolog to run quietly until it reaches a spypoint. Use the spy(pred) goal at the Prolog prompt to set a spypoint on the named predicate, nospy(pred) to remove one. You can also add a spypoint on the predicate of the cur- rent debugger port with the + command, and remove it with -.
– section 12 slide 15 –
Managing nondeterminism
This is a common mistake in defining factorial:
fact(0, 1).
fact(N, F) :-
N1 is N – 1,
fact(N1, F1),
F is N * F1.
fact(5,F) has only one solution, why was Prolog look- ing for another?
– section 12 slide 16 –
Correctness
The second clause promises that for all n, n! = n×(n− 1)!. This is wrong for n < 1.
?- fact(5, F).
F = 120 ;
ERROR: Out of local stack
Prolog is not like Haskell: even if one clause applies, later clauses are still tried. After finding 0! = 1, Prolog thinks 0! = 0 × −1!; tries to compute −1!, −2!, . . .
The simple solution is to ensure each clause is a correct (part of the) definition.
fact(0, 1).
fact(N, F) :-
N > 0,
N1 is N – 1,
fact(N1, F1),
F is F1 * N.
– section 12 slide 17 –
Choicepoints
This definition is correct, but it could be more efficient.
When a clause succeeds but there are later clauses that could possibly succeed, Prolog will leave a choicepoint so it can later backtrack and try the later clause.
In this case, backtracking to the second clause will fail unless N > 0. This test is quick. However, as long as the choicepoint exists, it inhibits the very impor- tant last call optimisation (discussed later). Therefore, where efficiency matters, it is important to make your recursive predicates not leave choicepoints when they should be deterministic.
In this case, N = 0 and N > 0 are mutually exclusive, so at most one clause can apply, so fact/2 should not leave a choicepoint.
NOTE Actually, you will probably never take the fac- torial of a very large number, so the loss in efficiency from this definition is unlikely to make a detectable difference. For deeper arithmetic recursions or other recursions that are not structural inductions, however, this technique is useful.
– section 12 slide 18 –
If-then-else
We can avoid the choicepoint with Prolog’s if-then-else construct:
fact(N, F) :-
(N =:=0->
F=1 ; N>0,
N1 is N – 1,
fact(N1, F1),
F is F1 * N
).
The -> is treated like a conjunction (,), except that when it is crossed, any alternative solutions of the goal before the ->, as well as any alternatives following the ; are forgotten. Conversely, if the goal before the -> fails, then the goal after the ; is tried. So this is deterministic whenever both the code between -> and ;, and the code after the ;, are.
– section 12 slide 19 –
If-then-else caveats
However, you should prefer indexing (discussed next time) and avoid if-then-else, when you have a choice. If-then-else usually leads to code that will not work smoothly in multiple modes. For example, append could be written with if-then-else:
ap(X, Y, Z) :-
( X = [] ->
Z=Y
; X = [U|V],
ap(V, Y, W),
Z = [U|W] ).
This may appear correct, and may follow the logic you would use to code it in another language, but it is not appropriate for Prolog.
– section 12 slide 20 –
If-then-else caveats With that definition of ap:
?- ap([a,b,c], [d,e], L). L = [a, b, c, d, e].
?- ap(L, [d,e], [a,b,c,d,e]). false.
?- ap(L, M, [a,b,c,d,e]). L = [],
M = [a, b, c, d, e].
Because the if-then-else commits to binding the first argument to [] when it can, this version of append will not work correctly unless the first argument is bound when append is called.
– section 12 slide 21 –
Section 13: Logic and Resolution
Interpretations
In the mind of the person writing a logic program,
• each constant (atomic term) stands for an entity in the “domain of discourse” (world of the program);
• each functor (function symbol of arity n where n > 0) stands for a function from n entities to one entity in the domain of discourse; and
• each predicate of arity n stands for a particular relationship between n entities in the domain of discourse.
This mapping from the symbols in the program to the world of the program (which may be the real world or some imagined world) is called an interpretation.
The obvious interpretation of the atomic formula parent(queen elizabeth, prince charles) is that Queen Elizabeth II is a parent of Prince Charles, but other interpretations are also possible.
NOTE Another interpretation would use the function symbol queen elizabeth to refer to George W Bush, the function symbol prince charles to refer to Barack Obama, and the predicate symbol parent to refer to the notion “succeeded by as US President”. However, any programmer using this interpretation, or pretty much any interpretation other than the obvious one, would be guilty of using a horribly misleading programming style.
Terms using non-meaningful names such as f(g, h) do not lead readers to expect a particular interpretation, so these can have many different non-misleading inter- pretations.
– section 13 slide 1 –
Two views of predicates
As the name implies, the main focus of the predicate
calculus is on predicates.
You can think of a predicate with n arguments in two
equivalent ways.
• You can view the predicate as a function from all possible combinations of n terms to a truth value (i.e. true or false).
• You can view the predicate as a set of tuples of n terms. Every tuple in this set is implicitly mapped to true, while every tuple not in this set is implic- itly mapped to false.
The task of a predicate definition is to define the map- ping in the first view, or equivalently, to define the set of tuples in the second view.
– section 13 slide 2 –
The meaning of clauses The meaning of the clause
grandparent(A, C) :- parent(A, B), parent(B, C).
is: for all the terms that A and C may stand for, A is the grandparent of C if there is a term B such that A is the parent of B and B is the parent of C.
In mathematical notation: ∀A∀C : grandparent(A, C) ←
∃B : parent(A, B) ∧ parent(B, C)
The variables appearing in the head are universally quantified over the entire clause, while variables appear- ing only in the body are existentially quantified over the body.
NOTE ∀ is “forall”, the universal quantifier, while ∃ is “there exists”, the existential quantifier. The sign ∧ denotes the logical “and” operation, while the sign ∨ denotes the logical “or” operation, and the sign ¬ denotes the logical “not” operation.
– section 13 slide 3 –
The meaning of predicate definitions
A predicate is defined by a finite number of clauses, each of which is in the form of an implication. A fact such as parent(queen elizabeth, prince charles) repre- sents this implication:
∀A∀B : parent(A, B) ←
(A = queen elizabeth ∧ B = prince charles)
To represent the meaning of the predicate, create a dis- junction of the bodies of all the clauses:
∀A∀B : parent(A, B) ← (A=queenelizabeth∧B=princecharles) ∨ (A=princephilip∧B=princecharles) ∨ (A=princecharles∧B=princewilliam) ∨ (A=princecharles∧B=princeharry) ∨ (A=princessdiana∧B=princewilliam) ∨ (A = princess diana ∧ B = prince harry)
NOTE Obviously, this definition of the parent relation- ship is the correct one only if you restrict the universe of discourse to this small set of people.
– section 13 slide 4 – The closed world assumption
To implement the closed world assumption, we only need to make the implication arrow go both ways (if and only if ):
∀A∀B : parent(A, B) ↔ (A=queenelizabeth∧B=princecharles) ∨ (A=princephilip∧B=princecharles) ∨ (A=princecharles∧B=princewilliam) ∨ (A=princecharles∧B=princeharry) ∨ (A=princessdiana∧B=princewilliam) ∨ (A = princess diana ∧ B = prince harry)
This means that A is not a parent of B unless they are one of the listed cases.
Adding the reverse implication this way creates the Clark completion of the program.
– section 13 slide 5 – Semantics of logic programs
A logic program P consists of a set of predicate defi- nitions. The semantics of this program (its meaning) is the set of its logical consequences as ground atomic formulas.
A ground atomic formula a is a logical consequence of a program P if P makes a true.
A negated ground atomic formula ¬a, written in Prolog as \+a, is a logical consequence of P if a is not a logical consequence of P.
For most logic programs, the set of ground atomic for- mulas it entails is infinite (as is the set it does not en- tail). As logicians, we do not worry about this any more than a mathematician worries that there are an infinite number of solutions to a + b = c.
– section 13 slide 6 –
Finding the semantics
You can find the semantics of a logic program by work- ing backwards. Instead of reasoning from a query to find a satisfying substitution, you reason from the pro- gram to find what ground queries will succeed.
.
it is sufficient to show any of
– section 13 slide 9 –
SLD resolution in action
Procedural Interpretation The logical reading of the clause
grandparent(X, Z) :-
parent(X, Y), parent(Y, Z).
says“forallX,Y,Z,ifX isparentofY andY is parent of Z, then X is grandparent of Z ”.
The procedural reading says “to show that X is a grand- parent of Z, it is sufficient to show that X is a parent ofY andY isaparentofZ”.
SLD resolution, used by Prolog, implements this strat- egy.
– section 13 slide 8 –
SLD Resolution
The consequences of a logic program are determined through a simple but powerful deduction strategy called resolution.
SLD resolution is an efficient version of resolution. The basic idea is: given this program, to show this goal is true
q :- b1a, b1b.
q :- b2a, b2b.
The immediate consequence operator TP takes a set of ground unit clauses C and produces the set of ground unit clauses implied by C together with the program P .
This always includes all ground instances of all unit clauses in P. Also, for each clause H : −G1,…,Ga. in P, if C contains instances of G1,…Gn, then the corresponding instance of H is also in the result.
Eg, if P = {q(X,Z) : −p(X,Y),p(Y,Z)} and C = {p(a, b).p(b, c).p(c, d).}, then TP (C) = {q(a, c).q(b, d).}.
The semantics of program P is always TP (TP (TP (· · · (∅) · · ·)))
(TP applied infinitely many times to the empty set).
– section 13 slide 7 –
?- p, q, r.
?- p, b1a, b1b, r. ?- p, b2a, b2b, r.
.
E.g., to determine if Queen Elizabeth is Prince Harry’s grandparent:
with this program
grandparent(X, Z) :-
parent(X, Y), parent(Y, Z).
we unify query goal grandparent(queen elizabeth, prince harry) with clause head grandparent(X, Z), apply the resulting substitution to the clause, yielding the resolvent. Since the goal is identical to the resolvent head, we can replace it with the resolvent body, leaving:
parent(prince_charles, prince_harry). parent(princess_diana, prince_harry).
This does not mean we are through: we must backtrack and try the first matching clause. This leaves
There is one matching program clause, leaving nothing more to prove. The query succeeds.
– section 13 slide 12 –
Resolution
?- grandparent(queen_elizabeth, prince_harry).
?- parent(queen_elizabeth, prince_charles).
?- parent(queen_elizabeth, Y),| parent(Y, prince_harry).
– section 13 slide 10 –
SLD resolution can fail
Now we must pick one of these goals to resolve; we select the second.
The program has several clauses for parent, but only two can successfully resolve with parent(Y, prince harry):
parent(prince_charles, prince_harry). parent(princess_diana, prince_harry).
We choose the second. After resolution, we are left with the query (note the unifying substitution is applied to both the selected clause and the query):
No clause unifies with this query, so resolution fails. Sometimes, it may take many resolution steps to fail.
– section 13 slide 11 –
SLD resolution can succeed
Selecting the second of these matching clauses led to failure:
This derivation can be shown as an SLD tree:
grandparent(queen_elizabeth, prince_harry).
parent(queen_elizabeth, Y), parent(Y, prince_harry).
parent(queen_elizabeth, prince_charles).
success
parent(queen_elizabeth, princess_diana).
failure
– section 13 slide 13 –
Order of execution
The order in which goals are resolved and the order in which clauses are tried does not matter for correctness (in pure Prolog), but it does matter for efficiency. In this example, resolving parent(queen elizabeth, Y) before parent(Y, prince harry) is more efficient, be- cause there is only one clause matching the former, and two matching the latter.
grandparent(queen_elizabeth, prince_harry).
parent(queen_elizabeth, Y), parent(Y, prince_harry).
parent(prince_charles, prince_harry).
success
– section 13 slide 14 –
?- parent(queen_elizabeth, princess_diana).
SLD resolution in Prolog
At each resolution step we must make two decisions:
1. which goal to resolve
2. which clauses matching the selected goal to pursue
(though there may only be one choice for either or both).
Our procedure was somewhat haphazard when deci- sions needed to be made. For pure logic programming, this does not matter for correctness. All goals will need to be resolved eventually; which order they are resolved in does not change the answers. All matching clauses may need to be tried; the order in which we try them determines the order solutions are found, but not which solutions are found.
Prolog always selects the first goal to resolve, and al- ways selects the first matching clause to pursue first. This gives the programmer more certainty, and control, over execution.
– section 13 slide 15 –
Backtracking
When there are multiple clauses matching a goal, Pro- log must remember which one to go back to if necessary. It must be able to return the computation to the state it was in when the first matching clause was selected, so that it can return to that state and try the next matching clause. This is all done with a choicepoint.
When a goal fails, Prolog backtracks to the most recent choicepoint, removing all variable bindings made since the choicepoint was created, returning those variables to their unbound state. Then Prolog begins resolution with the next matching clause, repeating the process until Prolog detects that there are no more matching clauses, at which point it removes that choicepoint. Subsequent failures will then backtrack to the next most recent choicepoint.
– section 13 slide 16 –
Indexing
Indexing can greatly improve Prolog efficiency
Most Prolog systems will automatically create an in- dex for a predicate such as parent/2 (Prolog uses name/arity to refer to predicates) with multiple clauses the heads of which have distinct constants or functors. This means that, for a call with the first argument bound, Prolog will immediately jump to the first clause that matches. If backtracking occurs, the index allows Prolog to jump straight to the next clause that matches, and so on.
If the first argument is unbound, then all clauses will have to be tried.
