Artificial Intelligence 9. Resolution Theorem Proving

1. Artificial Intelligence 9. Resolution Theorem Proving Course V231 Department of Computing Imperial College Jeremy Gow

2. The Full Resolution Rule • If Unify(Pj, ¬Qk) =  (¬ makes them unifiable) P1 …  Pm, Q1  …  Qn Subst(, P1  … (no Pj) …  Pm  Q1  … (no Qk) ... Qn) • Pj and Qk are resolved • Arbitrary number of disjuncts • Relies on preprocessing into CNF

3. A More Concise Version E.g. for A = {1, 2, 7} first clause is L1 L2  L7

4. Resolution Proving • Knowledge base of clauses • Start with the axioms and negation of theorem in CNF • Resolve pairs of clauses • Using single rule of inference (full resolution) • Resolved sentence contains fewer literals • Proof ends with the empty clause • Signifies a contradiction • Must mean the negated theorem is false • (Because the axioms are consistent) • Therefore the original theorem was true

5. Empty Clause means False • Resolution theorem proving ends • When the resolved clause has no literals (empty) • This can only be because: • Two unit clauses were resolved • One was the negation of the other (after substitution) • Example: q(X) and ¬q(X) or: p(X) and ¬p(bob) • Hence if we see the empty clause • This was because there was an inconsistency • Hence the proof by refutation

6. Resolution as Search • Initial State: Knowledge base (KB) of axioms and negated theorem in CNF • Operators: Resolution rule picks 2 clauses and adds new clause • Goal Test: Does KB contain the empty clause? • Search space of KB states • We want proof (path) or just checking (artefact)

7. Aristotle’s Example (Again) • Socrates is a man and all men are mortal Therefore Socrates is mortal • Initial state 1) is_man(socrates) 2) is_man(X)  is_mortal(X) 3) ¬is_mortal(socrates) (negation of theorem) • Resolving (1) & (2) gives new state (1)-(3) & 4) is_mortal(socrates)

8. Aristotle’s Example: Search Space 1) is_man(socrates) 2) is_man(X)  is_mortal(X) 3) ¬is_mortal(socrates) 4) is_mortal(socrates) 1) is_man(socrates) 2) is_man(X)  is_mortal(X) 3) ¬is_mortal(socrates) 1) is_man(socrates) 2) ¬is_man(X)  is_mortal(X) 3) ¬is_mortal(socrates) 4) ¬is_man(socrates) 1) is_man(socrates) 2) is_man(X)  is_mortal(X) 3) ¬is_mortal(socrates) 4) is_mortal(socrates) 5) False 1) is_man(socrates) 2) is_man(X)  is_mortal(X) 3) ¬is_mortal(socrates) 4) ¬is_man(socrates) 5) False

9. Resolution Proof Tree (Proof 1)

10. Resolution Proof Tree (Proof 2)

11. Reading Proof Tree 2 You said that all men were mortal. That means that for all things X, either X is not a man, or X is mortal [CNF step]. If we assume that Socrates is not mortal, then, given your previous statement, this means Socrates is not a man [first resolution step]. But you said that Socrates is a man, which means that our assumption was false [second resolution step], so Socrates must be mortal.

12. Russell & Norvig Example

13. Reminder: Kowalski NF • Can reintroduce  to CNF, e.g. ¬A  ¬C  B becomes (A  C)  B • Kowalski normal form (A1… An)  (B1 … Bn) • Resolve in KNF using ‘KNF style’ rules • e.g. Binary resolution… AB, BC AC

14. R&N Example: Kowalski NF

15. R&N Example: Proof Tree

16. R&N Example: Prover9 Input

17. R&N Example: Prover9 Proof

18. Equality Axioms • is_pres(obama) and is_pres(b_obama) • will not unify (syntactically different) • unification algorithm does not allow this • Even if we add to the knowledge base: • obama = b_obama • Solution: add equality axioms to KB • X=X, X=YY=X, etc. • Special axiom for every predicate/function: • X = Y  P(X) = P(Y)

19. Equality & Demodulation • Alternative solution: rewrite with equalities • Demodulation inference rule X=Y, A[S] Subst(, A[Y]) • Two input clauses (one an equality X=Y) • Unify X with a subterm S of other • Apply unifier to clause with subterm Y (not S) • Also works unifying with Y and putting in X Unify(X, S) = 

