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Logic Programming and Prolog

LOGIC PROGRAMMING

AND PROLOG

Logic programming languages are not proce-

dural or functional.

• Specify relations between objects

- larger(3,2)

- father(tom,jane)

Reading:

• Sebesta, chapter 16

• Separate logic from control:

- Programmer declares what facts and relations

are true.

- System determines how to use facts to solve

problems.

- System instantiates variables in order to make

relations true!

References:

• Clocksin and Mellish, Ch. 1-4, 6, 8

• Online Resources (tutorials, SWI page, etc.) • Computation engine: theorem-proving and recur-

sion (Unification, Resolution, Backward Chaining,

Backtracking)

- Higher-level than imperative languagesome material ©Diane Horton 2000, and

©Suzanne Stevenson 200l.

Modified by Sheila McIlraith 2004.

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Prolog vs Scheme Logic Programming

In Scheme, we program with functions (" pro-

cedures" ) .

• A function's arguments are different from

the function's value.

• Give a single Scheme function, we can only

ask one kind of question:

• A program consists of facts and rules.

• Running a program means asking queries.

• The language tries to find one way

(or more) to prove that the query is true.

Here are the argument values; tell me

what is the function's value.

• This may have the side effect of freezing

variable values.

• The language determines how to do all of

th is, not the prog ram.

In Prolog, we program with relations.

• There is no bias; all arguments are the

same.

• Given a single Prolog predicate, we can ask

many kinds of question:

• How does the language do it? Using unifi-

cation, resolution, and backtracking.

Here are some of the argument val-

ues; tell me what the others have to

be in order to make a true statement.

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?- notrace.

Ye s

cdf% pI

W elc ome to S WI- Pro log (M ult i- thre aded , Ve rsi on 5.2 .11 )

?- parent(Person, edward). For help, use ?- h elp (To pic) . or ?- apropos(Word).

Person

Person

No

albert;

victoria;

?- [family].

[ fa mi ly l oa de d]

Ye s

?- p ar en t( Pe rs on , e dw ar d) .

?- halt.

cdf% Person

Person

No

albert;

victoria;

edit family.P and remove parent(albert,edward). - --

?- ['family'].

% family compiled 0.00 sec, 5,200 bytes

Ye s

?- p ar en t( Pe rs on , e dw ar d) .

Person = victoria;

No

?- halt.

cdf%

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Some Prolog Syntax

Lexica I Rules:

• Variables are capitalized.

• Constants begin with a lower case letter.

• Predicate names begin with a lower case

letter.

Prolog Queries

A query is a proposed fact that is to be proven.

Simplified Grammar:• Ifthe query has no variables, returns yes/no.

<clause> ::= <pred>

<pred> :- <pred> { <pred>}. • If the query has variables, returns appro-

priate values of variables (called a substi-

tution).pred> ::= <pname>'(' <term> { , <term> } C ) '

<term> ::= <int> I <atom> I <var>

Note: No blank between predicate name and

opening bracket.

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Horn Clauses

(Rules)

Horn Clause Terminology

• Horn Clause = Clause

A Horn Clause is: C f-hl 1\ h2 1\ h3 1\ ... 1\ hn • Consequent = Goal = Head

• Antecedents = Subgoals = Tail

• Antecedents: conjunction of zero or more

conditions which are atomic formulae in

predicate logic

• Horn Clause with No Tail = Fact

• Horn Clause with Tail = Rule

• Consequent: an atomic formula in predi-

cate logic

In Prolog, a Horn clause

is written

Meaning of a Horn clause:

Syntax elements:r , ,

• "The consequent is true if the antecedents

are all true"

• c is true if hl. ba , h3..... and hn are all

true

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Prolog Horn Clause Examples Meaning of Prolog Rules

Without VariablesA Horn clause with no tail:

A prolog rule must have this form:

male(albert) .

I.e., a fact: albert is a male dependent on no

other conditions

which means in logic:

A Horn clause with a tail:

father( albert,edward):-

male(albert), parent(albert,edward). Restrictions

I.e., a rule: albert is the father of edward if al-

bert is male and albert is a parent of edward's.