– section 13 slide 17 –
Indexing
If some clauses have variables in the first argument of the head, those clauses will be tried at the appropriate time regardless of the call. Indexing changes perfor- mance, not behaviour. Consider:
p(a, z).
p(b, y).
p(X, X).
p(a, x).
For the call p(I, J), all clauses will be tried, in or- der. For p(a, J), the first clause will be tried, then the third, then fourth. For p(b, J), the second, then third, clause will be tried. For p(c, J), only the third clause will be tried.
– section 13 slide 18 –
Indexing
Some Prolog systems, such as SWI Prolog, will con- struct indices for arguments other than the first. For
parent/2, SWI Prolog will index on both arguments, so finding the children of a parent or parents of a child both benefit from indexing.
Just as important as jumping directly to the first matching clause, indexing tells Prolog when no further clauses could possibly match the goal, allowing it to re- move the choicepoint, or even to avoid creating the choi- cepoint in the first place. Even with only two clauses, such as for append/3, indexing can substantially im- prove performance.
– section 13 slide 19 –
Section 14: Tail Recursion
Tail recursion
A predicate (or function, or procedure, or method, or. . . ) is tail recursive if the only recursive call on any execution of that predicate is the last code executed before returning to the caller. For example, the usual definition of append/3 is tail recursive, but rev1/2 is not:
append([], C, C). append([A|B], C, [A|BC]) :-
append(B, C, BC).
rev1([], []). rev1([A|BC], CBA) :-
rev1(BC, CB), append(CB, [A], CBA).
– section 14 slide 1 – Tail recursion optimisation
Like most declarative languages, Prolog performs tail recursion optimisation (TRO). This is important for declarative languages, since they use recursion more than non-declarative languages. TRO makes recursive predicates behave as if they were loops.
Note that TRO is more often directly applicable in Pro- log than other languages because more Prolog code is tail recursive. For example, while append/3 in Prolog is tail recursive, ++ in Haskell is not, because the last operation performed is (:), not ++.
[] ++ lst = lst
(h:t) ++ lst = h:(t++lst)
However another optimisation can permit TRO for this code.
– section 14 slide 2 –
The stack
To understand TRO, it is important to understand how programming languages (not just Prolog) implement call and return using a stack (Haskell is a rare excep- tion). While a is executing, it stores its local variables, and where to return to when finished, in a stack frame or activation record. When a calls b, it creates a fresh stack frame for b’s local variables and return address, preserving a’s frame, and similarly when b calls c, as shown below.
Growth
But if all b will do after calling c is return to a, then there is no need to preserve its local variables. Prolog can release b’s frame before calling c, as shown below. When c is finished, it will return directly to a. This is called last call optimisation, and can save significant stack space.
Growth
– section 14 slide 3 –
TRO and choicepoints
Tail recursion optimisation is a special case of last call optimisation where the last call is recursive. This is especially beneficial, since recursion is used to replace looping. Without TRO, this would require a stack
frame for each iteration, and would quickly exhaust the stack. With TRO, tail recursive predicates execute in constant stack space, just like a loop.
However, if b leaves a choicepoint, it sits on the stack above b’s frame, “freezing” that and all earlier frames so that they are not reclaimed. This is necessary because when Prolog backtracks to that choicepoint, b’s argu- ments must be ready to try the next matching clause for b. The same is true if c or any predicate called later leaves a choicepoint, but choicepoints before the call to b do not interfere.
Growth
– section 14 slide 4 –
Making code tail recursive
Our factorial predicate was not tail recursive, as the last thing it does is perform arithmetic.
fact(N, F) :-
( N =:= 0 ->
F=1 ; N>0,
N1 is N – 1,
fact(N1, F1),
F is F1 * N
).
Note that Prolog’s if-then-else construct does not leave a choicepoint. A choicepoint is created, but is removed as soon as the condition succeeds or fails. So fact would be subject to TRO, if only it were tail recursive.
– section 14 slide 5 – Adding an accumulator
a
b
Choicepoint
c
a
b
c
a
c
We make factorial tail recursive by introducing an ac- cumulating parameter, or just an accumulator. This is an extra parameter to the predicate that holds a partially computed result.
Usually the base case for the recursion will specify that the partially computed result is actually the result. The recursive clause usually computes more of the partially computed result, and passes this in the recursive goal.
The key to getting the implementation correct is speci- fying what the accumulator means and how it relates to the final result. To see how to add an accumulator, de- termine what is done after the recursive call, and then respecify the predicate so it performs this task, too.
– section 14 slide 6 –
Adding an accumulator
For factorial, we compute fact(N1, F1), F is F1 * N last, so the tail recursive version will need to per- form the multiplication too. We must define a predicate fact1(N, A, F) so that F is A times the factorial of N. In most cases, it is not difficult to see how to transform the original definition to the tail recursive one.
fact(N, F) :-
( N =:= 0 ->
F=1 ; N>0,
N1 is N – 1,
fact(N1, F1),
F is F1 * N
).
fact(N, F) :- fact1(N, 1, F). fact1(N, A, F) :-
( N =:= 0 -> F=A
; N>0,
N1 is N – 1,
A1 is A * N,
fact1(N1, A1, F)
).
– section 14 slide 7 –
Adding an accumulator
Finally, define the original predicate in terms of the new one. Again, it is usually easy to see how to do that.
Another way to think about writing a tail recursive im- plementation of a predicate is to realise that it will es- sentially be a loop, so think of how you would write it as a while loop, and then write that loop in Prolog.
A=1; while (N A*= N–;
> 0) { N;
}if (N == else FAIL;
transforms to ⇓
0) return A;
translates to ⇓
fact(N, F) :- fact1(N, 1, F). fact1(N, A, F) :-
( N > 0 ->
A1 is A * N,
N1 is N – 1,
fact1(N1, A1, F)
; N =:= 0 ->
F=A ).
– section 14 slide 8 –
Transformation
Another approach is to systematically transform the non-tail recursive version into an equivalent tail recursive predicate. Start by defining a predicate to do the work of the recursive call to fact/2 and everything following it. Then replace the call to fact(N, F2) by the definition of fact/2. This is called unfolding.
fact1(N, A, F) :-
fact(N, F2),
F is F2 * A
transforms to ⇓
fact1(N, A, F) :-
( N =:= 0 ->
F2 = 1
; N>0,
N1 is N – 1,
fact(N1, F1),
F2 is F1 * N ),
F is F2 * A.
– section 14 slide 9 –
Transformation
Next we move the final goal into both the then and
else branches.
fact1(N, A, F) :- (N =:=0->
F2 = 1 ; N>0,
N1 is N – 1,
fact(N1, F1),
F2 is F1 * N
),
F is F2 * A.
transforms to ⇓
fact1(N, A, F) :-
( N =:= 0 ->
F2 = 1,
F is F2 * A
; N>0,
N1 is N – 1,
fact(N1, F1),
F2 is F1 * N,
F is F2 * A
).
Transformation
The next step is to simplify the arithmetic goals.
fact1(N, A, F) :-
( N =:= 0 ->
F2 = 1,
F is F2 * A N>0,
N1 is N – 1, fact(N1, F1), F2 is F1 * N, F is F2 * A
).
fact1(N, A,
( N =:= 0 ->
;
F) :-
transforms to ⇓
F=A
N>0,
N1 is N – 1, fact(N1, F1), Fis(F1*N)*A
).
;
– section 14 slide 11 –
Transformation
– section 14 slide 10 –
transforms to ⇓
Now we utilise the associativity of multiplication. This is the insightful step that is necessary to be able to make the next step.
fact1(N, A, F) :-
( N =:= 0 ->
F=A ; N>0,
N1 is N – 1,
fact(N1, F1),
F is (F1 * N) * A
).
fact1(N, A, F) :- (N =:=0->
F=A ; N>0,
N1isN-1, fact(N1, F1), FisF1*(N*A)
).
fact1(N, A, (N
;
).
fact1(N, A,
( N =:= 0 ->
F=A ; N>0,
– section 14 slide 12 –
Transformation
F) :-
=:= 0 -> F=A
N>0, N1isN-1, A1isN*A, fact(N1, F1), FisF1*A1
transforms to ⇓
Now part of the computation can be moved before the call to fact/2.
fact1(N, A, (N
;
).
fact1(N, A, F) :-
( N =:= 0 ->
F=A ; N>0,
N1 is N – 1, A1isN*A, fact1(N1, A1, F)
).
F) :-
=:=0->
F=A
N>0,
N1 is N – 1, fact(N1, F1),
F is F1 * (N * A)
– section 14 slide 14 –
Accumulating Lists
F) :-
).
N1 is N – 1,
A1 is N * A,
fact(N1, F1),
F is F1 * A1
– section 14 slide 13 –
Transformation
transforms to ⇓
The tail recursive version of fact is a constant factor more efficient, because it behaves like a loop. Sometimes accumulators can make an order difference, if it can replace an operation with a computation of lower asymptotic complexity, for example replacing append/3 (linear time) with list construction (constant time).
rev1([], []). rev1([A|BC], CBA) :-
rev1(BC, CB), append(CB, [A], CBA).
This definition of rev1/2 is of quadratic complexity, because for the nth element from the end of the first argument, we append a list of length n − 1 to a singleton list. Doing this for each of the n elements gives time proportional to n(n−1) .
2
– section 14 slide 15 –
The final step is to recognise that the last two goals look very much like the body of the original definition of fact1/3, with the substitution
{N → N1, F2 → F1, A → A1}. So we replace those two goals with the clause head with that substitution applied. This is called folding.
Tail recursive rev1/2
The first step in making a tail recursive version of rev1/2 is to specify the new predicate. It must combine the work of rev1/2 with that of append/3. The specification is:
% rev(BCD, A, DCBA)
% DCBA is BCD reversed, with A appended
We could develop this by transformation as we did for fact1/3, but we implement it directly here. We begin with the base case, for BCD = []:
rev([], A, A).
– section 14 slide 16 –
Tail recursive rev1/2 For the recursive case, take BCD = [B|CD]:
rev([B|CD], A, DCBA) :-
the result, DCBA, must be the reverse of CD, with [B] appended to the end, and A appended after that. In Haskell notation this is
(rev cd ++ [b]) ++ a.
Because append is associative, this is the same as
rev cd ++ ([b] ++ a) ≡ rev cd ++ (b:a). We can use our rev/3 predicate to compute that:
rev([B|CD], A, DCBA) :- rev(CD, [B|A], DCBA).
– section 14 slide 17 –
Tail recursive rev1/2
rev([], A, A). rev([B|CD], A, DCBA) :-
rev(CD, [B|A], DCBA).
At each recursive step, this code removes an element from the head of the input list and adds it to the head of the accumulator. The cost of each step is therefore a constant, so the overall cost is linear in the length of the list.
Accumulator lists work like a stack: the last element of the accumulator is the first element that was added to it, and so on. Thus at the end of the input list, the accumulator is the reverse of the original input.
– section 14 slide 18 –
Difference pairs
The trick used for a tail recursive reverse predicate is often used in Prolog: a predicate that generates a list takes an extra argument specifying what should come after the list. This avoids the need to append to the list.
In Prolog, if you do not know what will come after at the time you call the predicate, you can pass an unbound variable, and bind that variable when you do know what should come after. Thus many predicates intended to produce a list have two arguments, the first is the list produced, and the second is what comes after. This is called a difference pair, because the predicate generates the difference between the first and second list.
flatten(empty, List, List). flatten(node(L,E,R), List, List0) :-
flatten(L, List, List1), List1 = [E|List2], flatten(R, List2, List0).
– section 14 slide 19 –
Section 15:
All solutions and impurity
All solutions
Sometimes one would like to bring together all solutions to a goal. Prolog’s all solutions predicates do exactly this.
setof(Template, Goal, List) binds List to sorted list of all distinct instances of Template satisfying Goal
Template can be any term, usually containing some of the variables appearing in Goal. On completion, setof/3 binds its List argument, but does not further bind any variables in the Template.
List = [prince_harry, prince_william] ; P = prince_philip,
List = [prince_charles] ;
P = prince_william,
List = [prince_george] ;
P = princess_diana,
List = [prince_harry, prince_william] ; P = queen_elizabeth,
List = [prince_charles].
– section 15 slide 2 –
Existential quantification
Use existential quantification, written with infix caret (^), to collect solutions for a template regardless of the bindings of some of the variables not in the Template.
E.g., to find all the people in the database who are parents of any child:
– section 15 slide 3 –
Unsorted solutions
The bagof/3 predicate is just like setof/3, except that it does not sort the result or remove duplicates.
?- bagof(P, C^parent(P, C), Parents).
Parents = [queen_elizabeth, prince_philip, prince_charles, prince_charles, princess_diana, princess_diana, prince_william, duchess_kate].
Solutions are collected in the order they are produced. This is not purely logical, because the order of solutions should not matter, nor should the number of times a solution is produced.
?- setof(P, C^parent(P, C), Parents).
Parents = [duchess_kate, prince_charles, prince_philip, prince_william, princess_diana, queen_elizabeth].
?- setof(P-C, parent(P, C), List). List = [duchess_kate-prince_george, prince_charles-prince_harry, prince_charles-prince_william, prince_philip-prince_charles, prince_william-prince_george, princess_diana-prince_harry, princess_diana-prince_william, queen_elizabeth-prince_charles].
– section 15 slide 1 –
All solutions
If Goal contains variables not appearing in Template, setof/3 will backtrack over each distinct binding of these variables, for each of them binding List to the list of instances of Template for that binding.
