Lesson 15 - Module 6 Knowledge Representation and Logic...

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Module 6 Knowledge Representation and Logic – (First Order Logic) Version 1 CSE IIT, Kharagpur
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Lesson 15 Inference in FOL - I Version 1 CSE IIT, Kharagpur
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6.2.8 Resolution We have introduced the inference rule Modus Ponens. Now we introduce another inference rule that is particularly significant, Resolution. Since it is not trivial to understand, we proceed in two steps. First we introduce Resolution in the Propositional Calculus, that is, in a language with only truth valued variables. Then we generalize to First Order Logic. 6.2.8.1 Resolution in the Propositional Calculus In its simplest form Resolution is the inference rule: {A OR C, B OR (NOT C)} ---------------------- A OR B More in general the Resolution Inference Rule is: Given as premises the clauses C1 and C2, where C1 contains the literal L and C2 contains the literal (NOT L), infer the clause C, called the Resolvent of C1 and C2, where C is the union of (C1 - {L}) and (C2 -{(NOT L)}) In symbols: {C1, C2} --------------------------------- (C1 - {L}) UNION (C2 - {(NOT L)}) Example: The following set of clauses is inconsistent: 1. (P OR (NOT Q)) 2. ((NOT P) OR (NOT S)) 3. (S OR (NOT Q)) 4. Q In fact: 5. ((NOT Q) OR (NOT S)) from 1. and 2. 6. (NOT Q) from 3. and 5. 7. FALSE from 4. and 6. Notice that 7. is really the empty clause [why?]. Theorem: The Propositional Calculus with the Resolution Inference Rule is sound and Refutation Complete. NOTE: This theorem requires that clauses be represented as sets, that is, that each element of the clause appear exactly once in the clause. This requires some form of membership test when elements are added to a clause. C1 = {P P} Version 1 CSE IIT, Kharagpur
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C = {P (NOT P)} From now on by resolution we just get again C1, or C2, or C. 6.2.8.2 Resolution in First Order Logic Given clauses C1 and C2, a clause C is a RESOLVENT of C1 and C2 , if 1. There is a subset C1' = {A1, . ., Am} of C1 of literals of the same sign, say positive, and a subset C2' = {B1, . ., Bn} of C2 of literals of the opposite sign, say negative, 2. There are substitutions s1 and s2 that replace variables in C1' and C2' so as to have new variables, 3. C2'' is obtained from C2 removing the negative signs from B1 . . Bn 4. There is an Most General Unifier s for the union of C1'.s1 and C2''.s2 and C is ((C1 - C1').s1 UNION (C2 - C2').s2).s In symbols this Resolution inference rule becomes: {C1, C2} -------- C If C1' and C2' are singletons (i.e. contain just one literal), the rule is called Binary Resolution . Example: C1 = {(P z (F z)) (P z A)} C2 = {(NOT (P z A)) (NOT (P z x)) (NOT (P x z)) C1' = {(P z A)} C2' = {(NOT (P z A)) (NOT (P z x))} C2'' = {(P z A) (P z x)} s1 = [z1/z] s2 = [z2/z] C1'.s1 UNION C2'.s2 = {(P z1 A) (P z2 A) (P z2 x)} s = [z1/z2 A/x] C = {(NOT (P A z1)) (P z1 (F z1))} Notice that this application of Resolution has eliminated more than one literal from C2, i.e. it is not a binary resolution. Theorem: First Order Logic, with the Resolution Inference Rule, is sound and
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This note was uploaded on 09/20/2010 for the course MCA DEPART 501 taught by Professor Hemant during the Fall '10 term at Institute of Computer Technology College.

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Lesson 15 - Module 6 Knowledge Representation and Logic...

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