• There can be zero or more antecedents,

but they are conjoined; we cannot disjoin

them .

• There cannot be more than 1 consequent.

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Bending the Restrictions Why Can't We

Disjoin Consequents?

Getting disjoined antecedents

Why did the designers of Prolog disallow this?

Solution:

Getting more than 1 consequent, conjoined

Solution:

Getter more than 1 consequent, disjoined

Solution:

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Logic Review Horn Clauses with Variables

Variables may appear in the antecedents and

consequent of a Horn clause:

"For all values of Xl, ... ,Xn, the formula

C(Xl, ... ,Xn) is true if the formula h(Xl, ... ,Xn)

is true"

"For all values of Xl, ... ,Xn, the formula

C(Xl, ... ,Xn) is true if there exist values of

Yl,· .. ,Yk such that the formula h(Xl,· .. ,Xn,Yl,· .

is true"

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Meaning of Prolog Rules

With Variables

Sample run

cdf% pI

Welcome to SWI- Prolog (Multi-threaded, Version 5.2.11) ...

Example: ?- ['family'].

isaMother(X) ._ female (X) , parent (X, Y).W a rn i ng : ( . /f a mi l y. p l: 5 0) :

S i ng l et o n v a ri a bl e s: [y]

% family compiled 0.00 sec, 5,528 bytes

Logic:

parent(X, Y) 1\ female(X) :J isaM other(X).

But this is meaningless without quantifiers for

the variables.

?- isaMother(X).

X = victoria;

X = victoria;

No

The rule

A Prolog rule of this form (n ~ O,m::; n,k ~ 0) :

means:

VX1,···Xn

[::::lYl,·.. Yk [a(Xl,··· Xm, Yl,· .. Yk) :J C(Xl,· .. Xn)]

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Rule Ordering and Unification How Prolog Handles a Query

1. rule ordering used in search

2. unification requires two instances of the

same variable in the same rule to get the

same value

Example 1

Database:

3. unification does not require differently named

variables to get different values: hence,

sibling (edward ,edward).

4. all rules searched if requested by'·'

1 ) m al e(t om ).

2 ) m al e( pe te r) .

3) m al e(d ou g).

4 ) f em al e( su sa n) .

5 ) m al e( da vi d) .

6) parent (doug, susan).

7) parent (tom, william).

8) parent (doug, david).

9) parent (doug, tom).

10) grandfather(GP, GC) :- male (GP),

parent (GP, X) ,

p ar en t( X, G C) .

Query:

I?- grandfather(X,Y).

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Trace it by handTrace it in Prolog

[trace]Call:

Call:

Exit:

Call:

Exit:

Call:

Fail:

Redo:

Exit:

Call:

Fail:

Redo:

Exit:

Call:

Exit:

Call:

Fail:

Redo:

Exit:

Call:

Fail:

Redo:

Exit:

Call:

Exit:Exit:

X doug

Y = william

Ye s

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?- grandfather(X,Y).( 7) g ra nd fa th er (_ G2 83 , _ G2 84 ) ? creep