?- setof(C, parent(P, C), List). P = duchess_kate,
List = [prince_george] ;
P = prince_charles,
– section 15 slide 4 – Higher Order Programming
The call/1 built-in predicate executes a term as a goal. This capitalises on the fact that Prolog terms look just like Prolog goals. Many Prologs, including SWI Prolog, will call a variable used as a goal.
map(_, [], []).
map(P, [X|Xs], [Y|Ys]) :-
call(P, X, Y),
map(P, Xs, Ys).
This is in the SWI Prolog library with the name maplist/2, 3, 4, and 5.
– section 15 slide 7 –
Input/Output
Prolog’s Input/Output facility does not even try to be pure. I/O operations are executed when they are reached according to Prolog’s simple execution order. I/O is not “undone” on backtracking.
Prolog has builtin predicates to read and write arbitrary Prolog terms. Prolog also allows users to define their own operators. This makes Prolog very convenient for applications involving structured I/O.
– section 15 slide 8 –
Input/Output write/1 is handy for printing messages:
?- map(plus(1), [3,4,5], L). L = [4, 5, 6] ;
false.
?- map(plus(1), L, [3,4,5]). L = [2, 3, 4] ;
false.
?- X=append([1,2],[3],L), call(X). X = append([1, 2], [3], [1, 2, 3]), L = [1, 2, 3].
?- X=append(A, B, [1]), X.
X = append([], [1], [1]),
A = [],
B = [1] ;
X = append([1], [], [1]),
A = [1],
B = [] ;
false.
– section 15 slide 5 –
Currying
Currying in Prolog is done by simply omitting some final arguments of a higher-order goal. To support this, many Prologs, including SWI, support versions of call of higher arity. All arguments to call/n after the goal (first) argument are added as extra arguments to the goal.
– section 15 slide 6 –
Writing higher-order code
It is fairly straightforward to write higher order predicates using call/n. For example, here is map/3 in Prolog:
?- op(800, xfy, wibble).
true.
?- read(X).
|: p(x,[1,2],X>Y wibble z).
X = p(x, [1, 2], _1274>_1276 wibble z). ?- write(p(x,[1,2],X>Y wibble z)). p(x,[1,2],_1464>_1466 wibble z)
true.
?- X=append([1,2],[3]), call(X, L). X = append([1, 2], [3]),
L = [1, 2, 3].
?- write(’hello ’), write(’world!’). hello world!
true.
?- write(’world!’), write(’hello ’). world!hello
true.
?- sort([h,e,l,l,o], L). L = [e, h, l, o].
?- msort([h,e,l,l,o], L). L = [e, h, l, l, o].
?- keysort([7-a, 3-b, 3-c, 8-d, 3-a], L). L = [3-b, 3-c, 3-a, 7-a, 8-d].
This demonstrates that Prolog’s input/output predicates are non-logical. These should be equivalent, because conjunction should be commutative.
Code that performs I/O must be handled carefully — you must be aware of the modes. It is recommended to isolate I/O in a small part of the code, and keep the bulk of your code I/O-free. (This is a good idea in any language.)
– section 15 slide 9 –
Comparing terms
All Prolog terms can be compared for ordering using the built-in predicates @<, @=<, @>, and @>=. Prolog, somewhat arbitrarily, uses the ordering
Variables < Numbers < Atoms < CompoundTerms
but most usefully, within these classes, terms are ordered as one would expect: numbers by value and atoms are sorted alphabetically. Compound terms are ordered first by arity, then alphabetically by functor, and finally by arguments, left-to-right. It is best to use these only for ground terms.
– section 15 slide 10 –
Sorting
There are three SWI Prolog builtins for sorting ground lists according to the @< ordering: sort/2 sorts a list, removing duplicates, msort/2 sorts a list, without removing duplicates, and keysort/2 stably sorts list of X-Y terms, only comparing X parts:
– section 15 slide 11 – Determining term types
integer/1 holds for integers and fails for anything else. It also fails for variables.
Similarly, float/1 recognises floats, number recognises either kind of number, atom/1 recognises atoms, and compound/1 recognises compound terms. All of these fail for variables, so must be used with care.
– section 15 slide 12 – Recognising variables
var/1 holds for unbound variables, nonvar/1 holds for any term other than an unbound variable, and ground/1 holds for ground terms (this requires traversing the whole term). Using these or the predicates on the previous slide can make your code behave differently in different modes.
But they can also be used to write code that works in multiple modes.
Here is a tail-recursive version of len/2 that works whenever the length is known:
?- integer(3).
true.
?- integer(a).
false.
?- integer(X).
false.
?- hello @< hi.
true.
?- X @< 7, X = foo.
X = foo.
?- X = foo, X @< 7.
false.
len2(L, N) :-
( N =:= 0
-> L = []
; N1 is N – 1,
L = [_|L1],
len2(L1, N1)
).
– section 15 slide 13 –
Recognising variables
This version works when the length is unknown:
len1([], N, N). len1([_|L], N0, N) :-
N1 is N0 + 1,
len1(L, N1, N).
This code chooses between the two:
len(L, N) :-
( integer(N)
-> len2(L, N)
; nonvar(N)
-> throw(error(type_error(integer, N),
context(len/2, ’’))) ; len1(L, 0, N)
highest-performing declarative languages. Although Mercury’s learning curve is somewhat steeper than Prolog’s, it is a more realistic choice for serious development.
– section 15 slide 15 –
).
– section 15 slide 14 –
Mercury
The Mercury language was developed at The University of Melbourne as a purely declarative successor to Prolog. Mercury is strongly typed, with a type system very similar to Haskell’s. It is also strongly moded, which means that the binding state of all variables is determined at compile-time. Mercury has a strong determinism system, as well, which allows the compiler to determine, for each predicate mode, whether it will always be deterministic.
All these properties, along with aggressive compiler optimisations, make Mercury among the
Section 16: Constraint (logic) programming
Constraint (logic) programming
An imperative program specifies the exact sequence of actions to be executed by the computer.
A functional program specifies how the result of the program is to be computed at a more abstract level. One can read function definitions as suggesting an order of actions, but the language implementation can and sometimes will deviate from that order, due to lazyness, parallel execution, and various optimizations.
A logic program is in some ways more declarative, as it specifies a set of equality constraints that the terms of the solution must satisfy, and then searches for a solution.
A constraint program is more declarative still, as it allows more general constraints than just equality constraints. The search for a solution will typically follow an algorithm whose relationship to the specification can be recognized only by experts.
– section 16 slide 1 –
Problem specification
The specification of a constraint problem consists of
• a set of variables, each variable having a known domain,
• a set of constraints, with each constraint involving one or more variables, and
• an optional objective function.
The job of the constraint programming system is to find a solution, a set of assignments of values to variables (with the value of each variable being drawn from its domain) that satisfies all the constraints.
The objective function, if there is one, maps each solution to a number. If this number represents a cost, you want to pick the cheapest solution; if this number represents a profit, you want to pick the most profitable solution.
NOTE The objective function is optional because sometimes you care only about the existence of a solution, and sometimes all solutions are equally good.
The set of constraints may be given in advance, or it may be discovered piecemeal, as you go along. The latter occurs reasonably often in planning problems.
– section 16 slide 2 –
Kinds of constraint problems
There are several kinds of constraints, of which the following four are the most common. Most CP systems handle only one or two kinds.
In Herbrand constraint systems, the variables represent terms, and the basic constraints are unifications, i.e. they have the form term1 = term2. This is the constraint domain implemented by Prolog.
In finite domain or FD constraint systems, each variable’s domain has a finite number of elements.
In boolean satisfiability or SAT systems, the variables represent booleans, and each constraint asserts the truth of an expression constructed using logical operations such as AND, OR, NOT and implication.
In linear inequality constraint systems, the variables represent real numbers (or sometimes integers), and the constraints are of the form ax + by ≤ c (where x and y are variables, and a, b and c are constants).
NOTE You can view boolean satisfiability (SAT) problems as a subclass of finite domain problems, but there are specialized algorithms that work only on booleans and not on finite domains with more than two values, Research into solving SAT problems has
made great progress over the last 20 years, and modern SAT solvers are surprisingly efficient at solving NP complete problems. So in practice, the two classes of problems should be handled differently.
– section 16 slide 3 –
Constraints and Prolog
Prolog can be viewed as Constraint Logic Programming (CLP) over Herbrand terms. Recall that in a Herbrand constraint system, the constraints that we can model are equality and disequality over terms. We can view a goal in a Prolog program as representing a constraint.
Consider the parent/2 predicate defined by the following facts.
parent(joe, bob). parent(joe, julie).
We can view the goal parent(A, B) as constraining the values of variables A and B. This constraint is resolved via unification. The values for A and B that satisfy this constraint are:
A=joe, A=joe,
the function definitions. To simplify the process, we assume the program has been “normalised” into an equivalent definition expressed in a restricted form. In particular, all nested expressions have been replaced by let-bound temporary variables, and deconstructions have been replaced by case constructs.
– section 16 slide 5 –
Type inference
For example, the rule for handling function application states that if the type of f is a → b and the type of x is t then the type of f x is b, where t = a. The case construct is handled by unifying the type of the expression with the type of all of the cases.
Consider the normalised definition of the Haskell map function:
map f l = case l of
[] -> []
B = bob;
B = julie.
– section 16 slide 4 –
Type inference
The form of the equation tells you that the type of map,tmap,isatypeoftheformt1 →t2 →tr.
tmap , t1 , t2 and tr are all constraint variables whose domain is the set of possible Haskell types, and
tmap = t1 → t2 → tr is a constraint on those variables.
– section 16 slide 6 –
Type inference: an example
We can use Prolog’s Herbrand terms to infer types:
map f l = case l of
[] -> []
(e:es) -> let fes = map f es fe = f e
in fe:fes
?- % define arrow operator for convenience | op(200,xfy,’=>’).
?- Map = F => L => Result, L = [_],
| Result = [_], L = [E], Es = [E],
| Map = F => Es => FEs, F = E => FE,
(e:es) ->
let t = map f es h=fe
in h:t
One application of constraint programming using Herbrand terms is type inference for the Hindley-Milner type system (the Haskell type system). The process of inferring the types of variables inside functions can be viewed as a matter of solving a set of constraints over variable types. Herbrand terms can be used to neatly represent types. and (if not declared) the types of the functions and predicates themselves can be viewed as a Herbrand constraint problem.
This approach represents the type of each variable in the program as a Herbrand term, and applies rules to produce equality constraints for each subexpression in
| Result = [FE], FEs = [FE]. Map = (E=>FE)=>[E]=>[FE],
F = E=>FE,
L = Es, Es = [E],
Result = FEs, FEs = [FE].
– section 16 slide 7 –
Linear inequality constraints
Suppose you want to make banana and/or chocolate cakes for a bake sale, and you have 10 kg of flour, 30 bananas, 1.2 kg of sugar, 1.5 kg of butter, and 700 grams of cocoa on hand. You can charge $4.00 for a banana cake and $6.50 for a chocolate one. Each kind of cake requires a certain quantity of each ingredient. How do you determine how many of each cake to make so as to maximise your profit?
To solve such a problem, you need to set up a system of constraints saying, for example, that the number of each kind of cake times the amount of flour needed for that kind of cake must add to no more than the amount of flour you have, and so on.
You also need to specify that the number of each kind of cake must be non-negative. Finally, you need to define your revenue as the sum of the number of each kind of cake times its price, and specify that you would like to maximise revenue.
– section 16 slide 8 –
Linear inequality constraints
We can use SWI Prolog’s library(clpr) to solve such problems. This library requires constraints to be enclosed in curly braces.
?- 250*B + 200*C =< 10000,
| 2*B =< 30,
| 75*B + 150*C =< 1200,
| 100*B + 150*C =< 1500,
| 75*C =< 700,
| B>=0, C>=0,
| Revenue = 4*B + 6.5*C, maximize(Revenue). B = 12.0,
C = 2.0,
Revenue = 61.0
So we can make $61.00 by making 12 Banana and 2 Chocolate cakes.
– section 16 slide 9 –
Sudoku
Sudoku is a class of puzzles played on a 9×9 grid. Each grid position should hold a number between 1 and 9. The same integer may not appear twice
• in a single row,
• in a single column, or
• in one of the nine 3×3 boxes.
The puzzle creator provides some of the numbers; the challenge is to fill in the rest.
This is a classic finite domain constraint satisfaction problem.
r1
r2 r3 r4 r5 r6 r7 r8 r9
5
3
7
6
1
9
5
9
8
6
8
6
3
4
8
3
1
7
2
6
6
2
8
4
1
9
5
8
7
9
c1 c2 c3 c4 c5 c6 c7 c8 c9
– section 16 slide 10 –
Sudoku as finite domain constraints
You can represent the rules of sudoku as a set of 81 constraint variables (r1c1, r1c2 etc) each with the domain 1..9, and 27 all-different constraints: one for each row, one for each column, and one for each box. For example, the constraint for the top left box would be all different([r1c1, r1c2, r1c3, r2c1, r2c2, r2c3, r3c1, r3c2, r3c3]).
Initially, the domain of each variable is [1..9]. If you fix the value of a variable e.g. by setting r1c1 = 5, this means that the other variables that share a row, column or box with r1c1 (and that therefore appear in an all-different constraint with it) cannot be 5, so their domain can be reduced to [1..4, 6..9].
This is how the variables fixed by our example puzzle reduce the domain of r3c1 to only [1..2], and the domain of r5c5 to only [5].
