Chap16 謂詞演算中的歸結(jié) 人工智能課程 上海交大

1、2lUnificationlPredicate-Calculus ResolutionlCompleteness and SoundnesslConverting Arbitrary wffs to Clause FormlUsing Resolution to Prove TheoremslAnswer ExtractionlThe Equality PredicatelAdditional Readings and Discussion3lAbbreviating wffs of the form by , where are literals that might contain occ

2、urrences of the variableslSimply dropping the universal quantifiers and assuming universal quantification of any variables in the lClauses: WFFs in the abbreviated form lIf two clauses have matching but complementary literals, it is possible to resolve them Example: ,4lUnification: A process that co

3、mputes the appropriate substitution lSubstitution instance of an expression is obtained by substituting terms for variables in that expression. Four substitution instances of are The first instance is called an alphabetic variant. The last of the four different variables is called a ground instance

4、(A ground term is a term that contains no variables). 5lAny substitution can be represented by a set of ordered pairs The pair means that term is substituted for every occurrence of the variable throughout the scope of the substitution. No variables can be replaced by a tern containing that same var

5、iable. The substitutions used earlier in obtaining the four instances of ws denotes a substitution instance of an expression w, using a substitution s. Thus, The composition s1 and s2 is denoted by s1s2, which is that substitution obtained by first applying s2 to the terms of s1 and then adding any

6、pairs of s2 having variables not occurring among the variables of s1. Thus, 6l Let w be P(x,y), s1 be f(y)/x, and s2 be A/y then, and Substitutions are not, in general, commutativelUnifiable: a set of expressions is unifiable if there exists a substitution s such that unifies , to yield 7lMGU (Most

7、general (or simplest) unifier) has the property that if s is any unifier of yielding , then there exists a substitution such that . Furthermore, the common instance produced by a most general unifier is unique except for alphabetic variants.lUNIFY Can find the most general unifier of a finite set of

8、 unifiable expressions and that report failure when the set cannot be unified. Works on a set of list-structured expressions in which each literal and each term is written as a list. Basic to UNIFY is the idea of a disagreement set. The disagreement set of a nonempty set W of expressions is obtained

9、 by locating the first symbol at which not all the expressions in W have exactly the same symbol, and then extracting from each expression in W the subexpression that begins with the symbol occupying that position.89l are two clauses. Atom in and a literal in such that and have a most general unifie

10、r, , then these two clauses have a resolvent, . The resolvent is obtained by applying the substitution to the union of and , leaving out the complementary literals.lExamples:10lPredicate-calculus resolution is sound If is the resolvent of two clauses and , then , |= lCompleteness of resolution It is

11、 impossible to infer by resolution alone all the formulas that are logically entailed by a given set. In propositional resolution, this difficulty is surmounted by using resolution refutation. 111. Eliminate implication signs.2. Reduce scopes of negation signs.3. Standardize variables Since variable

12、s within the scopes of quantifiers are like “dummy variables”, they can be renamed so that each quantifier has its own variable symbol.4. Eliminate existential quantifiers.12 Skolem function, Skolemization: Replace each occurrence of its existentially quantified variable by a Skolem function whose a

13、rguments are those universally quantified variables Function symbols used in Skolem functions must be “new”.13 Skolem function of no arguments Skolem form: To eliminate all of the existentially quantified variables from a wff, the proceding procedure on each subformula is used in turn. Eliminating t

14、he existential quantifiers from a set of wffs produces what is called the Skolem form of the set of formulas. The skolem form of a wff is not equivalent to the original wff. . What is true is that a set of formulas, is satisfiable if and only if the Skolem form of is. Or more usefully for purpose of

15、 resolution refutations, is unsatisfiable if and only if the Skolem form of is unsatifiable.145. Convert to prenex formAt this stage, there are no remaining existential quantifiers, and each universal quantifier has its own variable symbol. A wff in prenex form consists of a string of quantifiers ca

16、lled a prefix followed by a quantifier-free formula called a matrix. The prenex form fof the example wff marked with an * earlier is 6. Put the matrix in conjunctive normal formWhen the matrix of the preceding example wff is put in conjunctive normal form, it became 157. Eliminate universal quantifi

17、ers Assume that all variables in the matrix are universally quantified.8. Eliminate symbols The explicit occurrence of symbols may be eliminated by replacing expressions of the form with the set of wffs . 169. Rename variables Variable symbols may be renamed so that no variable symbol appears in mor

18、e than one clause . 17lTo prove wff from , proceed just as in the propositional calculus. 1. Negate , 2. Convert this negation to clause form, and 3. Add it to the clause form of . 4. Then apply resolution until the empty clause is deduced. 18lProblem: the package delivery robot. Suppose this robot knows that all of the packages in room 27 are smaller than any of the o