ai present and future · 2019. 1. 22. · prolog syntax variables written with initial capital...

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AI Present and Future

Alan Smaill

University of Edinburgh, School of Informatics

[email protected]

22/01/19

Alan Smaill AI Present and Future 22/01/19 1/41

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Admin

Drop-in Lab sessions 9–11 on Mondays start next week.

Some material to work through for next week will be on-linelater today.

You should work through the material yourself;Learn Prolog Now will help here.

The lab session next week gives you a chance to get feedbackand let you discuss any problems you may have.


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Today

Logic Programming: the idea

Overview of the “pure” part of the language.

Inbuilt search strategy, and consequences.

Reasoning with lists, trees

See Learn Prolog Now, chs 1–4http:

//www.learnprolognow.org/lpnpage.php?pageid=online


http://www.learnprolognow.org/lpnpage.php?pageid=online

http://www.learnprolognow.org/lpnpage.php?pageid=online

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The Ideal

Program specifications can be written in logic.

Specifications are independent of computers.

Rules of logic can prove that a specification can be realised,even if computers didn’t exist.

But proof can also be done by a computer smart enough tofind the right proof.


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The Ideal





So specifications and programs are . . . the same.


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The Ideal





So specifications and programs are nearly the same.


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Grammar for first-order logic

. . . aka predicate calculusDefine terms by

term ::= constant

| var| fn symbol ( term list )

term list ::= term

| term , term list


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Modelling the world

We suppose that the world can be describe in terms of

set of objects

functions

relations

where

functions map objects to objects

relations with n arguments describe properties of n objects.


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Example

Look at a blocks world:

base

a b

c

d

e

Describe using objects with chosen names (constants):

base, a, b, c, d, e.

Say which object is on which other with a 2-place relation, on:on(e, c), on(c, a), on(e, d), on(d, b), on(b, base), on(a, base)


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Definite clause subset

Rather than dealing with full predicate logic, take a subset, like thepropositional definite clauses.

Drop ∨,¬ and explicit quantifiers.

statements: pred ( term list )

Definite clauses:

statement(statement ∧ · · · ∧ statement)→ statement

Queries: statement ∧ · · · ∧ statement

Definite clauses are implicitly universally quantified (∀)Queries are implicitly existentially quantified (∃)


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Getting started

We’ll use SWI Prolog

Available on all DICE machines

Assessed coursework will require up-to-date version of swipl.

Freely available: see

http://www.swi-prolog.org/

On-line documentation:http: // www. sics. se/ isl/ sicstuswww/ site/

Manual:http: // www. swi-prolog. org/ pldoc/ doc_ for?

object= manual


http://www.swi-prolog.org/

http://www.sics.se/isl/sicstuswww/site/

http://www.swi-prolog.org/pldoc/doc_for?object=manual

http://www.swi-prolog.org/pldoc/doc_for?object=manual

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Prolog syntax

Variables written with initial capital letterX, X15, ListOfInts, . . .

Constants, function symbols, predicates start with lower caseletterbartSimpsonJnr, a123, b′ . . .

Clause written in form:a :- b, c.


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Unification

The equation t = u is a basic goalwith a special meaning

What happens if we ask:

?- X = c.

?- f(X,g(Y,Z)) = f(c,g(X,Y)).

?- f(X,g(Y,f(X))) = f(c,g(X,Y)).

And how does it do that?

This shows how a successful query returns a witness for theexistential variable: a term that makes the query provable whenplugged into the query.


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Unification II

?- X = c.

X=c

yes

?- f(X,g(Y,Z)) = f(c,g(X,Y)).

X=c

Y=c

Z=c

yes

?- f(X,g(Y,f(X))) = f(c,g(X,Y)).

no


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Unification III

A substitution is a mapping from variables to terms

X1 = t1, . . . ,Xn = tn

Given two terms t and u

with free variables X1, . . . ,Xn,

a unifier is a substitution that makes t and u identical whenapplied to t and u.