Fixing r5c5 to be 5 gives us a chance to further reduce the domains of the other variables linked to r5c5 by constraints, e.g. r7c5.
– section 16 slide 11 – Sudoku in SWI Prolog
Using SWI’s library(clpfd), Sudoku problems can be solved:
sudoku(Rows) :- length(Rows, 9),
maplist(same_length(Rows), Rows), append(Rows, Vs), Vs ins 1..9, maplist(all_distinct, Rows), transpose(Rows, Columns), maplist(all_distinct, Columns), Rows = [A,B,C,D,E,F,G,H,I], blocks(A, B, C), blocks(D, E, F), blocks(G, H, I).
blocks([], [], []).
blocks([A,B,C|Bs1], [D,E,F|Bs2], [G,H,I|Bs3]) :-
all_distinct([A,B,C,D,E,F,G,H,I]), blocks(Bs1, Bs2, Bs3).
– section 16 slide 12 –
Sudoku in SWI Prolog
?- Puzzle=[[5,3,_, _,7,_, _,_,_],
| [6,_,_, 1,9,5, _,_,_],
| [_,9,8, _,_,_, _,6,_], |
| [8,_,_, _,6,_, _,_,3],
| [4,_,_, 8,_,3, _,_,1],
| [7,_,_, _,2,_, _,_,6], |
| [_,6,_, _,_,_, 2,8,_],
| [_,_,_, 4,1,9, _,_,5],
| [_,_,_, _,8,_, _,7,9]],
| sudoku(Puzzle),
| write(Puzzle).
In less than 1 20
– section 16 slide 13 –
Sudoku solution
of a second, this produces the solution:
Puzzle=[[5,3,4, 6,7,8, 9,1,2], [6,7,2, 1,9,5, 3,4,8], [1,9,8, 3,4,2, 5,6,7],
[8,5,9, 7,6,1, 4,2,3], [4,2,6, 8,5,3, 7,9,1], [7,1,3, 9,2,4, 8,5,6],
[9,6,1, 5,3,7, 2,8,4], [2,8,7, 4,1,9, 6,3,5], [3,4,5, 2,8,6, 1,7,9]]
– section 16 slide 14 –
Search
Prolog normally employs a strategy known as “generate and test” to search for variable bindings that satisfy constraints. Nondeterministic goals
generate potential solutions; later goals test those solutions, imposing further constraints and rejecting some candidate solutions.
For example, in
?- between(1,9,X), 0 =:= X mod 2, | X=:=X*Xmod10.
The first goal generates single-digit numbers, the second tests that it is even, and the third that its square ends in the same digit.
Constraint logic programming uses the more efficient “constrain and generate” strategy. In this approach, constraints on variables can be more sophisticated than simply binding to a Herbrand term. This is generally accomplished in Prolog systems with attributed variables, which allow constraint domains to control unification of constrained variables.
– section 16 slide 15 –
Propagation
The usual algorithm for solving a set of FD constraints
consists of two steps: propagation and labelling.
In the propagation step, we try to reduce the domain
of each variable as much as possible.
For each constraint, we check whether the constraint rules out any values in the current domains of any of the variables in that constraint. If it does, then we remove that value from the domain of that variable, and schedule the constraints involving that variable to be looked at again.
The propagation step ends
• if every variable has a domain of size one, which represents a fixed value, since this represents a solution;
• if some variable has an empty domain, since this represents failure; or
• if there are no more constraints to look at, in which case propagation can do no more.
– section 16 slide 16 –
Labelling
If propagation cannot do any more, we go on to the
labelling step, which
• picks a not-yet-fixed variable,
• partitions its current domain (of size n) into k parts d1, …, dk, where usually k = 2 but may be any value satisfying 2 ≤ k ≤ n, and
• recursively invokes the whole constraint solving algorithm k times, with invocation i restricting the domain of the chosen variable to di.
Each recursive invocation also consists of a propagation step and (if needed) a labelling step. The whole computation therefore consists of alternating propagation and labelling steps.
The labelling steps generate a search tree. The size of the tree depends on the effectiveness of propagation: the more effective propagation is at removing values from domains, the smaller the tree will be, and the less time searching it will take.
NOTE The root node of the tree represents the computation for solving the whole constraint problem. Every other node represents the choice within the computation represented by its parent node. If it is the ith child of its parent node, then it represents the choice in the labelling step to set the domain of the selected variable to di.
If the current domain of the chosen variable is e.g. [1..5], then the selected partition maybe contain two to five parts. The only partition that divides [1..5] into five parts is of course [1], [2], [3], [4], [5], but there are many partitions that divide [1..5] into two parts, with [1..2], [3..5] and [1, 3, 5], [2, 4] being two of them.
Note that one can view the labelling step as imposing an extra constraint on each branch of the search tree, with the extra constraint for branch i requiring the chosen variable to have a value in di of its current domain.
– section 16 slide 17 –
Section 17: More on Constraints
Prolog Arithmetic Revisted
In earlier lectures we introduced a number of built-in arithmetic predicates, including (is)/2, (=:=)/2, (=\=)/2, and (=<)/2. Recall that these predicates could only be used in certain modes. The predicate (is)/2, for example, only works when its second argument is ground.
The CLP(FD) library provides replacements for these lower-level arithmetic predicates. These new predicates are called arithmetic constraints and can be used in both directions (i.e., in,out and out,in).
– section 17 slide 1 –
CLP(FD) Arithmetic Constraints
CLP(FD) provides the following arithmetic constraints:
?- 25 #= X * 5.
X = 5.
– section 17 slide 2 –
Propagation and Labelling with CLP(FD)
Recall that the domain of a CLP(FD) variable is the set of all integers. We reduce or restrict the domain of these variables with the use of CLP(FD) constraints. When a constraint is posted, the library automatically revises the domains of relevant variables if necessary. This is called propagation.
As we saw in the Sudoku example, sometimes propagation alone is enough to reduce the domains of each variable to a single element. In other cases, we need to tell Prolog to perform the labelling step.
label/1 is an enumeration predicate that searches for an assignment to each variable in a list that satisfies all posted constraints.
– section 17 slide 3 –
N-Queens
Before we look at some real-world applications of constraint solving, let’s consider one more puzzle. In this puzzle, we have a N by N chessboard. We want to place N queens on this board so that no queen is under attack. Each column, row, and diagonal, can contain only a single queen.
?- 25 is X * X.
ERROR: Arguments are not sufficiently instantiated
?- X is 5 * 5.
X = 25.
?- 25 #= X * X, label([X]). X = -5;
X = 5.
Expr1 #= Expr2 Expr1 #\= Expr2 Expr1 #> Expr2 Expr1 #< Expr2 Expr1 #>= Expr2 Expr1 #=< Expr2
?- 25 #= X * X.
X in -5\/5.
Expr1 equals Expr2
Expr1 is not equal to Expr2
Expr1 is greater than Expr2
Expr1 is less than Expr2
Expr1 is greater than or equal to Expr2 Expr1 is less than or equal to Expr2
Q Q
?- n_queens(8,Qs), label(Qs). Qs = [1, 5, 8, 6, 3, 7, 2, 4] ; Qs = [1, 6, 8, 3, 7, 4, 2, 5] ; ...
There are 92 distinct solutions to the 8-Queens puzzle! – section 17 slide 6 –
Constraint Programming: Applications
Constraint Programming is a state-of-the-art approach for solving a range of combinatorial optimisation problems, including:
• Timetabling;
• Vehicle Routing;
• Project Scheduling (Resource Constrained Scheduling Problems);
Q
Q
Q Q
Q
– section 17 slide 4 –
Solving N-Queens
n_queens(N, Qs) :-
length(Qs, N), Qs ins 1..N, safe_queens(Qs).
safe_queens([]). safe_queens([Q|Qs]) :-
safe_queens(Qs, Q, 1), safe_queens(Qs).
safe_queens([], _, _). safe_queens([Q|Qs], Q0, D0) :-
Q0 #\= Q, abs(Q0 - Q) #\= D0, D1 #= D0 + 1,
safe_queens(Qs, Q0, D1).
Source: https://www.metalevel.at/queens/ – section 17 slide 5 –
Solving N-Queens
?- n_queens(8, Qs). _4010 in 1..8, abs(_4010-_4052)#\=7, _4010#\=_4052, abs(_4010-_4046)#\=6, ...
_4016#\=_4022,
_4016 in 1..8.
Q
• Rostering.
– section 17 slide 7 –
Timetabling
Timetabling is a finite domain problem of great practical importance.
To timetable university lectures, you need two variables for every lecture:
• The domain of timeCOMP30048L1 represents all the available lecture time slots from mon8am to fri7pm.
• The domain of locCOMP30048L1 represents all the available lecture theatres that are big enough for COMP30048’s enrolment.
There are many kinds of constraints, including:
• all the lectures for a subject must be at different times
• all the lectures taught by a single lecturer must be at different times
• all the rooms of lectures taught at the same time must be different
• the room in which each lecture is taught must be big enough for its anticipated enrolment
– section 17 slide 8 –
The search space
Timetabling happens to be very hard to do well, partly because it generates a huge search space if you impose only absolutely necessary constraints such as the previous two.
Imposing the constraint that timeCOMP30048L1 must be earlier in the week than timeCOMP30048L2 breaks a symmetry and thus halves the number of solutions. Imposing such a constraint on each of say 1000 subjects reduces the number of solutions by a factor of 21000.
Despite this astronomical reduction, the search space is still very large.
Most constraint problems are NP-complete, so in the worst case, their solution requires time that is exponential in the size of the problem.
Nevertheless, heuristics can often reduce the solution time to manageable levels, and new and better heuristics are being developed all the time.
NOTE Another essential constraint is that you can schedule at most one lecture in a particular theatre in a particular time slot. The university has violated this constraint in the past, which in one case led to a theatre overflowing with three sets of students, and three lecturers at the lectern energetically discussing whose subject the theatre was really reserved for.
One example of another kind of essential constraint applies to masters subjects intended for part-time
students. For these subjects, the usual arrangement is that the two lectures in a week are delivered back to back, from 5pm to 7pm, in the same lecture theatre, with the tutorial following from 7 to 8pm. This allows people who work during the day to take this subject without either having to take (much) time off work or having to commute to campus more than once a week.
A solution in which COMP30048 has
• a lecture called COMP30048L1 on tuesdays at 3pm and
• a lecture called COMP30048L2 on thursdays at 3pm,
and a solution in which COMP30048 has
• a lecture called COMP30048L2 on tuesdays at 3pm and
• a lecture called COMP30048L1 on thursdays at 3pm,
does not make any difference in practice, but to a constraint system, these are different solutions, and it will try to find them both, unless you add a constraint to rule out one of them.
– section 17 slide 9 –
Optional constraints
Another reason why timetabling is very hard to do well is because there are many objectives that are desirable but not essential:
• Different lectures in the same subject should not be on the same day.
• Schedule as few lectures for 8am, 1pm, 6pm and 7pm as possible.
• If a lecturer or a cohort of students have lectures in consecutive time slots, it should be possible to walk from the first lecture theatre to the second in less than ten minutes. (This should be essential, but apparently isn’t.)
• If a lecturer or a cohort of students spend most of their time in a building, then their lectures should be in that building.
Requiring all these constraints to be fulfilled would result in a horribly overconstrained problem and thus have no solution. However, if you have several solutions that satisfy the essential constraints, you can choose the solution that satisfies the most optional constraints.
– section 17 slide 10 –
Vehicle Routing
Given a logistics network of potentially thousands of nodes, and a potentially heterogeneous fleet of capacitated vehicles, find a set of routes that services all customers at minimum cost. We must determine which customers each vehicle services and in what order.
Further complications include the presence of time windows (individual customers must be serviced within a specific time period), packing and compatibility constraints, and transshipment.
– section 17 slide 11 –
Resource Constrained Scheduling Problems
In this type of problem, we have a set of activities or jobs that need to be completed, and we want to minimise makespan (time required to complete all jobs). Precedence constraints require that certain jobs be completed before others, and each job requires certain resources. In a job shop setting – our resources are machines; each job is assigned to a machine; and no two jobs can be scheduled on a machine at the same time.
– section 17 slide 12 –
Rostering and Employee Scheduling
The Nurse Scheduling or Rostering Problem is an example of a class of rostering problems for which CP techniques are well suited. We must assign shifts (day, night, and late-night shifts) and holidays to nurses.
Each shift must be adequately staffed by nurses with an appropriate combination of qualifications, while each nurse cannot be assigned to multiple shifts on a single day.
– section 17 slide 13 –
Further study
The University has developed a series of MOOCs on discrete optimisation, and basic and advanced modelling for discrete optimisation. These courses explore constraint problems and related algorithms in much more detail.
– section 17 slide 14 –
Section 18: Monads
any and all Other useful higher-order functions are
any :: (a -> Bool) -> [a] -> Bool all :: (a -> Bool) -> [a] -> Bool
For example, to see if every word in a list contains the letter ’e’:
To check if a word contains any vowels:
– section 18 slide 1 –
flip
If the order of arguments of elem were reversed, we could have used currying rather than the bulky \x -> elem x “aeiou”. The flip function takes a function and returns a function with the order of arguments flipped.
flip :: (a -> b -> c) -> (b -> a -> c) flip f x y = f y x
The ability to write functions to construct other functions is one of Haskell’s strengths.