20. Heuristic Strategies • Pure resolution search tends to be slow • For interesting problems • Many clauses in the initial knowledge base • Each step adds a new clause (which can be used) • Num. of possible resolution combinations explodes • Selection Heuristics • Intelligently choose which pair to resolve • Pruning Heuristics • Forbid certain pairs

21. Unit Preference Strategy • Prefer to resolve unit clauses • Contain only a single literal • Selection heuristic • Searching for smallest (empty) clause • Resolving with the unit clauses keeps small • Very effective early on for simple problems • Doesn’t reduce branching rate for medium problems

22. Set of Support Strategy • Distinguished subset of KB clauses • Set of support (SOS) clauses • Every step must involve SOS (pruning heuristic) • Must be careful not to lose completeness • Example SOS strategy: • Initial SOS is negated theorem • Add new clauses to SOS • Hence False will be deduced (strategy is complete) • Many provers use SOS, e.g. Prover9

23. Input Resolution Strategy • Special case of SOS strategy • SOS = clauses in the initial knowledge base • Clearly reduces search space • Every resolution must involve an original clause • So number of possible resolutions grows slowly • Not complete for first order logic • But complete for Horn-clauses, e.g. Prolog

24. Subsumption • Clause C subsumes clause D • if C is more ‘general’ (D is more specific) • Naive check for subsumption • Select C2, a subset of literals of C • Find Unify(C2, D) =  •  does not add anything to D (only renames vars) • Example: • p(george) ∨ q(X) subsumed by p(A) ∨ q(B) ∨ r(C) • Substitution: {A/george, X/B} • Second clause is more general

25. Subsumption Strategy • Check each new clause is not subsumed by KB • Complete strategy • Specific clauses can be inferred from general ones • So we can throw specific clauses away • Reduced search space still contains False • Can be inefficient • expense must be outweighed by the reduction in the search space

26. Applications: Axioms for Algebras • Bill McCune and Larry Wos • Argonne National Laboratories • FO resolution provers: EQP, Otter, Prover9 • Robbins Problem (axioms of Boolean algebras) • Stated 60+ years ago, mathematicians failed • 1996: EQP solved in 8 days in 1996 (+human work) • General application to algebraic axiomatisations • Generate possible axioms for algebras • Prove new axioms equivalent to old

27. Applications: Theory Formation • Simon’s HR system: Automated Theory Formation • Used in mathematical (and bioinformatics) domains • Theories = concepts, examples, conjectures, proofs • HR uses Otter to prove conjectures it makes • Effective in algebraic domains • See notes for anti-associative algebra results • Otter not so effective in number theory • Used as a ‘triviality’ filter (discard theorems it can prove) • Example conjectures made by HR (and proved by Simon): • Sum of divisors is prime → number of divisors is prime • Sum of divisors of a square is an odd number • Perfect numbers are pernicious [and many more…..]

28. Inductive Theorem Proving • Deduction by mathematical induction • Induction over many different structures • Allows reasoning about recursion/iteration • Useful for hardware/software verification • Don’t confuse inductive learning (next lecture)

29. Interactive Theorem Proving • Necessary to interact with humans in order to prove theorems of any difficulty • Mathematician’s assistant • Let a theorem prover do simple tasks while you develop a theory (e.g., Buchberger’s Theorema) • Guided theorem prover • User follows and guides computer proof attempt • Needs visualisation tools for proof trees

30. Higher Order Theorem Proving • Deduction in higher order logics • See lecture 4 • Allows more natural and succinct statements • Logics much less well-behaved • HOL theorem prover • Larry Paulson’s group in Cambridge • Has been used for verification tasks • E.g. verification of crytographic protocols • Uses induction and interactive control

31. Proof Planning • Initially Alan Bundy’s group in Edinburgh • Human proofs often follow a similar structure • Express this as a outline plan • Methods represent a patterns of deduction • Outline plan guides proof search • Results in specific plan for theorem • Critics deal with common problems • Particularly useful for inductive theorems • Proof of base case and step case follow pattern

32. Databases & Competitions • TPTP library (Sutcliffe & Suttner) • Thousands of Problems for Theorem Provers • Benchmarks for first order provers • HR is only non-human to add to this library • Annual CASC competition (Sutcliffe et al.) • Which is fastest/most accurate FO prover on planet? • Uses blind selection from the TPTP library • 2002-08 champion: Vampire (Voronkov & Riazonov)