程序代写代做代考 prolog algorithm LPN2

LPN2

Exercises
Exercises of LPN chapter 2
Practical work

© Patrick Blackburn, Johan Bos & Kristina Striegnitz
Aim of this lecture
Discuss unification in Prolog
Show how Prolog unification differs from standard unification

Explain the search strategy that
Prolog uses when it tries to deduce new information from old,
using modus ponens

© Patrick Blackburn, Johan Bos & Kristina Striegnitz
Unification
Recall previous example, where we said that Prolog unifies

woman(X)

with

woman(mia)

thereby instantiating the variable X with the atom mia.

Terms
Simple Terms
Complex Terms
Constants
Variables
Atoms
Numbers
Recall Prolog Terms

Working definition:

Two terms unify if they are the same term or if they contain variables that can be uniformly instantiated with terms in such a way that the resulting terms are equal

mia and mia unify
42 and 42 unify
woman(mia) and woman(mia) unify

This also means that:

vincent and mia do not unify
woman(mia) and woman(jody) do not unify

mia and X

mia and X
woman(Z) and woman(mia)

mia and X
woman(Z) and woman(mia)
loves(mia,X) and loves(X,vincent)

© Patrick Blackburn, Johan Bos & Kristina Striegnitz
Instantiations
When Prolog unifies two terms it performs all the necessary instantiations, so that the terms are equal afterwards
This makes unification a powerful programming mechanism

© Patrick Blackburn, Johan Bos & Kristina Striegnitz
Revised Definition 1/3
If T1 and T2 are constants, then
T1 and T2 unify if they are the same atom, or the same number.

© Patrick Blackburn, Johan Bos & Kristina Striegnitz
Revised Definition 2/3
If T1 and T2 are constants, then
T1 and T2 unify if they are the same atom, or the same number.
If T1 is a variable and T2 is any type of term, then T1 and T2 unify, and T1 is instantiated to T2. (and vice versa)

© Patrick Blackburn, Johan Bos & Kristina Striegnitz
Revised Definition 3/3
If T1 and T2 are constants, then
T1 and T2 unify if they are the same atom, or the same number.
If T1 is a variable and T2 is any type of term, then T1 and T2 unify, and T1 is instantiated to T2. (and vice versa)
If T1 and T2 are complex terms then they unify if:
They have the same functor and arity, and
all their corresponding arguments unify, and
the variable instantiations are compatible.

Why? After working through the first goal, Prolog has instantiated X with mia, so that it cannot unify it with vincent anymore. Hence the second goal fails.

© Patrick Blackburn, Johan Bos & Kristina Striegnitz
Example with complex terms
?- k(s(g),Y) = k(X,t(k)).
X=s(g)
Y=t(k)
yes
?-

© Patrick Blackburn, Johan Bos & Kristina Striegnitz
Example with complex terms
?- k(s(g),t(k)) = k(X,t(Y)).
X=s(g)
Y=k
yes
?-

© Patrick Blackburn, Johan Bos & Kristina Striegnitz
Prolog and unification
Prolog does not use a standard unification algorithm
Consider the following query:

?- father(X) = X.

Do these terms unify or not?

© Patrick Blackburn, Johan Bos & Kristina Striegnitz
Infinite terms
?- father(X) = X.
X=father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(father(

© Patrick Blackburn, Johan Bos & Kristina Striegnitz
Infinite terms
?- father(X) = X.
X=father(father(father(…))))
yes
?-

© Patrick Blackburn, Johan Bos & Kristina Striegnitz
Occurs Check
A standard unification algorithm carries out an occurs check
If it is asked to unify a variable with another term it checks whether the variable occurs in the term
In Prolog:
?- unify_with_occurs_check(father(X), X).
no

© Patrick Blackburn, Johan Bos & Kristina Striegnitz
Programming with Unification
vertical( line(point(X,Y),
point(X,Z))).

horizontal( line(point(X,Y),
point(Z,Y))).

© Patrick Blackburn, Johan Bos & Kristina Striegnitz
Programming with Unification
vertical( line(point(X,Y),
point(X,Z))).

horizontal( line(point(X,Y),
point(Z,Y))).
?-

© Patrick Blackburn, Johan Bos & Kristina Striegnitz
Programming with Unification
vertical( line(point(X,Y),
point(X,Z))).

horizontal( line(point(X,Y),
point(Z,Y))).
?- vertical(line(point(1,1),point(1,3))).
yes
?-

© Patrick Blackburn, Johan Bos & Kristina Striegnitz
Programming with Unification
vertical( line(point(X,Y),
point(X,Z))).

horizontal( line(point(X,Y),
point(Z,Y))).
?- vertical(line(point(1,1),point(1,3))).
yes
?- vertical(line(point(1,1),point(3,2))).
no
?-

© Patrick Blackburn, Johan Bos & Kristina Striegnitz
Programming with Unification
vertical( line(point(X,Y),
point(X,Z))).

horizontal( line(point(X,Y),
point(Z,Y))).
?- horizontal(line(point(1,1),point(1,Y))).
Y = 1;
no
?-

© Patrick Blackburn, Johan Bos & Kristina Striegnitz
Programming with Unification
vertical( line(point(X,Y),
point(X,Z))).

horizontal( line(point(X,Y),
point(Z,Y))).
?- horizontal(line(point(2,3),Point)).
Point = point(_554,3);
no
?-

© Patrick Blackburn, Johan Bos & Kristina Striegnitz
Proof Search
Now that we know about unification, we are in a position to learn how Prolog searches a knowledge base to see if a query is satisfied.
In other words: we are ready to learn about proof search

© Patrick Blackburn, Johan Bos & Kristina Striegnitz
Example
f(a).
f(b).
g(a).
g(b).
h(b).
k(X):- f(X), g(X), h(X).
?- k(Y).