– section 18 slide 2 –
Monads
Monads build on this strength. A monad is a type constructor that represents a computation. These computations can then be composed to create other computations, and so on. The power of monads lies in the programmer’s ability to determine how the computations are composed. Phil Wadler, who introduced monads to Haskell, describes them as “programmable semicolons”.
A monad M is a type constructor that supports two operations:
• A sequencing operation, denoted >>=, whose type is M a -> (a -> M b) -> M b.
• An identity operation, denoted return, whose typeisa -> M a.
– section 18 slide 3 –
Monads
Think of the type M a as denoting a computation that produces an a, and possibly carries something extra. For example, if M is the Maybe type constructor, that something extra is an indication of whether an error has occurred so far.
• You can take a value of type a and use the (misnamed) identity operation to wrap it in the monad’s type constructor.
• Once you have such a wrapped value, you can use the sequencing operation to perform an operation on it. The >>= operation will unwrap its first argument, and then typically it will invoke the function given to it as its second argument, which will return a wrapped up result.
Prelude> all (elem ’e’) [“eclectic”, “ele phant”, “legion”]
True
Prelude> any (\x -> elem x “aeiou”) “sky” False
Prelude> any (flip elem “aeiou”) “hmmmm” False
NOTE You can apply the sequencing operation to any value wrapped up in the monad’s type constructor; it does not have to have been generated by the monad’s identity function.
– section 18 slide 4 –
The Maybe and MaybeOK monads
The obvious ways to define the monad operations for the Maybe and MaybeOK type constructors are these (MaybeOK is not in the library):
maybe_head :: [a] -> MaybeOK a
maybe_head [] = Error “head of empty list” maybe_head (x:_) = OK x
maybe_sqrt :: Int -> MaybeOK Double maybe_sqrt x =
if x >= 0 then
OK (sqrt (fromIntegral x))
else
Error “sqrt of negative number”
How can you encode a sequence of operations such as taking the head of a list and computing its square root?
NOTE The fromIntegral function can convert an integer to any other numeric type.
– section 18 slide 6 –
Simplifying code with monads
maybe_head :: [a] -> MaybeOK a maybe_sqrt :: Int -> MaybeOK Double maybe_sqrt_of_head ::
[Int] -> MaybeOK Double
— definition not using monads maybe_sqrt_of_head l =
let mh = maybe_head l in case mh of
Error msg -> Error msg OK h -> maybe_sqrt h
— simpler definition using monads maybe_sqrt_of_head l =
maybe_head l >>= maybe_sqrt
NOTE The monadic version is simpler because in that version, the work of checking for failure and handling it if found is done once, in the MaybeOK monad’s sequence operator.
In the version without monads, this code would have to be repeated for every step in a sequence of possibly-failing operations. If this sequence has ten operations, you will be grow bored of writing the pattern-match code way before writing the tenth copy. The steady increase in indentation required by the
— monad ops
data Maybe t
for Maybe
= Just t
| Nothing
return x = Just x
(Just x) >>= f = f x Nothing >>= _ = Nothing
— monad ops
data MaybeOK
return x
for MaybeOK
t = OK t
| Error String = OK x
=fx
(OKx)>>=f
(Error m) >>= _ = Error m
In a sequence of calls to >>=, as long as all invocations of f succeed, (returning Just x or OK x), you keep going.
Once you get a failure indication, (returning Nothing or Error m), you keep that failure indication and perform no further operations.
– section 18 slide 5 –
Why you may want these monads
Suppose you want to encode a sequence of operations that each may fail. Here are two such operations:
offside rule also makes the code ugly, since the code of the tenth iteration would have to be squashed up against the right margin. Longer sequences of operations may even require the use of more columns than your screen actually has.
Note that the two occurrences of Error m in the first definition are of different types; the first is type MaybeOK Int, while the second is of type MaybeOK Double. The two occurrences of Error m in the definition of the sequence operation for the MaybeOK monad are similarly of different types, MaybeOK a for the first and MaybeOK b for the second.
– section 18 slide 7 –
I/O actions in Haskell
Haskell has a type constructor called IO. A function that returns a value of type IO t for some t will return a value of type t, but can also do input and/or output. Such functions can be called I/O functions or I/O actions.
Haskell has several functions for reading input, including
getChar :: IO Char
getLine :: IO String
Haskell has several functions for writing output, including
putChar :: Char -> IO ()
putStr :: String -> IO () putStrLn :: String -> IO ()
print :: (Show a) => a -> IO ()
The type (), called unit, is the type of 0-tuples (tuples containing zero values). This is similar to the void type in C or Java. There is only one value of this type, the empty tuple, which is also denoted ().
NOTE The notion that a function that returns a value of type IO t for some t actually does input and/or output is only approximately correct; we will get to the fully correct notion in a few slides.
Since there is only one value of type (), variables of this type carry no information.
Haskell represents a computation that takes a value of type a and computes a value of type b as a function of type a -> b.
Haskell represents a computation that takes a value of type a and computes a value of type b while also possibly performing input and/or output (subject to the caveat above) as a function of type a -> IO b.
All the functions listed on this slide are defined in the prelude.
getChar returns the next character in the input. getLine returns the next line in the input, without
the final ’\n’.
putChar prints the given character.
putStr prints the given string.
putStrLn prints the given string, and appends a newline character.
print uses the show function (which is defined for every type in the Show type class) to turn the given value into a string; it then prints that string and appends a newline character.
– section 18 slide 8 –
Operations of the I/O monad The type constructor IO is a monad.
The identity operation: return val just returns val (inside IO) without doing any I/O.
The sequencing operation: f >>= g
1. calls f, which may do I/O, and which will return
a value rf that may be meaningful or may be (),
2. calls g rf (passing the return value of f to g), which may do I/O, and which will return a value rg that also may be meaningful or may be (),
3. returns rg (inside IO) as the result of f >>= g.
You can use the sequencing operation to create a chain of any number of I/O actions.
– section 18 slide 9 –
Example of monadic I/O: hello world
hello :: IO ()
hello =
putStr “Hello, ”
>>=
\_ -> putStrLn “world!”
This code has two I/O actions connected with >>=.
1. The first is a call to putStr. This prints the first
half of the message, and returns ().
2. The second is an anonymous function. It takes as argument and ignores the result of the first action, and then calls putStrLn to print the second half of the message, adding a newline at the end.
The result of the action sequence is the result of the last action.
NOTE In this case, the resulf of the last action will always be ().
Actually, there is a third monad operation besides return and >>=: the operation >>. This is identical to >>=, with the exception that it ignores the return value of its first operand, and does not pass it to its second operand. This is useful when the second operand would otherwise have to explicitly ignore its argument, as in this case.
With >>, the code of this function could be slightly simpler:
hello :: IO ()
hello =
putStr “Hello, ”
>>
putStrLn “world!”
– section 18 slide 10 –
Example of monadic I/O: greetings
greet :: IO ()
greet =
putStr “Greetings! What is your name? ” >>=
\_ -> getLine
>>=
\name -> (
putStr “Where are you from? ” >>=
\_ -> getLine
>>=
\town ->
)
let msg = “Welcome, ” ++ name ++ ” from ” ++ town
in putStrLn msg
NOTE This example shows how each call to getLine is followed by a lambda expression that captures the value of the string returned getLine: the name for the first call, and the town for the second. In the case of the first call, the return value is not used immediately; we do not want to pass it to the next call putStr. That is why we need to make sure that the scope of the lambda expression that takes name as its argument includes all the following actions, or at least all the following actions that need access to the value of name (which in this case is the same thing).
– section 18 slide 11 –
do blocks
Code written using monad operations is often ugly, and writing it is usually tedious. To address both concerns, Haskell provides do blocks. These are merely syntactic sugar for sequences of monad operations, but they make the code much more readable and easier to write.
A do block starts with the keyword do, like this:
hello = do
putStr “Hello, ”
putStrLn “world”
– section 18 slide 12 –
do block components Each element of a do block can be
• an I/O action that returns an ignored value, usually of type (), such as the calls to putStr and putStrLn below (just call the function);
• an I/O action whose return value is used to bind a variable, (use
var <- expr to bind the variable);
• bind a variable to a non-monadic value (use let var = expr (no in)).
greet :: IO ()
greet = do
putStr
"Greetings! What is your name? "
name <- getLine
putStr "Where are you from? " town <- getLine
let msg = "Welcome, " ++ name ++
" from " ++ town
putStrLn msg
NOTE Each do block element has access to all the variables defined by previous actions but not later ones.
Let clauses in do blocks do not end with the keyword in, since the variable defined by such a let clause is available in all later operations in the block, not just the immediately following opeation.
This notation is so much more convenient to use than raw monad operations that raw monad operations occur in real Haskell programs only rarely.
– section 18 slide 13 –
Operator priority problem
Unfortunately, the following line of code does not work:
The reason is that due to its system of operator priorities, Haskell thinks that the main function being invoked here is not putStrLn but ++, with its left argument being putStrLn "Welcome, ".
This is also the reason why Haskell accepts only the second of the following equations. It parses the left hand side of the first equation as (len x):xs, not as len (x:xs).
len x:xs =
len (x:xs)
1+lenxs
= 1 + len xs
– section 18 slide 14 –
Working around the operator priority problem There are two main ways to fix this problem:
putStrLn ("Welcome, " ++ name ++ " from " ++ town)
putStrLn $ "Welcome, " ++ name ++ " from " ++ town
The first simply uses parentheses to delimit the possible scope of the ++ operator.
The second uses another operator, $, which has lower priority than ++, and thus binds less tightly.
The main function invoked on the line is thus $. Its first argument is its left operand: the function putStrLn, which is of type String -> IO (). Its second argument is its right operand: the expression “Welcome, ” ++ name ++ ” from ” ++ town, which is of type String.
$isoftype(a -> b) -> a -> b. Itappliesitsfirst argument to its second argument, so in this case it invokes putStrLn with the result of the concatenation.
– section 18 slide 15 –
return
If a function does I/O and returns a value, and the code that computes the return value does not do I/O, you will need to invoke the return monad operation as the last operation in the do block.
putStrLn “Welcome, ” ++ name ++ ” from ” ++ town
main :: IO ()
main = do
putStrLn “Please input a string” len <- readlen
putStrLn $
"The length of that string is " ++ show len
readlen :: IO Int
readlen = do
str <- getLine
return (length str)
– section 18 slide 16 –
Section 19: More on monads
I/O actions as descriptions
Haskell programmers usually think of functions that return values of type IO t as doing I/O as well as returning a value of type t. While this is usually correct, there are some situations in which it is not accurate enough.
The correct way to think about such functions is that they return two things:
• a value of type t, and
• a description of an I/O operation.
The monadic operator >>= can then be understood as taking descriptions of two I/O operations, and returning a description of those two operations being executed in order.
The monadic operator return simply associates a description of a do-nothing I/O operation with a value.
– section 19 slide 1 –
Description to execution: theory
Every complete Haskell program must have a function
named main, whose signature should be main :: IO ()
As in C, this is where the program starts execution. Conceptually,
• the OS starts the program by invoking the Haskell runtime system;
• the runtime system calls main, which returns a description of a sequence of I/O operations; and
• the runtime system executes the described sequence of I/O operations.
– section 19 slide 2 –
Description to execution: practice
In actuality, the compiler and the runtime system together ensure that each I/O operation is executed as soon as its description has been computed,
• provided that the description is created in a context which guarantees that the description will end up in the list of operation descriptions returned by main, and
• provided that all the previous operations in that list have also been executed.
The provisions are necessary since
• you don’t want to execute an I/O operation that the program does not actually call for, and
• you don’t want to execute I/O operations out of order.
– section 19 slide 3 –
Example: printing a table of squares directly
main ::IO ()
main = do
putStrLn “Table of squares:” print_table 1 10
print_table :: Int -> Int -> IO () print_table cur max
| cur > max = return () | otherwise = do
putStrLn (table_entry cur) print_table (cur+1) max
table_entry :: Int -> String table_entry n = (show n) ++ “^2 = ”
++ (show (n*n))
NOTE The definition of print table uses guards. If cur > max, then the applicable right hand side is the one that follows that guard expression; if cur <= max, then the applicable right hand side is the one that follows the keyword otherwise.
– section 19 slide 4 –
Non-immediate execution of I/O actions
Just because you have created a description of an I/O action, does not mean that this I/O action will eventually be executed.
Haskell programs can pass around descriptions of I/O operations. They cannot peer into a description of an I/O operation, but they can nevertheless do things with them, such as
• build up lists of I/O actions, and
• put I/O actions into binary search trees as values.
Those lists and trees can then be processed further, and programmers can, if they wish, take the descriptions of I/O actions out of those data structures, and have them executed by including them in the list of actions returned by main.
– section 19 slide 5 –
Example: printing a table of squares indirectly
main = do
putStrLn "Table of squares:" let row_actions =
map show_entry [1..15] execute_actions (take 10 row_actions)
table_entry :: Int -> String table_entry n = (show n) ++ “^2 = ”
++ (show (n*n))
show_entry :: Int -> IO ()
show_entry n = do putStrLn (table_entry n)
execute_actions :: [IO ()] -> IO ()
execute_actions [] = return () execute_actions (x:xs) = do
x
execute_actions xs
NOTE The take function from the Haskell prelude returns the first few elements of a list; the number of elements it should return is specified by the value of its first argument.
– section 19 slide 6 –
Input, process, output
A typical batch program reads in its input, does the required processing, and prints its output.
A typical interactive program goes through the same three stages once for each interaction.
In most programs, the vast majority of the code is in the middle (processing) stage.
