prolog substitutions unification

8/10/2019 Prolog Substitutions Unification

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PROLOG.

Substitutions and Unification

Antoni Ligeza

Katedra Automatyki, AGH w Krakowie

2011

Antoni Ligeza Prolog 1/1


2/19

References

[1] Ulf Nilsson, Jan Mauszynski:Logic, Programming and Prolog, John Wiley &

Sons Ltd., pdf,http://www.ida.liu.se/ ulfni/lpp

[2] Dennis Merritt: Adventure in Prolog, Amzi, 2004

http://www.amzi.com/AdventureInProlog

[3] Quick Prolog:

http://www.dai.ed.ac.uk/groups/ssp/bookpages/quickprolog/quickprolog.html

[4] W. F. Clocksin, C. S. Mellish: Prolog. Programowanie. Helion, 2003

[5] SWI-Prologs home:http://www.swi-prolog.org

[6] Learn Prolog Now!: http://www.learnprolognow.org

[7] http://home.agh.edu.pl/ ligeza/wiki/prolog

[8] http://www.im.pwr.wroc.pl/ przemko/prolog



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Substitutions

The role of substitutions

Substitutionis an operation allowing to replace some variables occurring in

a formula with terms.

The goal of applying a substitution is to make a certain formula more specific so

that it matches another formula. Substitutions allow forunificationof formulae

(or terms).

Definition

A substitution is any finite mapping of variables into terms of the form

: VTER.

Notation of substitutions

Any (finite) substitution can be presented as

={X1/t1,X2/t2, . . . ,Xn/tn},

whereti is a term to be substituted for variable Xi ,i = 1, 2, . . . , n.

is the formula (or term) resulting fromsimultaneousreplacement of thevariables of with the appropriate terms of.



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Extending substitutions over Terms and Formulae

Any substitution ( : V TER) is extended to operate on terms and formulae sothat a finite mapping of the form

: TER FOR TER FOR

satisfying the following conditions is induced:

(c) =cfor anyc C;

(X

)TER, and

(X

)=Xfor a certain finite number of variables only;

iff(t1, t2, . . . , tn) TER, then

(f(t1, t2, . . . , tn)) =f((t1), (t2), . . . , (tn));

ifp(t1, t2, . . . , tn) ATOM, then

(p(t1, t2, . . . , tn)) =p((t1), (t2), . . . , (tn));

( ) =() ()for any two formulae, FORand for {, , , };



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Instances, Ground Terms, Ground Formulae

An Instance

Any formula()resulting from application of substitution to the variables ofwill be denoted as

and it will be called asubstitution instanceor simply aninstanceof.

A Ground Instance, Term FormulaIf no variables occur in, it will be called aground instance(aground formulaor aground term, respectively).

Example

Let ={X/a, Y/f(b)}, and let =p(X, Y, g(X)). Then

=p(a,f(b), g(a))

and it is a ground formula.



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Composition of Substitutions

Since substitutions are mappings, acompositionof substitutions is well defined.

Having two substitutions, say and, the composed substitutioncan be obtained

from by: simultaneousapplication of to all the terms of,

deletion of any pairs of the formX/twheret=X(identity substitutions), and

enclosing all the pairsX/tof, such that does not substitute for (operate on)X.

More formally:

Let ={X1/t1,X2/t2, . . . ,Xn/tn} and let ={Y1/s1, Y2/s2, . . . , Ym/sm}. Thecomposition of the above substitutions is obtained from the set

{X1/t1,X2/t2 , . . . ,Xn/tn, Y1/s1, Y2/s2, . . . , Ym/sm}

by:

removing all the pairsXi/tiwhereXi =ti, and

removing all the pairsYj/sj whereYj {X1,X2, . . . ,Xn}.



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An Example

Example

Consider the following substitutions:

={X/g(U), Y/f(Z), V/W,Z/c}

and

={Z/f(U), W/V, U/b}.

The composition of them is defined as: ???


A E l


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An Example

Example

Consider the following substitutions:

={X/g(U), Y/f(Z), V/W,Z/c}

and={Z/f(U), W/V, U/b}.

The composition of them is defined as:

={X/g(b), Y/f(f(U)),Z/c, W/V, U/b}.


R i d I S b tit ti


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Renaming and Inverse Substitutions

A Renaming Substitution

Substitution

is arenaming substitutioniff it is off the form

={X1/Y1,X2/Y2, . . . ,Xn/Yn} (1)

Moreover, it is a one-to-one mapping ifYi =Yj fori =j,i,j {1, 2, . . . , n}.

An Inverse SubstitutionAssumeis a renaming, one-to-one substitution . The inverse substitution for it isgiven by

1 ={Y1/X1, Y2/X2, . . . , Yn/Xn, }.

Composition of inverse substitutions

The composition of a renaming substitution and the inverse one leads to an empty

substitution, traditionally denoted with; we have

1 =.


S ti


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Some properties

An Instance

LetEdenote an expression (formula or term), denote an empty substitution, and letbe a one-to-one renaming substitution; and denote any substitutions.The following properties are satisfied for any substitutions:

E() = (E), () = ()(associativity),

E= E,

= = .

Note that, in general, the composition of substitutions is not commutative.


Unification


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Unification

Substitutions are applied tounifyterms and formulae. Unification is a process of

determining and applying a certain substitution to a set of expressions (terms or

formulae) in order to make them identical. We have the following definition of

unification.

Definition

LetE1,E2, . . . ,En TER FORare certain expressions. We shall say thatexpressionsE1,E2, . . . ,En areunifiableif and only if there exists a substitution,such that

{E1,E2, . . . ,En} ={E1,E2 , . . . ,En}

is a single-element set.

