Resolution Theorem Proving in Predicate Calculus

1. Resolution Theorem Proving in Predicate Calculus Lecture No 10 By Zahid Anwar

2. Resolution • Resolution is a technique for proving theorems in predicate calculus • Resolution is a sound inference rule that, when used to produce a refutation, is also complete • In an important practical application resolution theorem proving particularly the resolution refutation system, has made the current generation of Prolog interpreters possible

3. Resolution Refutation • The resolution principle, describes a way of finding contradictions in a data base of clauses with minimum substitution • Resolution Refutation proves a theorem by negating the statement to be proved and adding the negated goal to the set of axioms that are known or have been assumed to be true • It then uses the resolution rule of inference to show that this leads to a contradiction

4. Resolution Refutation • Once the theorem prover shows that the negated goal is inconsistent with the given set of axioms, it follows that the original goal must be consistent . • This proves the theorem

5. Steps in Resolution Refutation Proofs • Put the premises or axioms into clause form • Add the negations of what is to be proved in clause form, to the set of axioms • Resolve these clauses together, producing new clauses that logically follow from them • Produce a contradiction by generating the empty clause • The substitutions used to produce the empty clause are those under which the opposite of the negated goal is true

6. Discussion about Steps • Resolution Refutation proofs require that the axioms and the negation of the goal be placed in a normal form called the clause form • Clausal form represents the logical database as a set of disjunctions of literals • Resolution is applied to two clauses when one contains a literal and the other its negation

7. Discussion about Steps • If these literals contain variables, they must be unified to make them equivalent • A new clause is then produced consisting of the disjunction of all the predicates in the two clauses minus the literal and its negative instance (which are said to have been “resolved away”)

8. An Example • We wish to prove that “Fido will die” from the statements that “Fido is a dog” and “all dogs are animals” and “all animals will die”

9. Equivalent Reasoning by Resolution • Convert these predicates to clause form

10. Apply Resolution • Negate the conclusion that fido will die ¬die(fido) ¬dog(x) V animal(x) ¬animal(y) V die(y) [Y/X] dog(fido) ¬dog(y) V die(y) [fido/Y] die(fido) ¬die(fido) a Null clauseHence fido will die

11. Another Example Anyone passing the Artificial Intelligence exam and winning the lottery is happy. But anyone who studies or is lucky can pass all their exams. Ali did not study be he is lucky. Anyone who is lucky wins the lottery. Is Ali happy?

12. Another Example Anyone passing the AI Exam and winning the lottery is happy X:[pass(x,AI) Λ win(x, lottery) happy(x)] Anyone who studies or is lucky can pass all their exams X Y [studies(x) V lucky(x) pass(x,y)] Ali did not study but he is lucky ¬ study(ali) Λ lucky(ali) Anyone who is lucky wins the lottery X: [lucky(x) win(x,lottery)]

13. Change to clause Form • ¬pass(X,AI) V ¬win(X,lottery) V happy(X) • ¬study(Y) V pass(Y,Z) • ¬lucky(W) V pass(W,V) • ¬study(ali) • Lucky(ali) • ¬lucky(u) V win(u,lottery) • Add negation of the conclusion ¬happy(ali)

14. ¬pass(X,AI) V ¬win(X,lottery) V happy(X) win(u,lottery) V ¬lucky(u) u/v ¬pass(u,AI) V happy(u) V ¬lucky(u) ali/u ¬happy(ali) ¬pass(ali,AI) V ¬lucky(ali) lucky(ali) ¬ pass(ali,AI) Ali/v,AI/w ¬lucky(v) V pass(V,W) ¬lucky(ali) lucky(ali)