1. Resolution Refutation Formal Aspects of Computer Science - Week 10 An Automated Theorem Prover Lee McCluskey, room 2/07 Email lee@hud.ac.uklee@hud.ac.uk http://scom.hud.ac.uk/scomtlm/cia2326/

2. Resolution Refutation Recap Last Week -First Order Logic - Relationship with Prolog -How to get answers from Resolution Refutation (and Prolog) -The Martians Example This week – An Automated Theorem Prover (ATP)

3. Resolution Refutation Automated Theorem Proving -Using a tool to show that a wff1 logically follows from a wff2 -Of great interest in problem solving, proving theorems, proving software etc -NOTE proof and correctness is always RELATIVE. You can prove w1 follows from w2 is correct, but you can’t prove simply “w1 is correct”. So – you could prove a program is correct with respect to a specification, but never “a program is correct”.

4. Resolution Refutation Automated Theorem Proving Strategies A membership procedure is DECIDABLE if there is an algorithm which can deliver a solution (yes or no) in a finite no. of steps Theoretically, proof that w1 follows from w2 is “SEMI-DECIDABLE” in the sense that -if you know w1 follows from w2 then we can use resolution to deduce wffs from w2 and know that eventually w1 will be produced -If you DON’T KNOW that w1 follows from w2, then there is no effective procedure that can let us know the answer for all possible inputs 

5. Resolution Refutation Automated Theorem Proving Strategy We try to embed strategies into help with the problem of deciding which of possibly many clause pairs (parents) should be resolved next. Some common ones: "Set of Support" Strategy: use a parent which is either an element of the negated query clause or an ancestor of one. “Unit Preference” Strategy use clauses which contain 1 literal if possible

6. Resolution Refutation Automated Theorem Prover :- [cf,ur,testmartians]. predicate prove(depth,query,wff). cf is the code to change to clausal form ur is the code that does resolution testmartians contains a particular problem “depth” is the search limit of the ATP

7. Resolution Refutation Automated Theorem Prover To use you must learn the syntax for writing the Wffs in- to do this look at examples.. p53:-prove(9, all(x1,green(x1)->friendly(x1) ), ( all(x2, green(x2)->antennae(x2)) & all(x3, all(x4, child(x4,x3)->antennae(x4)) -> friendly(x3)) & all(x5, exists(x6, child(x5,x6)&green(x6)) ->green(x5) ) ) ).

8. Resolution Refutation Automated Theorem Prover DEMO …………

9. Resolution Refutation Summary Proof is semi decidable Proof tools use heuristics and strategies such as set-of-support You have been emailed an ATP written in Prolog