In programs written in imperative languages like C, Java, and Python, the type of a function (or procedure, subroutine or method) does not tell you whether the function does I/O.
In Haskell, it does.
– section 19 slide 7 –
I/O in Haskell programs
In most Haskell programs, the vast majority of the functions are not I/O functions and they do no input or output. They merely build, access and transform data structures, and do calculations. The code that does I/O is a thin veneer on top of this bulk.
This approach has several advantages.
• A unit test for a non-IO function is a record of the values of the arguments and the expected value of the result. The test driver can read in those values, invoke the function, and check whether the result matches. The test driver can be a human.
• Code that does no I/O can be rearranged. Several optimizations exploit this fact.
• Calls to functions that do no I/O can be done in parallel. Selecting the best calls to parallelize is an active research area.
– section 19 slide 8 –
Debugging printfs
One standard approach for debugging a program written in C is to edit your code to insert debugging printfs to show you what input your buggy function is called with and what results it computes.
In a program written in Haskell, you can’t just insert printing code into functions not already in the IO monad.
Debugging printfs are only used for debugging, so you’re not concerned with where the output from debugging printfs appears relative to other output. This is where the function unsafePerformIO comes in: it allows you to perform IO anywhere, but the order of output will probably be wrong.
Do not use unsafePerformIO in real code, but it is useful for debugging.
– section 19 slide 9 –
unsafePerformIO
The type of unsafePerformIO is IO t -> t.
• You give it as argument an I/O operation, which means a function of type IO t. unsafePerformIO calls this function.
• The function will return a value of type t and a description of an I/O operation.
• unsafePerformIO executes the described I/O operation and returns the value.
Here is an example:
sum :: Int -> Int -> Int
sum x y = unsafePerformIO $ do
putStrLn (“summing ” ++ (showx)++”and”++(showy))
return (x + y)
NOTE As its name indicates, the use of unsafePerformIO is unsafe in general. This is because you the program does not indicate how the I/O operations performed by an invocation of unsafePerformIO fit into the sequence of operations executed by main. This is why unsafePerformIO is not defined in the prelude. This means that your code will not be able to call it unless you import the module that defines it, which is GHC.IOBase.
– section 19 slide 10 – The State monad
The State monad is useful for computations that need to thread information throughout the computation. It allows such information to be transparently passed around a computation, and accessed and replaced when needed. That is, it allows an imperative style of programming without losing Haskell’s declarative semantics.
This code adds 1 to each element of a tree, and does not need a monad:
data Tree a
= Empty
| Node (Tree a) a (Tree a)
deriving Show
type IntTree = Tree Int
incTree :: IntTree -> IntTree incTree Empty = Empty
incTree (Node l e r) =
Node (incTree l) (e + 1) (incTree r)
– section 19 slide 11 –
Threading state
If we instead wanted to add 1 to the leftmost element, 2 to the next element, and so on, we would need to
pass an integer into our function saying what to add, but also we need to pass an integer out, saying what to add to the next element. This requires more complex code:
incTree1 :: IntTree -> IntTree incTree1 tree = fst (incTree1’
incTree1’ :: IntTree -> Int -> incTree1’ Empty n = (Empty, n) incTree1’ (Node l e r) n =
tree 1)
(IntTree, Int)
ln
let (newl, n1) = incTree1’ (newr, n2) = incTree1’
r (n1 + 1) in (Node newl (e+n1) newr, n2)
NOTE Given a tuple of two elements, the fst builtin function returns its first element, and the snd builtin function returns its second element.
– section 19 slide 12 –
Introducing the State monad
The State monad abstracts the type s -> (v,s), hiding away the s part. Haskell’s do notation allows us to focus on the v part of the computation while ignoring the s part where not relevant.
incTree2 :: IntTree -> IntTree incTree2 tree =
fst (runState (incTree2’ tree) 1)
incTree2’ :: IntTree -> State Int IntTree incTree2’ Empty = return Empty
incTree2’ (Node l e r) = do
newl <- incTree2’ l
n <- get -- gets the current state put (n + 1) -- sets the current state newr <- incTree2’ r
return (Node newl (e+n) newr)
– section 19 slide 13 –
Abstracting the state operations
In this case, we do not need the full generality of being able to update the integer state in arbitrary ways; the
only update operation we need is an increment. We can therefore provide a version of the state monad that is specialized for this task. Such specialization provides useful documentation, and makes the code more robust.
type Counter = State Int
withCounter :: Int -> Counter a -> a withCounter init f = fst (runState f init)
nextCount :: Counter Int nextCount = do
n <- get
put (n + 1)
return n
– section 19 slide 14 –
Using the counter
Now the code that uses the monad is even simpler:
incTree3 :: IntTree -> IntTree
incTree3 tree = withCounter 1 (incTree3’ tree)
incTree3’ :: IntTree -> Counter IntTree incTree3’ Empty = return Empty incTree3’ (Node l e r) = do
newl <- incTree3’ l
n <- nextCount
newr <- incTree3’ r
return (Node newl (e+n) newr)
– section 19 slide 15 –
Section 20: Lazyness
Eager vs lazy evaluation
In a programming language that uses eager evaluation, each expression is evaluated as soon as it gets bound to a variable, either explicitly in an assignment statement, or implicitly during a call. (A call implicitly assigns each actual parameter expression to the corresponding formal parameter variable.)
In a programming language that uses lazy evaluation, an expression is not evaluated until its value is actually needed. Typically, this will be when
• the program wants the value as input to an arithmetic operation, or
• the program wants to match the value against a pattern, or
• the program wants to output the value. Almost all programming languages use eager
evaluation. Haskell uses lazy evaluation. – section 20 slide 1 –
Lazyness and infinite data structures
Lazyness allows a program to work with data structures that are conceptually infinite, as long as the program looks at only a finite part of the infinite data structure.
For example, [1..] is a list of all the positive numbers. If you attempt to print it out, the printout will be infinite, and will take infinite time, unless you interrupt it.
On the other hand, if you want to print only the first n positive numbers, you can do that with take n [1..].
Even though the second argument of the call to take is infinite in size, the call takes finite time to execute.
NOTE The expression [1..] has the same value as all ints from 1, given the definition
To find all primes up to n:
Lazyness allows the programmer of all primes to concentrate on the function’s task, without having to also pay attention to exactly how the program wants to decide how many primes are enough.
Haskell automatically interleaves the computation of the primes with the code that determines how many primes to compute.
– section 20 slide 4 –
Representing unevaluated expressions
In a lazy programming language, expressions are not evaluated until you need their value. However, until then, you do need to remember the code whose execution will compute that value.
In Haskell implementations that compile Haskell to C (this includes GHC), the data structure you need for that is a pointer to a C function, together with all the arguments you will need to give to that C function.
This representation is sometimes called a suspension, since it represents a computation whose evaluation is temporarily suspended.
It can also be called a promise, since it also represents a promise to carry out a computation if its result is needed.
Historically inclined people can also call it a thunk, because that was the name of this construct in the first programming language implementation that used it. That language was Algol-60.
NOTE The word “thunk” can actually refer to several programming language implementation constructs, of which this is only one. Think of “thunk” as the compiler writer’s equivalent of the mechanical engineer’s word “gadget”: they can both be used to refer to anything small and clever.
– section 20 slide 5 –
takeWhile (<= n) all_primes
all_ints_from
all_ints_from
:: Integer -> [Integer]
n = n:(all_ints_from (n+1))
– section 20 slide 2 –
The sieve of Eratosthenes
— returns the (infinite) list of all primes all_primes :: [Integer]
all_primes = prime_filter [2..]
prime_filter :: [Integer] -> [Integer] prime_filter [] = []
prime_filter (x:xs) =
x:prime_filter
(filter (not . (‘divisibleBy‘ x)) xs)
— n ‘divisibleBy‘ d means n is divisible by d divisibleBy n d = n ‘mod‘ d == 0
NOTE The input to prime filter is a list of integers which share the property that they are not evenly divisible by any prime that is smaller than the first element of the list.
The invariant is trivially true for the list [2..], since there are no primes smaller than 2.
Since the smallest element of such a list cannot be evenly divisible by any number smaller than itself, it must be a prime. Therefore filtering multiples of the first element out of the input sequence before giving the filtered sequence as input to the recursive call to prime filter maintains the invariant.
– section 20 slide 3 –
Using all primes To find the first n primes:
take n all_primes
Parametric polymorphism
Parametric polymorphism is the name for the form of polymorphism in which types like [a] and Tree k v, and functions like length and insert bst, include type variables, and the types and functions work identically regardless of what types the type variables stand for.
The implementation of parametric polymorphism requires that the values of all types be representable in the same amount of memory. Without this, the code of e.g. length wouldn’t be able to handle lists with elements of all types.
That “same amount of memory” will typically be the word size of the machine, which is the size of a pointer. Anything that does not fit into one word is represented by a pointer to a chunk of memory on the heap.
Given this fact, the arguments of the function in a suspension can be stored in an array of words, and we can arrange for all functions in suspensions to take their arguments from a single array of words.
NOTE Parametric polymorphism is the form of polymorphism on which the type systems of languages like Haskell are based. Object-oriented languages like Java are based on a different form of polymorphism, which is usually called “inclusion polymorphism” or “subtype polymorphism”.
The “same amount of memory” obviously does not include the memory needed by the pointed-to heap cells, or the cells they point to directly or indirectly. A value such as [1, 2, 3, 4, 5] will require several heap cells; in this case, these will be five cons cells and (depending on the details of the implementation) maybe one cell for the nil at the end. (The nonempty list constructor : is usually pronounced “cons”, while the empty list constructor [] is usually pronounced “nil”.) However, the whole value can be represented by a pointer to the first cons cell.
– section 20 slide 6 –
Evaluating lazy values only once
Many functions use the values of some variables more
than once. This includes takeWhile, which uses x twice:
takeWhile _ [] = [] takeWhile p (x:xs)
| p x = x : takeWhile p xs | otherwise = []
Youneedtoknowthevalueofxtodothetestp x, which requires calling the function in the suspension representing x; if the test succeeds, you will again need to know the value of x to put it at the front of the output list.
To avoid redundant work, you want the first call to x’s suspension to record the result of the call, and you want all references to x after the first to get its value from this record.
Therefore once you know the result of the call, you don’t need the function and its arguments anymore.
– section 20 slide 7 –
Call by need
Operations such as printing, arithmetic and pattern matching start by ensuring their argument is at least partially evaluated.
They will make sure that at least the top level data constructor of the value is determined. However, the arguments of that data constructor may remain suspensions.
For example, consider the match of the second argument of takeWhile against the patterns [] and (p:ps). If the original second argument is a suspension, it must be evaluated enough to ensure its top-level constructor is determined. If it is x:xs, then the first argument must be applied to x. Whether x needs to be evaluated will depend on what the first argument (function) does.
This is called “call by need”, because function arguments (and other expressions) are evaluated only when their value is needed.
– section 20 slide 8 –
Control structures and functions
(a) … if (x < y) f(x); else g(y); ... (b) ... ite(x < y, f(x), g(y)); ...
int ite(bool c, int t, int e)
{ if (c) then return t; else return e; }
In C, (a) will generate a call to only one of f and g, but (b) will generate a call to both.
(c) ... if x < y then f x else g y ... (d) ... ite (x < y) (f x) (g y) ...
ite :: Bool -> a -> a -> a
ite c t e = if c then t else e
In Haskell, (c) will execute a call to only one of f and g, and thanks to lazyness, this is also true for (d).
NOTE The Haskell implementation of if-then-else calls evaluate suspension on the suspension representing the condition. If the condition’s value is True, it will then call evaluate suspension on the suspension representing then part, and return its value; if the condition’s value is False, it will call
evaluate suspension on the suspension representing the else part, and return its value. In each case, the suspension for the other part will remain unevaluated. Roughly speaking, both (c) and (d) are implemented this way.
– section 20 slide 9 –
Implementing control structures as functions
Without lazyness, using a function instead of explicit code such as a sequence of if-then-elses could get unnecessary non-termination or at least unnecessary slowdowns.
Lazyness’ guarantee that an expression will not be evaluated if its value is not needed allows programmers to define their own control structures as functions.
For example, you can define a control structure that returns the value of one of three expressions, with the expression chosen based on whether an expression is less than, equal to or greater than zero like this:
ite3 :: (Ord a, Num a) => a -> b -> b -> b -> b ite3 x lt eq gt
|x<0 =lt | x == 0 = eq |x>0 =gt
NOTE This ability to define new control structures can come in handy if you find yourself repeatedly writing code with the same nontrivial decision-making structure. However, such situations are pretty rare.
The kind of three-way branch shown by this example was the main way to implement choice in the first widely-used high level programming language, the original dialect of Fortran. That version of the if statement specified three labels; which one control jumped to next depended on the value of the control expression.
– section 20 slide 10 –
Using lazyness to avoid unnecessary work
min = head . sort
On the surface, this looks like a very wasteful method for computing the minimum, since sorting is usually done with an O(n2) or O(n logn) algorithm, and min should be doable with an O(n) algorithm.
However, in this case, the evaluation of the sorted list can stop after the materialization of the first element.
If sort is implemented using selection sort, this is just a somewhat higher overhead version of the direct code for min.
– section 20 slide 11 – Multiple passes
output_prog chars = do let anno_chars =
annotate_chars 1 1 chars let tokens = scan anno_chars let prog = parse tokens
let prog_str = show prog putStrLn prog_str
This function takes as input one data structure (chars) and calls for the construction of four more (anno chars, tokens, prog and prog str).
