1. The Law of Resolution Formal Aspects of Computer Science - Week 7 The Law of Resolution Lee McCluskey, room 2/07 Email lee@hud.ac.uklee@hud.ac.uk http://scom.hud.ac.uk/scomtlm/cia2326

2. The Law of Resolution Recap and Motivation We want to be able to reason with logic using tools in order to prove properties about specifications / requirements / code, and enable reasoning in intelligent agents. Just study ONE law of inference to do this Resolution. This law is often embodied is a “proof procedure” called Resolution Refutation which is SOUND and COMPLETE.

3. The Law of Resolution SOME JARGON To carry out Resolution, we need to learn about: The Clausal form of wffs (a little bit like clauses in Prolog) The Process of Unification (very similar to Prolog matching) Resolution Form: Parent Clause1 Parent Clause2 Child Clause

4. The Law of Resolution SOME JARGON ABOUT CLAUSES A LITERAL is a predicate - called a positive literal, or predicate preceded by a “~”, called a negative literal. 3 Example literals: on(x,y), ~on(x,table), green(table). A CLAUSE is a disjunction of literals in which any variable is assumed to be universally quantified. Sometimes we think of a clause as a SET of literals implicitly “disjuncted”. Example: on(x,y) v ~on(x,table) v green(table) Or in set notation {on(x,y), ~on(x,table), green(table)} A set of wffs w1, w2,... wn is in CLAUSAL FORM if each wi (i=1 to n) is a clause

5. The Law of Resolution To change any Wff to Clausal Form: Those Eight Steps …….. 1. Replace any occurrences of using the law: A B = (A => B) & (A <= B) 2. Replace all occurrences of => and <= using the laws: A => B = ~A v B …and... A <= B = A v ~B 3. Reduce the scope of every “~” so that they all operate on predicates. To do this use you may need the laws: ~(Ax wff) = Ex (~wff) and ~(Ex wff) = Ax (~wff) etc 4. Standardise variables - make sure all quantified variables are different 5. Eliminate existential quantifiers - change into Skolem Constants [TRICKY STEP] 6. Universal quantifiers can now all be removed, making every variable implicitly universally quantified. 7. Use the laws to convert to conjunctions of disjunctive literals (A & B) V C = (AVC) & (B V C) 8. Making all the conjunctions implicit, we are left with a set of clauses.

6. The Law of Resolution Student Example from last week… S = student, D = academic, T = teaches Ax ( S(x)=>D(x) ) ; Ax ( (Ey (T(x,y) & D(y) ) => D(x) ) S(Fred) ; T(Jeff,Fred) CLAUSAL FORM: 1. ~S(x) V D(x) 2. ~T(x,y) V ~D(y) V D(x) 3. S(Fred) 4. T(Jeff,Fred)

7. The Law of Resolution Unification: the process of substitution The law of resolution depends on the idea of “unification”. This virtually the same as matching in Prolog. We first introduce the idea of substitution: A legal substitution is the consistent replacing of a variable symbol x by a term T on condition that T does not contain any occurrence of x. If W is a clause we may write: W[T/x] meaning perform the textual substitution T for x throughout W. IF we have a sequence of substitutions S, we write W/S meaning perform the following sequence of substitutions on W: S = { [T1/x1],..., [Tn/xn]}.

8. The Law of Resolution Unification - definition A set of literals unify if and only if a step by step application of a sequence (possibly null) of legal substitutions make them identical. Identical here means: - they all have the same predicate symbol; - they all have the same polarity; - they all have the same number of slots; - they all have identical terms in corresponding slots. Given a set of literals we can try to UNIFY them by applying substitutions systematically

9. The Law of Resolution The Law of (Binary) Resolution Two PARENT clauses w1 and w2 infer a CHILD clause wr if there are literals L in w1 and M in w2 such that {L,~M} unify under some substitution sequence S. Remembering that clauses are sets of literals, we can deduce wr = [ w1 U w2 minus { L, ~ M } ]/S. The law also assumes that each clause has unique variable letters. This does not restrict its generality because variables in separate clauses are independent.

10. The Law of Resolution Summary Resolution is a law of inference that is based on: - Wffs in CLAUSAL FORM - The process of UNIFICATION of literals It inputs TWO clauses and outputs a child clause.