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Example I

f(X,g(Y,Z)) = f(c,g(X,Y))

X=c

Y=X

Z=Y

f

X g

Y Z

f

c g

X Y


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Example I: apply the substitution

f(X,g(Y,Z)) = f(c,g(X,Y))

X=c

Y=c

Z=c

f

c g

c c

f

c g

c c


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Example II

f(X,g(Y,f(X))) = f(c,g(X,Y))

X=c

Y=X

f

X g

Y f

X

f

c g

X Y


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Example II: apply partial substitution

f(X,g(Y,f(X))) = f(c,g(X,Y))

X=c

Y=c

Y=f(X)

f(X)=c???

f

c g

c f

c

f

c g

c c


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Execution model

The query is run by trying to find a solution to the goal using theclauses:

Unification is used to match goals and clauses

There may be zero, one, or many solutions

Execution may backtrack


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Depth-first search

Basic Idea:

To solve atomic goal A:

If B is a fact in the program, and there is a substitution θsuch that θ(A) = θ(B), then return answer θ;

else,if B :-G1, ...,Gn is a clause in the program,and θ unifies A with B,

then solve θ(G1), . . . , θ(Gn)

else give up on this goal:

backtrack to last choice point

Clauses are tried in declaration order

Compound goals are tried in left-right order


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Depth-first search

Prolog tries clauses in order of appearance in the program.We look at a couple of search trees for query execution.Assume: foo(a). foo(b). foo(c).

then:

?- foo(X). foo(X)

X=a X=b X=c


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Depth-first search


then:

?- foo(X).

X=afoo(X)

X=a X=b X=c


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Depth-first search


then:

?- foo(X).

X=a ;

X=b

foo(X)

X=a X=b X=c


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Depth-first search


then:

?- foo(X).

X=a ;

X=b ;

X=c

foo(X)

X=a X=b X=c


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Depth-first search


then:

?- foo(X).

X=a ;

X=b ;

X=c ;

no

foo(X)

X=a X=b X=c


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Depth first search ctd

Prolog backtracks to the last choice point if a sub-goal fails.Assume: bar(b). bar(c). baz(c). then:

?- bar(X),baz(X). bar(X),baz(X)

X=b

baz(b)

X=c

baz(c)


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?- bar(X),baz(X).

X = cbar(X),baz(X)

X=b

baz(b)

X=c

baz(c)


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?- bar(X),baz(X).

X = c ;

no

bar(X),baz(X)

X=b

baz(b)

X=c

baz(c)


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Recursion

In recursive use, the same predicate is used in the head (lhs) of therule as in the body (rhs)(in the second clause below):

ancestor(X,Y) :- parent(X,Y).

ancestor(X,Y) :- parent(X,Z),

ancestor(Z,Y).

This is a fine declarative description of what it is to be an ancestor.

But watch out for the traps!!!


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Recursion: order can matter

Take the program for ancestor/2 with clauses in the oppositeorder:

ancestor(X,Y) :- parent(X,Z),

ancestor(Z,Y).

ancestor(X,Y) :- parent(X,Y).

This may be less efficient – looks for longest path first.

More likely to loop – if the parent/2 relation has cycles.

HEURISTIC: write base cases first (ie non-recursive cases).


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Rule order affects search

parent(a,b).

parent(b,c).ancestor(a,b)

parent(a,Z),ancestor(Z,b)

ancestor(b,b)

Z=b

parent(a,b)


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Rule order affects search

parent(a,b).

parent(b,a).ancestor(a,b)

parent(a,Z),ancestor(Z,b)

ancestor(b,b)

parent(b,W),ancestor(W,b)

ancestor(a,b)

...

W=a

Z=b


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Recursion again

Goal order can matter!

ancestor3(X,Y) :- parent(X,Y).

ancestor3(X,Y) :- ancestor3(Z,Y),

parent(X,Z)

This returns all solutions, then loops, eg with the following facts:

parent(a,b).

parent(b,c).

Heuristic: keep recursive calls late in the body


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Recursion and terms

Terms can be arbitrarily nested

Example: unary natural numbers

nat(z).

nat(s(X)) :- nat(X).