( 8 ) m a le ( _G 28 3 ) ? creep

( 8 ) m a le ( to m) ? creep

( 8) p ar en t ( to m, _ L2 05 ) ? creep

( 8) p ar en t ( to m, w il li am ) ? creep

( 8) p ar en t ( wi ll ia m, _ G2 84 ) ? creep

( 8) p ar en t ( wi ll ia m, _ G2 84 ) ? creep

( 8 ) m a le ( _G 28 3 ) ? creep

( 8 ) m a le ( pe te r ) ? creep

( 8) p ar en t ( pe te r, _ L2 05 ) ? creep

( 8) p ar en t ( pe te r, _ L2 05 ) ? creep

( 8 ) m a le ( _G 28 3 ) ? creep

( 8 ) m a le ( do ug ) ? creep

( 8) p ar en t ( do ug , _ L2 05 ) ? creep

(8 ) p aren t (do ug, su san ) ? creep

( 8) p ar en t ( su sa n, _ G2 84 ) ? creep

( 8) p ar en t ( su sa n, _ G2 84 ) ? creep


( 8) p ar en t ( do ug , d av id ) ? creep

( 8) p ar en t ( da vi d, _ G2 84 ) ? creep

( 8) p ar en t ( da vi d, _ G2 84 ) ? creep


(8 ) p aren t (do ug, t om) ? creep

( 8) p ar en t ( to m, _ G2 84 ) ? creep

( 8) p ar en t ( to m, w il li am ) ? creep( 7) g ra nd fa th er ( do ug , w il li am ) ? creep

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Prolog Search Trees Example 2

• Each node is an ordered list of goals. Database:

• Each edge is labelled with the variable bind-

ings that occurred due to applying a rule.

(The binding are in effect throughout the

subtree.)

1 ) m al e( al be rt ).

2 ) f em al e( al ic e) .

3 ) m al e( ed wa rd ).

4 ) f em al e( vi ct or ia ).

5 ) p ar e nt ( al be r t, e dw a rd ) .

6 ) p ar e nt ( vi ct o ri a ,e d wa r d) .

7 ) p ar e nt ( al be r t, a li c e) .

8 ) p ar e nt ( vi ct o ri a ,a l ic e ).

9) sibl ing( X, y) :- p ar en t( P, X ), p ar en t( P, y).

• Each leaf represents either success or fail-

ure.

Query:

?- sibling(alice,Asib).

Asib = edward ;

Asib = alice ;

Asib = edward;

Asib = alice ;

No

?- s i bl i ng ( As ib , a li c e) .

Asib = edward ;

Asib = edward ;

Asib = alice

Asib = alice

No

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Trace it by hand Trace it in Prolog

[trace]

Call:

Call:

Exit:

Call:

Exit:

Exit:

?- sibling(alice,Asib).

( 7) s ib li ng (a li ce , _ G2 84 ) ? creep

( 8) p ar en t ( _L 20 5, a li ce ) ? creep

( 8) p ar en t ( al be rt , a li ce ) ? creep

( 8) p ar en t ( al be rt , _ G2 84 ) ? creep

( 8) p ar en t ( al be rt , e dw ar d) ? creep

( 7) s ib li ng (a li ce , e dw ar d) ? creep

Asib = edward ;

Redo: (8 ) p ar en t ( al be rt , _G284) ? creep

Exit: (8 ) p ar en t ( al be rt , a li ce ) ? creep

Exit: (7) s ib li ng (a li ce , a li ce ) ? creep

Asib = alice

Redo : (8 ) p aren t (_L 205 , a lic e) ? creep

Exi t: (8 ) p aren t (vi cto ria , ali ce) ? creep

Ca ll: (8 ) p aren t (vi cto ria , _G2 84) ? creep

Exi t: (8 ) p aren t (vi cto ria , edw ard) ? creep

Exi t: (7 ) si bli ng( ali ce, ed war d) ? creep

Asib = edward ;

Redo : (8 ) p aren t (vi cto ria , _G2 84) ? creep

Exi t: (8 ) p aren t (vi cto ria , ali ce) ? creep

Exi t: (7 ) si bli ng( ali ce, al ice ) ? creep

Asib = alice

No

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Logic Programming vs. Prolog Logic Programming vs. Prolog

cousin(X,Y) :- parent(W,X), sister(W,Z),

parent(Z,Y).

I ?- cousin(X,jane). % a query

Logic Programming: Nondeterministic

• Arbitrarily choose rule to expand first

• Arbitrarily choose subgoal to to explore

first

• Results don't depend on rule and sub-

goal ordering

cousin(X,Y) :- parent(W,X), brother(W,Z),

parent(Z,Y).

Rule and Goal Ordering:

• There are two rules for cousin

• Which rule do we try first?

• Each rule for cousin has several subgoals

• Which subgoal do we try first?

Prolog: Deterministic

• Expand first rule first

• Explore first subgoal first

• Results may depend on rule and subgoal

ordering

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