1. AI - Week 16 Logic and Reasoning in AI: Resolution Refutation Lee McCluskey, room 2/07 Email lee@hud.ac.uklee@hud.ac.uk http://scom.hud.ac.uk/scomtlm/cha2555/

2. Logic and Reasoning in AI Recap First order logic is an expressive way of representing knowledge in general. Expressions in FOL are called “wffs”. Laws of Inference are methods for deducing wffs from wffs.They are SOUND if they only ever deduce wffs that also logically follow. We want to embed sound laws of inference in software that will be able to perform automated reasoning efficiently.

3. Logic and Reasoning in AI Aside: Proof by Contradiction (Sometimes called Proof by Refutation or Reductio ad Absurdum) This is an efficient way of reasoning: assume what we are trying to prove is FALSE, then get a CONTRADICTION => what we were trying to prove is TRUE. Imagine we know Wff1 to be TRUE and we want to prove Wff2 logically follows from Wff1. If we can derive a contradiction from (Wff1 & ~Wff2) then assuming Wff1 is TRUE we know that Wff2 logically follows from Wff1, or written in logic: Wff1 |- Wff2

4. Logic and Reasoning in AI Resolution Refutation Resolution is a super law of inference which - can easily be automated - when used in refutation mode it is COMPLETE - it can deduce any Wff that logically follows. - is the basis for Prolog Resolution Refutation: To PROVE Wff2 FROM Wff1 1. Translate Wff1 to CLAUSAL FORM 2. Translate ~ Wff2 to CLAUSAL FORM 3. Get contradiction from 1 + 2 using Resolution …. If follows that Wff1 |- Wff2

5. Logic and Reasoning in AI SOME JARGON To do resolution refutation, we need to learn about: Clausal form (covered it a bit last week) Unification (just Prolog matching..) A LITERAL is a predicate which may be negated, called a negative literal, or a predicate itself, called a positive literal. A CLAUSE is a disjunction of literals which contain only universally quantified variables. Sometimes we think of a clause as a SET of literals implicitly “disjuncted”. A set of wffs w1, w2,... wn is in CLAUSAL FORM if each wi (i=1 to n) is a clause

6. Logic and Reasoning in AI 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 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.

7. Logic and Reasoning in AI Unification - substition 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 either is a clause we may write: W[T/x] meaning perform the textual substitution T for x throughout W, or 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. Logic and Reasoning in AI Unification - definition A set of literals unify if and only if each literal in the set is identical or a step by step application of a sequence 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. Logic and Reasoning in AI 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. Logic and Reasoning in AI Example 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) Goal: D(Jeff) CLAUSAL FORM of Wff1 & ~Wff2: 1. ~S(x) V D(x) 2. ~T(x,y) V ~D(y) V D(x) 3. S(Fred) 4. T(Jeff,Fred) 5. ~D(Jeff)

11. Logic and Reasoning in AI Example 1. ~S(x) V D(x)3. S(Fred) D(Fred) 2. ~T(x,y) V ~D(y) V D(x) ~T(x,Fred) V D(x)5. ~D(Jeff) ~T(Jeff,Fred)4. T(Jeff,Fred) Subs = Fred / x Subs = Fred / y Subs = Jeff / x..So D(Jeff) follows from our premises

12. Logic and Reasoning in AI Summary Resolution is a law of inference that is based on: - Wffs in CLAUSAL FORM - The method of UNIFICATION of literals Resolution Refutation is a deduction procedure that is COMPLETE amenable to AUTOMATION PROLOG works using “single literal depth-first” (SLD) resolution