Substitution satisfying the above condition is called a unifier(or aunifyingsubstitution) for expressionsE1,E2, . . . ,En.

Note that if there exists a unifying substitution for some two or more expressions

(terms or formulae), then there usually exists more than one such substitution (or

even infinitely many unifiers).


The Most General Unifier (mgu)


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The Most General Unifier (mgu)

It is useful to define the so-called most general unifier(mgu, for short), which,

roughly speaking, substitutes terms for variables only if it is necessary, leaving as

much place for possible further substitutions, as possible.

Definition

A substitution is amost general unifierfor a certain set of expressions if and onlyif, for any other unifier of this set of expressions, there exists a substitution , suchthat =.

The meaning of the above definition is obvious. Substitution is not a most generalunifier, since it is a composition of some simpler substitution with an auxiliarysubstitution.In general, for arbitrary expressions there may exist an infinite number of unifying

substitutions. However, it can be proved that any two most general unifiers can differ

only with respect to variable names. This is stated with the following theorem.

A Theorem

Let1and2 be two most general unifiers for a certain set of expressions. Then, thereexists a one-to-one renaming substitution such that1 =2and 2 =1

1.


An example


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An example

Example

As an example consider atomic formulaep(X,f(Y))and p(Z,f(Z)). The followingsubstitutions are all most general unifiers:

={X/U, Y/U,Z/U},

1 ={Z/X, Y/X},

2 ={X/Y,Z/Y},

3 ={X/Z, Y/Z}.

All of the above unifiers are equivalent each of them can be obtained from another

one by applying a renaming substitution. For example, =1for = {X/U}; on

the other hand obviously1 =1

.


Unification Algorithm An Idea


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Unification Algorithm An Idea

It can be proved that if the analyzed expressions are terms or formulae, then

there exists an algorithm for efficient generating the most general unifier,provided that there exists one; in the other case the algorithm terminates after

finite number of steps . Hence, the unification problem is decidable.

The basic idea of the unification algorithm can be explained as a subsequent

search through the structure of the expressions to be unified for inconsistent

relative components and replacing one of them, hopefully being a variable, with

the other.

In order to find inconsistent components it is useful to define the so-called

disagreementset.

LetWTER FORbe a set of expressions to be unified. A disagreement setD(W)for a nonempty set Wis the set of terms obtained through parallel search

of all the expressions ofW(from left to right), which are different with respectto the first symbol. Hence, the set D(W)specifies all the inconsistent relativeelements met first during the search.


Unification Algorithm


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Unification Algorithm

Algorithm for Unification

1 Seti = 0,Wi =W,i =.

2 IfWi is a singleton, then stop;i is the most general unifier forW.

3 FindD(Wi).

4 If there are a variableXD(Wi)and a term t D(Wi), such thatXdoes notoccur int, then proceed; otherwise stop Wis not unifiable.

5 Seti+1 ={X/t},Wi+1 =Wi{X/t}.

6 Seti = i+1 and go to 2.


Application Example


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Application Example

Consider two atomic formulae p(X,f(X, Y), g(f(Y,X)))and p(c,Z, g(Z)). The

following steps illustrate the application of the unification algorithm to these atomicformulae.

1 i= 0,W0 ={p(X,f(X, Y), g(f(Y,X))),p(c,Z, g(Z))},0 ={}.

2 D(W0) ={X, c}.

3 1 ={X/c},W1 ={p(c,f(c, Y), g(f(Y, c))),p(c,Z, g(Z))}.

4 D(W1) ={f(c, Y),Z}.

5 2 ={X/c}{Z/f(c, Y)}= {X/c,Z/f(c, Y)},W2 ={p(c,f(c, Y), g(f(Y, c))),p(c,f(c, Y), g(f(c, Y)))}.

6 D(W2) ={Y, c}.

7 3 ={X/c,Z/f(c, Y)}{Y/c}= {X/c,Z/f(c, c), Y/c},

W3 ={p(c,f(c, c), g(f(c, c))),p(c,f(c, c), g(f(c, c)))}.

8 Stop; the most general unifier is 3 ={X/c,Z/f(c, c), Y/c}.


Properties of the Unification Algorithm


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Properties of the Unification Algorithm

Theorem

1 IfWis a finite set of unifiable expressions, then

1 the Unification Algorithmalways terminatesat step 2 and

2 itproduces the most general unifierfor W.

2 Moreover, if the expressions ofWare not unifiable, then the algorithm

terminates at step 4.


Unification in Prolog


18/19

Unification in Prolog

From: R. Bartak:

http://kti.mff.cuni.cz/~bartak/prolog/data_struct.html

Unification Algorithm defined in Prolog

1 unify(A,B):-

2 atomic(A),atomic(B),A=B.

3 unify(A,B):-

4 var(A),A=B. % without occurs check

5 unify(A,B):-6 nonvar(A),var(B),A=B. % without occurs check

7 unify(A,B):-

8 compound(A),compound(B),

9 A=..[F|ArgsA],B=..[F|ArgsB],

10 unify_args(ArgsA,ArgsB).

11

12 unify_args([A|TA],[B|TB]):-13 unify(A,B),

14 unify_args(TA,TB).

15 unify_args([],[]).


A final problem
http://kti.mff.cuni.cz/~bartak/prolog/data_struct.htmlhttp://kti.mff.cuni.cz/~bartak/prolog/data_struct.html


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p

Question

AreXandf(X)unifiable?

Example

What is/should be the result of:

?- X=f(X).

?- X=a, X=f(X).

?- X=f(X), write(X).


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