CS计算机代考程序代写 Lambda Calculus interpreter CSC324 Assignment 4. Theorem Proving in miniKanren

CSC324 Assignment 4. Theorem Proving in miniKanren
In this assignment, we will write a basic proof checker for purely implicational minimal logic in miniKanren. Like how
we ran a miniKanren interpreter “backwards” to synthesize programs, we will run this proof checker “backwards” to
create a rudimentary theorem prover.

Before we start, remember these general guidelines for all labs and assignments:

• You may not import any libraries or modules unless explicitly told to do so.
• You may not use the function “eval” for any lab or assignment unless explicitly told otherwise.
• You may not use any iterative or mutating functionality unless explicitly allowed. Remember that a big goal of

this course is to learn about different models and styles of programming!
• You may write helper functions freely; in fact, you are encouraged to do so to keep your code easy to understand.

Breaking any of these above rules can result in a grade of 0.

• Code that cannot be imported (e.g., due to a syntax error, compilation error, or runtime error during import)
will receive a grade of zero! Please make sure to run all of your code before your final submission, and test it on
the Teaching Lab environment (which is the environment we use for testing).

• The (provide …) and module (…) where code in your files are very important. Please follow the
instructions, and don’t modify those lines of code! If you do so, your code will not be able to run and you will
receive a grade of zero.

Starter code
• a4.rkt
• mk.rkt

You will only be submitting a4.rkt and not any of the other files. We’ll be supplying our own version of the remaining
files to test your code.

Background: Purely Implicational Minimal Logic
A logic describes a way to write down propositions (statements that make some claim, which can be true or false)
and possible proofs or ways to explain how a proposition is true. The logic that we will be implementing is called
purely implicational minimal logic, chosen for its simplicity.

Propositions

A propositions is a logical statements that makes some claim. In our logic, the only types of propositions that we
allow are:

• Constants, like A, B, or other symbols.
• Implications, like (A -> B), which can be verbalized as “A implies B”.

The left-hand side of an implication is called its hypothesis, and the right-hand side is called its conclusion. For
example, the hypothesis of (A -> B) is A, and the conclusion of (A -> B) is B.

Propositions will be represented as data with a shape given by the following grammar:

:: = IDENTIFIER | ‘(‘ ‘->’ ‘)’

That is, a proposition may either be a Racket symbol, or a 3-element list where the second element is the symbol ->,
and the first and third elements are propositions. Note the recursive structure.

For example:

• A, B, foo, bar, are all propositions.
• (foo -> bar) is a proposition.
• (A -> ((B -> C) -> D)) is a proposition.

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https://en.wikipedia.org/wiki/Minimal_logic
a4.rkt
mk.rkt

Proofs

A Proof is a logical argument that explains how a proposition is true. You might intuitively see that the propositions
(A -> A) and (A -> (B -> A)) are both true. However, in a logic system, we use precise rules about the types of
reasoning allowed. In particular, there are three rules that we can use in a proof:

• The rule use makes use of a proposition that is already available as an assumption.
• The rule assume makes a new proposition available for use as an assumption.
• The rule modus-ponens uses the proof of an implication’s hypothesis to prove the implication’s conclusion. This

rule is sometimes called the elimination rule since it it is used to “eliminate” implications.

Intuitively, if a proposition is currently available as an assumption, we can prove that same proposition by using the
assumption. We call this rule “use”. For example, if A is available as an assumption, then we can simply prove A by
using that assumption.

If we can prove the conclusion of an implication while assuming its hypothesis, then we can prove the implication
itself. This follows the introduction rule for implication, which we call “assume”. For example, we can prove the
implication (A -> A) because we can prove the implication’s conclusion A by using the assumption A, which is the
implication’s hypothesis.

If both an implication and its hypothesis can be proven, then the implication’s conclusion can also be proven. This
follows the elimination rule for implication, which we call “modus ponens”. For example, if we’ve proved both (A ->
B) and A, then we can use those proofs to prove B by invoking “modus ponens”.

Proofs will be represented as data with a shape given by the following grammar:

:: = ‘(‘ use ‘)’
| ‘(‘ assume ‘)’
| ‘(‘ modus-ponens ‘)’

That is, a proof is always a list of at least two elements, where the first element is a symbol indicating which proof
rule is being used. Depending on the kind of rule being used, a proof may reference propositions and/or sub-proofs.

For example:

• (assume A (use A)) is a proof of (A -> A).
• (assume A (assume (A -> B) (modus-ponens (use A) (use (A -> B))))) is a proof of (A -> ((A -> B)

-> B)).
• (assume (C -> D) (assume C (modus-ponens (use C) (use (C -> D))))) is a proof of ((C -> D) -> (C

-> D)).
• (assume (C -> D) (use (C -> D))) is also a proof of ((C -> D) -> (C -> D)).

As always, we use lists in Racket to represent propositions and proofs. In Racket, if we want to represent the proof
(assume A (use A)), we write ‘(assume A (use A)) or (list ‘assume ‘A (list ‘use ‘A)).