To do interesting things, we need recursion.


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Addition, subtraction

Addition:

add(z,N,N).

add(s(N),M,s(P)) :- add(N,M,P).

Run in reverse to get all M,N that sum to P:

?- add(X,Y,s(s(s(z)))).

X=z,Y=s(s(s(z)));

X=s(Z),Y=s(s(z));

...

Use to define leq/2:

leq(M,N) :- add(M,_,N).

Here “_” is a so-called anonymous variable;use to avoid warning of singleton variable in Prolog programs.Can also use, for example, _X, _Anon.


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Multiplication

Now define multiplication:

multiply(z,N,z). % or: multiply(z,_,z).

multiply(s(N),M,P) :-

multiply(N,M,Q), add(M,Q,P).

square(N,M) :- multiply(N,N,M).


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lists

Lists are built-in (and very useful) data structures.

Syntax:

[1,2,3,4]

[a,[1,2,3],42,’forty-two’] % nested list

[a,b,c|Xs] % first three elements,

% and Rest

?- [1,2,3,4] = [A,B,C|D].

A = 1, B = 2, C = 3, D = [4].


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List Processing

Predicate to recognise lists:

list([]).

list([X|L]) :- list(L).

Examples: list append

append([],L,L).

append([X|L],M,[X|N]) :- append(L,M,N).


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append in action

Forward direction:

?- append([1,2],[3,4],X).

X = [1,2,3,4]

Backward direction

?- append(X,Y,[1,2,3,4]).

X=[], Y=[1,2,3,4];

X=[1],Y=[2,3,4];

...


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Mode annotations

There are recognised ways of indicating properties of Prologprocedures.

Notation: append(+,+,-)

Expect to be called with the first two arguments ground, andthird a variable (which we normally expect to bound after thecall)

Similarly, append(-,-,+)

Call with last argument ground, first two as variables(which we normally expect to be bound after the call).

Not “code”, but often used in annotations

“?” annotation used where any term may appear— i.e. ground, variable, or compound term with variables.


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List Processing ctd

When is something a member of a list?

member(X, [X|_]).

member(X, [_|T]) :- member(X, T).

Typical modes:member(+,+)

member(-,+)


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List processing ctd

Removing an element of a list:

remove(X, [X|L], L).

remove(X, [Y|L], [Y|M]) :- remove(X, L, M).

NB: removes one occurrence of X;fails if X is not a member of the list.

Typical mode:remove(+,+,-)


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List processing ctd

Zip: pairing of corresponding elements of lists:assumed to be of same length.

zip([],[],[]).

zip([X|L], [Y|M], [(X,Y)|N]) :- zip(L, M, N).

Typical modes:

zip(+,+,-).

zip(-,-,+). % unzip


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Records

Can use terms to define data structures:

pb([entry(alan, ’156-675’),...]).

and operations on them:

pb_lookup(pb(B), P, N) :-

member(entry(P,N), B).

pb_insert(pb(B), P, N, pb([entry(P,N) | B])).

pb_remove(pb(B), P, pb(B2)) :-

remove(entry(P,_), B, B2).


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Trees

We can define (binary) trees with data (at the nodes).

tree(leaf).

tree(node( Data, LT, RT )) :- tree(LT), tree(RT).

Data membership in a tree —using “;” for alternatives in the body of a clause.

mem_tree(X, node(X, _, _)).

mem_tree(X, node(_, LT, RT)) :-

mem_tree(X, LT) ;

mem_tree(X, RT).


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Preorder traversal

Pick up the data in a particular order:start at root, traverse recursively left subtree, then right subtree.

preorder(leaf, []).

preorder(node(X, LT, RT), [X|N]) :-

preorder(LT, LO),

preorder(RT, RO),

append( LO, RO, N).

What happens if we run this in reverse?


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Seen today

Logic Programming: the idea

Overview of the “pure” part of the language.

Inbuilt search strategy, and consequences.

Reasoning with lists, trees


ai present and future · 2019. 1. 22. · prolog syntax variables written with initial capital...

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