This kind of pass structure occurs frequently in real programs.
– section 20 slide 12 –
The effect of lazyness on multiple passes
With eager evaluation, you would completely construct each data structure before starting construction of the next.
The maximum memory needed at any one time will be the size of the largest data structure (say pass n), plus the size of any part of the previous data structure (pass n − 1) needed to compute the last part of pass n. All other memory can be garbage collected before then.
With lazy evaluation, execution is driven by putStrLn, which needs to know what the next character to print (if any) should be. For each character to be printed, the program will materialize the parts of those data structures needed to figure that out.
The memory demand at a given time will be given by the tree of suspensions from earlier passes that you need to materialize the rest of the string to be printed. The maximum memory demand can be significantly less than with eager evaluation.
NOTE The memory requirements analysis above assumes that the code that constructs pass n’s data structure uses only data from pass n − 1, and does not need access to data from pass n−2, n−3 etc.
If the last pass builds a data structure instead of printing out the data structure built by the second-last pass, then you will need to add the memory needed by the part of the last data structure constructed so far to the memory needed at any point in time. This will increase the maximum memory demand with lazy evaluation, but it can still be less than with eager evaluation.
– section 20 slide 13 –
Lazy input
In Haskell, even input is implemented lazily.
Given a filename, readFile returns the contents of the file as a string, but it returns the string lazily: it reads the next character from the file only when the rest of the program needs that character.
parse_prog_file filename = do fs <- readFile filename let tokens =
scan (annotate_chars 1 1 chars) return (parse_prog [] tokens)
When the main module calls parse prog file, it gets back a tree of suspensions.
Only when those suspensions start being forced will the input file be read, and each call to
evaluate suspension on that tree will cause only as much to be read as is needed to figure out the value of the forced data constructor.
NOTE The readFile function is defined in the prelude.
– section 20 slide 14 –
Section 21: Performance
Effect of lazyness on performance
Lazyness adds two sorts of overhead that slow down programs.
• The execution of a Haskell program creates a lot of suspensions, and most of them are evaluated, so eventually they also need to be unpacked.
• Every access to a value must first check whether the value has been materialized yet.
However, lazyness can also speed up programs by avoiding the execution of computations that take a long time, or do not terminate at all.
Whether the dominant effect is the slowdown or the speedup will depend on the program and what kind of input it typically gets.
The usual effect is something like lotto: in most cases you lose a bit, but sometimes you win a little, and in some rare cases you win a lot.
– section 21 slide 1 –
Strictness
Theory calls the value of an expression whose evaluation loops infinitely or throws an exception “bottom”, denoted by the symbol ⊥.
A function is strict if it always needs the values of all its arguments. In formal terms, this means that if any of its arguments is ⊥, then its result will also be ⊥.
The addition function + is strict. The function ite from earlier in the last lecture is nonstrict.
Some Haskell compilers including GHC include strictness analysis, which is a compiler pass whose job is to analyze the code of the program and figure out which of its functions are strict and which are nonstrict.
When the Haskell code generator sees a call to a strict function, instead of generating code that creates a suspension, it can generate the code that an imperative language compiler would generate: code that evaluates all the arguments, and then calls the function.
NOTE Eager evaluation is also called strict evaluation, while lazy evaluation is also called nonstrict evaluation.
– section 21 slide 2 –
Unpredictability
Besides generating a slowdown for most programs, lazyness also makes it harder for the programmer to understand where the program is spending most of its time and what parts of the program allocate most of its memory.
This is because small changes in exactly where and when the program demands a particular value can cause great changes in what parts of a suspension tree are evaluated, and can therefore cause great changes in the time and space complexity of the program. (Lazy evaluation is also called demand driven computation.)
The main problem is that it is very hard for programmers to be simultaneous aware of all the relevant details in the program.
Modern Haskell implementations come with sophisticated profilers to help programmers understand the behavior of their programs. There are profilers for both time and for memory consumption.
– section 21 slide 3 –
Memory efficiency (Revised) BST insertion code:
insert_bst :: Ord k => Tree k v -> k -> v -> Tree k v
insert_bst Leaf ik iv = Node ik iv Leaf Leaf insert_bst (Node k v l r) ik iv
| k == ik = Node ik iv l r
|k >ik =Nodekv(insert_bstlikiv)r | otherwise = Node k v l (insert_bst r ik iv)
As discussed earlier, this creates new data structures instead of destructively modifying the old structure.
The advantage of this is that the old structure can still be used.
The disadvantage is new memory is allocated and written. This takes time, and creates garbage that must be collected.
– section 21 slide 4 –
Memory efficiency
Insertion into a BST replaces one node on each level of the tree: the node on the path from the root to the insertion site.
In (mostly) balanced trees with n nodes, the height of the tree tends to be about log2(n).
Therefore the number of nodes allocated during an insertion tends to be logarithmic in the size of the tree.
• If the old version of the tree is not needed, imperative code can do better: it must allocate only the new node.
• If the old version of the tree is needed, imperative code will do worse: it must copy the entire tree, since without that, later updates to the new version would update the old one as well.
– section 21 slide 5 – Reusing memory
When insert bst inserts a new node into the tree, it allocates new versions of every node on the path from
the root to the insertion point. However, every other node in the tree will become part of the new tree as well as the old one.
This shows what happens when you insert the key ”h” into a binary search tree that already contains ”a” to ”g”.
”d”4•• ”d”4• •
”b”2• •
”f”6• •
”f”6• •
”g”7/ •
”h” 8 / /
”a”1/ /
”c”3/ /
”e”5/ /
”g”7/ /
NOTE In this small example, the old and the new tree share the entire subtree rooted at the node whose key is “b’, and the node whose key is “e’.
– section 21 slide 6 –
Deforestation
As we discussed earlier, many Haskell programs have code that follows this pattern:
• You start with the first data structure, ds1.
• You traverse ds1, generating another data
structure, ds2.
• You traverse ds2, generating yet another data structure, ds3.
If the programmer can restructure the code to compute ds3 directly from ds1, this should speed up the program, for two reasons:
• the new version does not need to create ds2, and • the new version does one traversal instead of two.
Since the eliminated intermediate data structures are often trees of one kind or another, this optimization idea is usually called deforestation.
– section 21 slide 7 –
filter map
filter_map :: (a -> Bool) -> (a -> b) -> [a] -> [b]
filter_map _ _ [] = [] filter_map f m (x:xs) =
let newxs = filter_map f m xs in if f x then (m x):newxs else newxs
one_pass xs = filter_map is_even triple xs two_pass xs = map triple (filter is_even xs)
The one pass function performs exactly the same task as the two pass function, but it does the job with one list traversal, not two, and does not create an intermediate list.
One can also write similarly deforested combinations of many other pairs of higher order functions, such as map and foldl.
– section 21 slide 8 –
Computing standard deviations
four_pass_stddev :: [Double] -> Double four_pass_stddev xs =
let
count = fromIntegral (length xs) sum = foldl (+) 0 xs
sumsq = foldl (+) 0
(map square xs)
in
Computing standard deviations in one pass
data StddevData =
SD Double Double Double
one_pass_stddev :: [Double] -> Double one_pass_stddev xs =
let
init_sd = SD 0.0 0.0 0.0 update_sd (SD c s sq) x =
SD (c + 1.0) (s + x) (sq + x*x) SD count sum sumsq =
foldl update_sd init_sd xs in
(sqrt (count * sumsq – sum * sum)) / count
NOTE This is an example of a call to foldl in which the base is of one type (StddevData) while the list elements are of another type (Double).
It is also an example of a let clause that defines an auxiliary function, in this case update sd, and one in which the last part picks up the values of the arguments of the SD data constructor in three variables.
– section 21 slide 10 –
Cords
Repeated appends to the end of a list take time that is quadratic in the final length of the list.
In imperative languages, you would avoid this quadratic behavior by keeping a pointer to the tail of the list, and destructively updating that tail.
In declarative languages, the usual solution is to switch from lists to a data structure that supports appends in constant time. These are usually called cords. This is one possible cord design; there are several.
(sqrt (count
count
square :: Double
square x = x * x
* sumsq – sum * sum)) /
-> Double
This is the simplest approach to writing code that computes the standard deviation of a list. However, it traverses the input list three times, and it also traverses a list of that same length (the list of squares) once.
– section 21 slide 9 –
data Cord a
= Nil
| Leaf a
| Branch
append_cords
append_cords
(Cord a) (Cord a)
:: Cord a -> Cord a -> Cord a a b = Branch a b
– section 21 slide 11 –
Converting cords to lists
The obvious algorithm to convert a cord to a list is
cord_to_list :: Cord a -> [a] cord_to_list Nil = [] cord_to_list (Leaf x) = [x] cord_to_list (Branch a b) =
(cord_to_list a) ++ (cord_to_list b)
Unfortunately, it suffers from the exact same performance problem that cords were designed to avoid.
The cord Branch (Leaf 1) (Leaf 2) that the last equation converts to a list may itself be one branch of a bigger cord, such as Branch (Branch (Leaf 1) (Leaf 2)) (Leaf 3).
The list [1], converted from Leaf 1, will be copied twice by ++, once for each Branch data constructor in whose first operand it appears.
– section 21 slide 12 –
Accumulators
With one exception, all leaves in a cord are followed by another item, but the second equation puts an empty list behind all leaves, which is why all but one of the lists it creates will have to be copied again. The other two equations make the same mistake for empty and branch cords.
Fixing the performance problem requires telling the conversion function what list of items follows the cord currently being converted. This is easy to arrange using an accumulator.
cord_to_list :: Cord a -> [a] cord_to_list c = cord_to_list’ c []
cord_to_list’ :: Cord a -> [a] -> [a] cord_to_list’ Nil rest = rest cord_to_list’ (Leaf x) rest = x:rest cord_to_list’ (Branch a b) rest =
cord_to_list’ a (cord_to_list’ b rest)
– section 21 slide 13 –
Sortedness check
The obvious way to write code that checks whether a list is sorted:
sorted1 :: (Ord a) sorted1 []
sorted1 [_] sorted1 (x1:x2:xs)
=> [a] -> Bool
= True
= True
= x1 <= x2 && sorted1 (x2:xs)
However, the code that looks at each list element handles three alternatives (lists of length zero, one and more).
It does this because each sortedness comparison needs two list elements, not one.
– section 21 slide 14 –
A better sortedness check
sorted2 :: (Ord a) => [a] -> Bool sorted2 [] = True
sorted2 (x:xs) = sorted_lag x xs
sorted_lag :: (Ord a) sorted_lag _ [] sorted_lag x1 (x2:xs)
=> a -> [a] -> Bool = True
= x1 <= x2
&& sorted_lag x2 xs
In this version, the code that looks at each list element handles only two alternatives. The value of the previous element, the element that the current element should be compared with, is supplied separately.
– section 21 slide 15 –
Optimisation
You can use :set +s in GHCi to time execution.
Prelude> :l sorted
[1 of 1] Compiling Sorted
( sorted.hs, interpreted ) Ok, modules loaded: Sorted.
*Sorted> :set +s
*Sorted> sorted1 [1..100000000] True
(50.11 secs, 32,811,594,352 bytes) *Sorted> sorted2 [1..100000000] True
(40.76 secs, 25,602,349,392 bytes)
The sorted2 version is about 20% faster and uses 22% less memory.
– section 21 slide 16 –
Optimisation
However, the Haskell compiler is very sophisticated. After doing
ghc -dynamic -c -O3 sorted.hs, we get this:
Section 22: Interfacing with foreign languages
Foreign language interface
Many applications involve code written in a number of different languages; declarative languages are no different in this respect. There are many reasons for this:
• to interface to existing code (especially libraries) written in another language;
• to write performance-critical code in a lower-level language (typically C or C++);
• to write each part of an application in the most appropriate language;
• as a way to gracefully translate an application from one language to another, by replacing one piece at a time.
Any language that hopes to be successful must be able to work with other languages. This is generally done through what is called a foreign language interface or foreign function interface.
– section 22 slide 1 – Application binary interfaces
In computer science, a platform is a combination of an instruction set architecture (ISA) and an operating system, such as x86/Windows 10, x86/Linux or SPARC/Solaris.
Each platform typically has an application binary interface, or ABI, which dictates such things as where the callers of functions should put the function parameters and where the callee function should put the result.
Prelude> :l sorted
Ok, modules loaded: Sorted.
Prelude Sorted> :set +s
Prelude Sorted> sorted1 [1..100000000] True
(2.89 secs, 8,015,369,944 bytes) Prelude Sorted> sorted2 [1..100000000] True
(2.91 secs, 8,002,262,840 bytes)
Compilation gives a factor of 17 speedup and a factor of 3 memory savings. It also removes the difference between sorted1 and sorted2. Always benchmark your compiled code when trying to speed it up.
– section 21 slide 17 –
By compiling different files to the same ABI, functions in one file can call functions in a separately compiled file, even if compiled with different compilers.
The traditional way to interface two languages, such as C and Fortran, or Ada and Java, is for the compilers of both languages to generate code that follows the ABI.
– section 22 slide 2 –
Beyond C
ABIs are typically designed around C’s simple calling pattern, where each function is compiled to machine language, and each function call passes some number of inputs, calls another known function, possibly returns one result, and is then finished.
This model does not work for lazy languages like Haskell, languages like Prolog or Mercury that support nondeterminism, languages like Prolog, Python, and Java that are implemented through an abstract machine, or even languages like C++ where function (method) calls may invoke different code each time they are executed.
