first order logic resolution dr. rogelio dávila pérez profesor-investigador división de posgrado...

First Order Logic Resolution

Dr. Rogelio Dávila PérezProfesor-InvestigadorDivisión de PosgradoUniversidad Autónoma de [email protected]

Unification

We can represent any substitution by a set of ordered pairs:

s = { t1/v1, t2/v2, …, tn/vn}

where: ti/vi means that the term i substitutes variable i throughout the expression.

e.g. s1 = {b/x, a/y, a/z}Note:

- every occurrence of a variable should be substituted by the same term.- no variable can be replaced by a term that contains the same variable.

Def. To denote a substitution instance of an expression E, using a substitution s, we write: Es.

Def. Finding substitutions of terms for variables to make two or more expressions identical is called unification.

e.g. {p(g(x,f(y)),a), p(g(b,f(z)),y)} ==> p(g(b,f(a)),a)

Def. A substitution instance of an expression is obtained by substituting terms for variables in that expression.

Def. A ground instance of an expression is one in which there are no variables (and consequently, no quantifiers).

Def. An instance of an expression which results from changing just the names of the variables in it, is called an alphabetic variant of the original.

e.g. x.p(x,f(x,b)) ==> z.p(z,f(z,b))

Unification

Def. A composition of two substitutions s1 and s2 is written s1s2, and it is accomplished following the steps:- apply s2 to the elements of s1.- add any pairs of s2 having variables not occurring in s1.

e.g. s1={g(x,y)/z}, s2={a/x, b/y, c/w, d/z} s1s2 ==> {g(a,b)/z, a/x, b/y, c/w}

s2s1 ==> {a/x, b/y, c/w, d/z} = s2Some properties of composition are:

1. (Es1)s2=E(s1s2)2. s1(s2s3)=(s1s2)s3

Unification

Def. We say that a set {Ei} of expressions is unifiable if there exists a substitution s that makes: E1s=E2s=…=Ens.

Def. If a substitution g has the property that if s is a unifier of {Ei} yielding {Ei} s, then there exists a substitution s ’ such that

{Ei} s={Ei} gs’, then g is called the most general unifier mgu.

For the unify procedure presented next, it is assumed that its arguments are two lists of expressions. e.g.

g(r(x,f(z)),y) ==> [g, [r, x, [f, z ]], y]

g(r(a, y), u) ==> [g, [r, a, y], u]

Unification

Procedure UNIFIY (E1, E2)If either E1 or E2 is an atom { if E1 and E2 are identical,

return NIL if E1 is a variable, do:

{ if E1 occurs in E2, return FAIL

return {E2/E1} } if E2 is a variable, return

{E1/E2} return FAIL }F1 the first element of E1T1 the rest of E1F2 the first element of E2 T2 the rest of E2Z1 UNIFY(F1, F2)

If Z1 = FAIL, return FAILG1 result of applying Z1 to T1G2 result of applying Z1 to T2Z2 UNIFY(G1, G2)If Z2 = FAIL, return FAILReturn the composition of Z1 and Z2

________________________

Procedure taken from Nilsson N.J. “Principles of Artificial Intelligence”, Tioga Publishing, California 1980.

Unification Algorithm

Conjunctive normal form: any formula of the predicate calculus can be transformed into a conjunctive normal form.

Def. A formula is said to be in conjunctive normal form if it consists in the conjunction of clauses.

A1 A2 … An where Ai is a clause.

Def. A formula is said to be a clause if it consists in a disjunction of literals. A clause has the following form:

L1 v L2 v … v Lm where Li is a literal.

Def. A literal is an atomic formula or the negation of an atomic formula.

Def. A formula is said to be in clausal form if it can be expressed as a set of clauses:

{C1 , … , Cn,} where Ci is a clause

Clausal Form

1. Eliminate implication symbols (), using the identity:

~ v 2. Introduce negation: reduce scopes of negation

symbols by repeatedly applying the De Morgan rules and the quantifier identities:

(i) ~( v ) ~ ~ (ii) ~( ) ~ v ~

(iii) ~x. (x) x. ~(x)

(iv) ~x. (x) x. ~(x)

Transforming to Clausal Form

3. Transform into prenex normal form:Def. A formula is in prenex normal form if it consists in a list of quantifiers called prefix, followed by a quantifier-free formula called the matrix.

Procedure(i) Standardize variables: re-name variables.(ii) Eliminate existential quantifiers: replace all existential quantifier’s variables by skolem functions of all the universal quantified variables which are over the scope of the existential quantifier.(iii) Move all quantifiers to the left: there remain only universal

quantifiers.


4. Put matrix in conjunctive normal form by repeatedly using the distributive laws:

(i) v ( ) ( v ) ( v )

(ii) ( v ) ( ) v ( )

5. Eliminate universal quantifiers.

6. Eliminate conjunction () symbols separating in clauses.

7. Rename variables.


Example:x. (p(x) {y. [p(y) p(f(x,y))] ~y. [q(x,y)p(y)]})

===> gets transformed into the following clauses:

c1: ~p(x1) v ~p(y) v p(f(x1,y))c2: ~p(x2) v q(x2, g(x2))c3: ~p(x3) v ~p(g(x3))

Predicate logic resolution

We can use production systems to show that a given formula, called the goal, is a theorem derivable from a set of formulas (they provide the state description).