Task 1: Using a proof checking function in Racket
This question came up in Prof. Michael Miljanovic’s CSC236 course:

Figure 1: Message from Prof. Miljanovic

Our first task is to help convince Prof. Miljanovic that he did not make a mistake, and that we can write a proof
for the theorem ((A -> (B -> C)) -> (B -> (A -> C))).

Write the proof in the grammar provided above, and assign it to the variable my-proof in the file a4.rkt.

Unfortunately, Prof. Miljanovic is not easily convinced, so we will use a proof checker to check our proof. In
particular, a proof checker function proof? is written for you in the starter code.

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The function proof? takes a proposition and a proof, and returns whether the second argument is a valid proof of
the first argument. Here are some example calls to proof?:

> (proof? ‘(A -> A) ‘(assume A (use A)))
#t
> (proof? ‘(A -> B) ‘(assume A (use A)))
#f
> (proof? ‘A ‘(use A))
#f ; (because `A` is not available as an assumption where it is used)

We will be testing to see whether (proof? ‘((A -> (B -> C)) -> (B -> (A -> C))) my-proof) returns true.

Advice: You may find it helpful to study some of the proofs that are provided to you. The modus-ponens
may be unintuitive to use. You may also choose to skip to Task 2, and synthesize the proof in Task
2 instead.

Task 2: A proof checking relation
In this part of the assignment, we will define the relation proofo in miniKanren. The miniKanren relation proofo is
analogous to the Racket procedure proof?.

• The first argument is a proposition.
• The second argument is a proof.
• The relation holds if the second argument is a valid proof of the first argument.

For example:

• (proofo ‘(A -> A) ‘(assume A (use A))) should succeed.
• (proofo ‘(A -> B) ‘(assume A (use A))) should fail.
• (proofo ‘A ‘(use A)) should also fail (because A is not available as an assumption where it is used).

Because proof? returns a boolean, proofo does not need an extra argument for representing the result. Instead,
proofo will succeed when proof? would return #t, and fail when proof? would return #f.

You should implement proofo in terms of a helper relation proof-helpero that, in addition to the proposition and
proof arguments, also maintains a list of the currently-valid assumptions.

We implemented relations by referencing their functional equivalents. You may find it helpful to think of
proof-helpero as the relational form of proof/asmpt, and think of the parameter prop of proof-helpero as the
return value of proof/asmpt.

Alternatively, you may choose to implement proof-helpero by reasoning about how the proposition, proof, and
assumptions should relate to one another:

• Start by considering the three different kinds of proofs. You will want to treat these separately. In the functional
setting, you may want to use cond, if, or pattern matching to identify these cases. What miniKanren construct
is equivalent?

• For use proofs, what would need to be true about the proposition that you are trying to prove (vs the proposition
that you are trying to use)? What would need to be true about the proposition that you are trying to use (vs
the assumptions that you have?)

• For assume proofs, what does your proposition need to look like? What sub-proof would you need to check?
What fresh variables do you need to introduce in order to express these constraints?

• For modus-ponens proofs, you will need to separately prove the two sub-proofs. But what propositions do the
hypothesis and conclusion sub-proofs prove? What assumptions can you use in these sub-proofs? What fresh
variables do you need to introduce in order to express these constraints?

As always trust the recursion!

Task 3: Prove theorems by running “backwards”
Define a theorem prover prove in Racket, which runs the proofo relation backwards to attempt to prove a theorem.
In other words, prove should be a procedure that takes a proposition and returns either a proof for the proposition,
or #f. When there is no proof for the proposition, the procedure may also fail to terminate.

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• The argument is a proposition.
• If the proposition has any proof, a proof is returned.
• If the proposition cannot be proven, then either #f is returned, or the procedure will fail to terminate.

You should define prove in terms of a miniKanren run 1 expression that makes use of proofo. Remember that run
returns a list of answers, so you will need to unwrap any answer produced.

You may wish to use prove to find proofs for the following theorems:

• (A -> (B -> A))
• (A -> (B -> B))
• ((A -> B) -> (B -> C) -> (A -> C))
• (((A -> (B -> C)) -> C) -> ((B -> (A -> C)) -> C))

Theorem Proving and Programming Languages
Very astute readers might see some connection between the theorem checker proof? (or even proofo) and the
interpreter for the Lambda Calculus that we wrote back in weeks 3-4 of class. There are three kinds of expressions in
the Lambda Calculus, and there are three kinds of theorems our minimal logic. The “use” rule is similar to identifier
lookup, the “assume” rule is similar to lambda expressions, and the “modus-ponens” rule is similar to application.

It turns out that there is a correspondence between proofs and programs—and in particular, proof checkers and type
checkers. You can find out more by reading about the Curry-Howard correspondence.

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https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence

CSC324 Assignment 4. Theorem Proving in miniKanren
Starter code
Background: Purely Implicational Minimal Logic
Propositions
Proofs

Task 1: Using a proof checking function in Racket
Task 2: A proof checking relation
Task 3: Prove theorems by running “backwards”
Theorem Proving and Programming Languages