In such languages, code is not compiled to the normal ABI. Then it becomes necessary to provide a mechanism to call code written in other languages. Typically, calling C code through the normal ABI is supported, but interfacing to other languages may also be supported.
– section 22 slide 3 –
Boolean functions
One application of a foreign interface is to use specialised data structures and algorithms that would be difficult or inefficient to implement in the host language.
Some applicatations need to be able to efficiently manipulate Boolean formulas (Boolean functions). This includes the following primitive values and operations:
• true, false
• Boolean variables: eg: a, b, c, . . .
• Operations: and (∧), or (∨), not (¬), implies
(→), iff (↔), etc.
• Tests: satisfiability (is there any binding for the variables that makes a formula true?), equivalence (are two formulas the same for every set of variable bindings?)
For example, is a ↔ b equivalent to ¬((a∧¬b)∨(¬a∧b))?
– section 22 slide 4 –
Binary Decision Diagrams
Deciding satisfiability or equivalence of Boolean functions is NP-complete, so we need an efficient implementation.
BDDs are decision graphs, based on if-then-else (ite) nodes, where each node is labeled by a boolean variable and each leaf is a truth value.
With a truth assignment for each variable, the value of the formula can be determined by traversing from the root, following then branch for true variables and else branch for false variables.
a
true
b
– section 22 slide 5 –
BDDs in Haskell
true
false
a
b
the BDD
T
T
T
T
F
T
F
T
T
F
F
F
We could represent BDDs in Haskell with this type:
data BDD label
= BTrue | BFalse
| Ite label (BDD label) (BDD label)
The meaning of a BDD is given by: meaning BTrue = true
meaning BFalse = false
meaning (Ite v t e) = (v ∧meaning t) ∨
(¬v ∧ meaning e)
So for example,
meaning (Ite a BTrue (Ite b BTrue BFalse))
= (a ∧true)∨(¬a∧(b∧true ∨(¬b∧false))) = a ∨(¬a ∧ b ∨ false)
= a ∨b
– section 22 slide 6 –
ROBDDs
Reduced Ordered Binary Decision Diagrams (ROBDDs) are BDDs where labels are in increasing order from root to leaf, no node has two identical children, and no two distinct nodes have the same semantics.
a
b c
By sharing the c node, the ROBDD is smaller than it would be if it were a tree. For larger ROBDDs, this can be a big savings.
– section 22 slide 7 –
Object Identity
ROBDD algorithms traverse DAGs, and often meet the same subgraphs repeatedly. They greatly benefit from caching: recording results of past operations to reuse without repeating the computation.
Caching requires efficiently recognizing when a node is seen again. Haskell does not have the concept of object identity, so it cannot distinguish
a
c
a
cb c
– section 22 slide 8 –
Structural Hashing
b
false
false
true
from
false
true
false
false
true
false
false
true
a
b
c
the BDD
T
T
T
F
T
T
F
T
T
F
T
F
T
F
F
T
Building a new ROBDD node ite(v, thn, els) must ensure that:
1. the two children of every node are different; and
2. for any Boolean formula, there is only one ROBDD node with that semantics.
F
T
T
F
F
T
F
T
F
F
T
F
F
F
F
F
The first is achieved by checking if thn = els, and if so, returning thn
The second is achieved by structural hashing (AKA hash-consing): maintaining a hash table of past calls ite(v, thn, els) and their results, and always returning the past result when a call is repeated.
Because of point 1 above, satisfiability of an ROBDD can be tested in constant time (meaning r is satisfiable iff r ̸= false)
Because of point 2 above, equality of two ROBDDs is also a constant time test (meaning r1 = meaning r2 iff r1 = r2)
– section 22 slide 9 –
Impure implementation of pure operations
(We haven’t cheated NP-completeness: we have only shifted the cost to building ROBDDs.)
Yet ROBDD operations are purely declarative. Constructing ROBDDs, conjunction, disjunction, negation, implication, checking satisfiability and equality, etc., are all pure.
This is an example of using impurity to build purely declarative code.
In fact, all declarative code running on a commodity computer does that: these CPUs work through impurity. Even adding two numbers on a modern CPU works by destructively adding one register to another.
If you are required to work in an imperative or object-oriented language, you can still use such languages to build declarative abstractions, and work with them instead of working directly with impure constructs.
– section 22 slide 10 –
robdd.h: C interface for ROBDD implementation
extern robdd *true_rep(void);
/* ROBDD true */
extern robdd *false_rep(void);
/* ROBDD false */
extern robdd *variable_rep(int var);
/* ROBDD extern robdd *conjoin(robdd *a, extern robdd *disjoin(robdd *a, extern robdd *negate(robdd *a); extern robdd *implies(robdd *a,
extern int is_true(robdd *f); /* ROBDD
extern int is_false(robdd *f); /* ROBDD
for var */
robdd *b);
robdd *b);
robdd *b);
== true? */
== false? */
extern int robdd_label(robdd *f);
/* label of f */
extern robdd *robdd_then(robdd *f);
/* then branch of f */
extern robdd *robdd_else(robdd *f);
/* else branch of f */
– section 22 slide 11 –
Interfacing Haskell to C
For simple cases, the Haskell foreign function interface is fairly simple. You can interface to a C function with a declaration of the form:
foreign import ccall “C name” Haskell name :: Haskell type
But how shall we represent an ROBDD in Haskell?
C primitive types convert to and from natural Haskell types, eg,
C int ←→ Haskell Int
In Haskell, we want to treat ROBDDs as an opaque type: a type we cannot peer inside, we can only pass it to, and receive it as output from, foreign functions.
The Haskell Word type represents a word of memory, much like an Int. However Word is not opaque, as we can confuse it with an integer, or any Word type.
– section 22 slide 12 –
newtype
Declaring type BoolFn = Word would not make BoolFn opaque; it would just be an alias for Word, and could be passed as a Word.
We can make it opaque with a data BoolFn = BoolFn Word declaration. We could convert a Word w to a BoolFn with BoolFn w, and convert a BoolFn b to a Word with
let BoolFn w = b in …
But this would box the word, adding an extra
indirection to operations. Instead we declare:
newtype BoolFn = BoolFn Word deriving Eq
We can only use newtype to declare types with only one constructor, with exactly one argument. This avoids the indirection, makes the type opaque, and allows it to be used in the foreign interface.
– section 22 slide 13 – The interface
foreign import ccall foreign import ccall
:: BoolBinOp
“negate” negation
:: BoolFn->BoolFn
“implies” implies
:: BoolBinOp
– section 22 slide 14 –
Using it
To make C code available, compile it and pass the
object file on the ghc or ghci command line.
(There is also code to show BoolFns in disjunctive normal form; all code is in BoolFn.hs, robdd.c and robdd.h in the examples directory.)
nomad% gcc -c -Wall robdd.c nomad% ghci robdd.o
GHCi, version 8.4.3: http://www.haskell.org/ghc/ Prelude> :l BoolFn.hs
[1 of 1] Compiling BoolFn
( BoolFn.hs, interpreted )
Ok, one module loaded.
*BoolFn> (variable 1) ‘disjoin‘ (variable 2) ((1) | (~1 & 2))
*BoolFn> it ‘conjoin‘ (negation $ variable 3) ((1 & ~3) | (~1 & 2 & ~3))
foreign import ccall
foreign import ccall
foreign import ccall
foreign import ccall
foreign import ccall
foreign import ccall
foreign import ccall
foreign import ccall
“true_rep” true
:: BoolFn
“false_rep” false
:: BoolFn “variable_rep” variable :: Int->BoolFn
“is_true” isTrue
:: BoolFn->Bool “is_false” isFalse :: BoolFn->Bool “robdd_label” minVar :: BoolFn->Int “robdd_then” minThen :: BoolFn->BoolFn “robdd_else” minElse :: BoolFn->BoolFn
– section 22 slide 15 –
Interfacing to Prolog
The Prolog standard does not standardise a foreign language interface. Each Prolog system has its own approach.
The SWI Prolog approach does most of the work on the C side, rather than in Prolog. This is powerful, but inconvenient.
In an SWI Prolog source file, the declaration
:- use_foreign_library(swi_robdd).
type BoolBinOp = BoolFn -> BoolFn -> BoolFn foreign import ccall “conjoin” conjoin
:: BoolBinOp foreign import ccall “disjoin” disjoin
will load a compiled C library file that links the code in swi_robdd.c, which forms the interface to Prolog, and the robdd.c file.
These are compiled and linked with the shell command:
swipl-ld -shared -o swi_robdd swi_robdd.c robdd.c
– section 22 slide 16 –
Connecting C code to Prolog
The swi robdd.c file contains C code to interface to Prolog:
install_t install_swi_robdd() { PL_register_foreign(“boolfn_node”, 4,
pl_bdd_node, 0); PL_register_foreign(“boolfn_true”, 1,
pl_bdd_true, 0); PL_register_foreign(“boolfn_false”, 1,
pl_bdd_false, 0); PL_register_foreign(“boolfn_conjoin”, 3,
pl_bdd_and, 0); PL_register_foreign(“boolfn_disjoin”, 3,
pl_bdd_or, 0); PL_register_foreign(“boolfn_negation”, 2,
pl_bdd_negate, 0); PL_register_foreign(“boolfn_implies”, 3,
pl_bdd_implies, 0);
}
This tells Prolog that a call to boolfn node/4 is implemented as a call to the C function pl bdd node, etc.
– section 22 slide 17 –
Marshalling data
A C function that implements a Prolog predicate needs to convert between Prolog terms and C data structures. This is called marshalling data.
static foreign_t pl_bdd_and(term_t f, term_t g,
term_t result_term) { void *f_nd, *g_nd;
if (PL_is_integer(f)
&& PL_is_integer(g)
&& PL_get_pointer(f, &f_nd)
&& PL_get_pointer(g, &g_nd)) {
robdd *result = conjoin((robdd *)f_nd, (robdd *)g_nd);
return PL_unify_pointer(result_term, (void *)result);
}
} else {
PL_fail;
}
– section 22 slide 18 –
Making Boolean functions abstract in Prolog
To keep Prolog code from confusing an ROBDD (address) from a number, we wrap the address in a boolfn/1 term, much like we did in Haskell. We must do this manually; it is most easily done in Prolog code.
% conjoin(+BFn1, +BFn2, -BFn)
% BFn is the conjunction of BFn1 and BFn2. conjoin(boolfn(F), boolfn(G), boolfn(FG)) :-
boolfn_conjoin(F, G, FG).
We can make Prolog print BDDs (or anything) nicely by adding a clause for user:portray/1:
:- multifile(user:portray/1). user:portray(boolfn(BDD)) :- !,
% definition is in boolfn.pl …
– section 22 slide 19 –
Using it
nomad% swipl-ld -shared -o swi_robdd \
> swi_robdd.c robdd.c
nomad% pl
Welcome to SWI-Prolog (threaded, 64 bits,
version 7.7.19)
1 ?- [boolfn].
true.
2
|
|
A = ((1)),
B ((2)),
C = ((3)),
AB = ((1) | (~1 & 2)),
NotC = ((~3)),
X = ((1 & ~3) | (~1 & 2 & ~3)).
?- variable(1,A), variable(2,B), variable(3,C), disjoin(A,B,AB), negation(C,NotC), conjoin(AB,NotC,X).
– section 22 slide 20 –
Impedance mismatch
Declarative languages like Haskell and Prolog typically use different representations for similar data. For example, what would be represented as a list in Haskell or Prolog would most likely be represented as an array in C or Java.
The consequence of this is that in each language (declarative and imperative) it is difficult to write code that works on data structures defined in the other language.
This problem, usually called impedance mismatch, is the reason why most cross-language interfaces are low level, and operate only or mostly on values of primitive types.
NOTE Impedance mismatch is the name of a problem in electrical engineering that has some similarity to this situation. In both cases, difficulties arise from trying to connect two systems that make incompatible assumptions.
– section 22 slide 21 –
Comparative strengths of declarative languages
• Programmers can be significantly more productive because they can work at a significantly higher level of abstraction. They can focus on the big picture without getting lost in details, such as whose responsibility it is to free a data structure.
• Processing of symbolic data is significantly easier due to the presence of algebraic data types and parametric polymorphism.
• Programs can be significantly more reliable, because
– you cannot make a mistake in an aspect of programming that the language automates (e.g. memory allocation), and
– the compiler can catch many more kinds of errors.
• What debugging is still needed is easier because you can jump backward in time.
• Maintenance is significantly easier, because
– the type system helps to locate what needs
to be changed, and
– the typeclass system helps avoid unwanted coupling in the first place.
• You can automatically parallelize declarative programs.
– section 22 slide 22 – Comparative strengths of imperative languages
• If you are willing to put in the programming time, you can make the final program significantly faster.
• Most existing software libraries are written in imperative languages. Using them in declarative languages is harder than using them in another imperative language (due to dissimilarity of basic concepts), while using them is easiest in the language they are written in. If the bulk of a program interfaces to an existing library, this argues for writing the program in the language of the library:
– Java for Swing
– Visual Basic.NET or C# for ASP.NET
• There is a much greater variety of programming tools to choose from (debuggers, profilers, IDEs etc).
• It is much easier to find programmers who know or can quickly learn the language.
NOTE ASP.NET is a .NET application, which means that it is native not to a single programming language but to the .NET ecosystem, of which the two principal languages are Visual Basic.NET and C#.
– section 22 slide 23 –