These production systems are called theorem proving systems or deduction systems.

Resolution Refutation Systems are based in the following principle:

= if and only if, {~} =

Resolution Refutation Systems

Resolution refutation procedureIn general a resolution refutation proving that an

arbitrary wff fallows from a set of wffs , |= , proceeds as follows:

1. Convert the wffs in to clausal form.

2. Negate the formula to be proved and convert the resulting formula, ~ , into to clausal form.

3. Combine the clauses resulting form steps 1 and 2 into a single set, .

4. Iteratively apply resolution to the clauses in and add the results to either until there are no more resolvents that can be added or until the empty clause is produced.


Def. Resolution Rule: p v q1 v q2 v … v qn

~p v r1 v r2 v … v rm

-------------------------------------------------

q1 v … v qn v r1 v … v rm

e.g.

1. Whoever can read is a literate. Dolphins are not literate.Some dolphins are intelligent. Hence, some who are intelligent can not read.

2. (c1) p(x,f(a)) v p(x,f(y)) v q(y)

(c2) ~p(z,f(a)) v ~q(z)


Example:

1. Tiger and savages animals are fierce or destructives.

2. If there is something destructive in the room, then something in the room will get damaged.

3. Tim is a tiger.4. Tim is a savage animal.5. Tim is in the room but it is not fierce.

Prove using resolution that something in the room will get damaged.


Important results Completeness of resolution refutation: the empty

clause will be produced by the resolution

refutation procedure if |= , thus we say that predicate calculus resolution is refutation complete.

Decidability of predicate calculus resolution refutation: We say that resolution refutation in predicate calculus is semi-decidable as it terminates by generating the

empty clause if |= , but it may never terminate if | .


Simplification strategiesSometimes a set of clauses can be simplified by elimination of certain clauses or elimination of certain literals in the clauses.

Elimination of tautologiesAny clause containing a literal and its negation may be eliminated. E.g.

p(x) v q(y) v ~ q(y) ==> T

Procedural attachmentSometimes it is possible to evaluate the truth values of literals:

v F ==> e.g.

p(x) V q(y) V ( 3 >7) ==> p(x) V q(y) since we already know that 3>7 is false.


Elimination by subsumptionDef. A clause {Li} subsumes a clause {Mi} if there exists a substitution s such that {Li}s is a subset of {Mi}. e.g. p(x) subsumes p(y) v q(z)

p(x) subsumes p(a)p(x) subsumes p(a) v q(z)p(x) v q(a) subsumes p(f(a)) v q(a) v r(y)

A clause in an unsatisfiable set that is subsumed by another clause in the set can be eliminated without affecting the unsatisfiability of the rest of the set.


Extracting answers from Resolution RefutationsMany applications of theorem-proving systems involve proving formulas containing existentially quantified variables and finding values or instances for these variables. To get an answer the proof method has to be constructive.

E.g. “If Fido goes wherever John goes and if John is at school,

where is Fido?”

To answer this question first we must have to proof of the goal: z. at(fido,z)

Next: we must extract the answer to the question from this refutation tree. In order to do that we proceed as follows:

(a) append to each clause arising from the negation of the goal wff its own negation.

(b) Following the structure of the refutation tree, perform the same resolutions as before until some clause is obtained at the root.

(c) Use the clause at the root as an answer statement.


Extracting the answer involves converting a refutation tree (with NIL at the root) to a proof tree with some statement at the root that can be used as an answer. The conversion involves converting every clause arising from the negation of the goal into a tautology. Another example:

- For all x, y, there exists a z, such that if- parent(x,y) & parent(y,z)

grandfather(x,z)- And, everyone has a parent: x.y.parent(x,y)- Question: “Do there exists individuals x and y such

that x is the grandparent of y.” it can be expressed as: x.y.g(x, y)


Universally quantified variables in goalsA problem when a universal quantified variable appears on the

goal is that after the goal is negated we have an existencially quantified variable which is Skolemized and a function is introduced. What is the meaning of this Skolem function and how it has to be interpreted when extracting the answer?

Steps:1. A resolution-refutation tree is found by some search

process. The unification subsets of the clauses of the tree are marked.

2. New variables are substituted for any Skolem functions occurring in the clauses that result from the negation of the goal wff.

3. The clauses resulting from the negation of the goal wff are converted into tautologies by appending to them their own negations.


4. A modified proof tree is produced modeling the structure of the original refutation tree. Each resolution in the modified tree uses a unification set determined by the unification set used by the corresponding resolution in the refutation tree.

5. The clause at the root of the modified tree is the answer statement extracted by these process.

It can be shown [Luckham and Nilsson, 1971] that in the answer-extracting process it is correct to replace any Skolem functions in the clauses coming from the negation of the goal wff with new variables.

_____________________Luckham, D.C., and Nilsson, N. J. 1971. Extracting information

from resolution proof trees. Artificial Intelligence, 2(1), 27-54.


first order logic resolution dr. rogelio dávila pérez profesor-investigador división de posgrado...

Documents