Ordered ResolutionComputational LogicLecture 13Michael GeneserethSpring 20022Unordered ResolutionOrdinary resolution is unordered. The rule can be applied toany literal in clause.The more opportunities for resolution, the larger the searchspace.{ϕ1,...,χ,...,ϕm}{ψ1,...,¬χ,...,ψn}{ϕ1,...,ϕm,ψ1,...,ψn}3ExampleNegated Goal{¬p3,¬q3,¬r3}Premises{p3,¬p2} {q3,¬q2} {r3,¬r2}{p2,¬p1} {q2,¬q1} {r2,¬r1}{p1} {q1} {r1}4Large Search Space{¬p3,¬q3,¬r3} {¬p2,¬q3,¬r3} {¬p1,¬q3,¬r3}{¬p3,¬q3,¬r2} {¬p2,¬q3,¬r2} {¬p1,¬q3,¬r2}{¬p3,¬q3,¬r1} {¬p2,¬q3,¬r1} {¬p1,¬q3,¬r1}{¬p3,¬q2,¬r3} {¬p2,¬q2,¬r3} {¬p1,¬q2,¬r3}{¬p3,¬q2,¬r2}. {¬p2,¬q2,¬r2} {¬p1,¬q2,¬r2}{¬p3,¬q2,¬r1} {¬p2,¬q2,¬r1} {¬p1,¬q2,¬r1}{¬p3,¬q1,¬r3} {¬p2,¬q1,¬r3} {¬p1,¬q1,¬r3}{¬p3,¬q1,¬r2} {¬p2,¬q1,¬r2} {¬p1,¬q1,¬r2}{¬p3,¬q1,¬r1} {¬p2,¬q1,¬r1} {¬p1,¬q1,¬r1}5To derive empty clause, every literal must be eliminated.Idea: work on first literal till it is gone before starting to workon other literals.Intuition for Ordered Resolution{ ¬p3, ¬q3,¬r3}| | |¬p2¬q2¬r2| | |¬p1¬q1¬r16ExampleA clause is a set of literals.{p,¬q,¬r}A chain is a sequence of literals〈p,¬q,¬r〉7Ordered Resolution〈ϕ,ϕ1,...,ϕm〉〈¬ψ,ψ1,...,ψn〉〈ϕ1,...,ϕm,ψ1,...,ψn〉σwhere σ = mgu(ϕ,ψ )8Smaller Search Space10. 〈¬p3,¬q3,¬r3〉11. 〈¬p2,¬q3,¬r3〉12. 〈¬p1,¬q3,¬r3〉13. 〈¬q3,¬r3〉14. 〈¬q2,¬r3〉15. 〈¬q1,¬r3〉16. 〈¬r3〉17. 〈¬r2〉18. 〈¬r1〉19. 〈 〉1. 〈p3,¬p2〉2. 〈p2,¬p1〉3. 〈p1〉4. 〈q3,¬q2〉5. 〈q2,¬q1〉6. 〈q1〉7. 〈r3,¬r2〉8. 〈r2,¬r1〉9. 〈r1〉9Does Not Play Well With OthersOrdered Resolution + Set of Support is not complete.Two Complete Answers and One Partial Answer:Semi-Ordered ResolutionContrapositivesHorn Restriction1. 〈p,q〉Premise2. 〈¬p,q〉Premise3. 〈¬q〉 Goal10Semi-Ordered Resolution〈ϕ1,...,ϕ,...,ϕm〉〈ψ,ψ1,...,ψn〉〈ϕ1,...,ϕm,ψ1,...,ψn〉σwhere σ = mgu(¬ϕ,ψ)11Example1. 〈p,q〉Premise2. 〈¬p,q〉Premise3. 〈¬q〉 Goal4. 〈p〉 1,35. 〈¬p〉 2,36. 〈 〉 4,512ContrapositivesA contrapositive of a chain is a permutation in which adifferent literal is placed at the front.Chain: 〈p,¬q,¬r〉Contrapositive: 〈¬q,p,¬r〉Contrapositive: 〈¬r,p,¬q〉The contrapositives of a chain are logically equivalent tothe original chain.13Example1. 〈p,q〉Premise2. 〈q, p〉Premise3. 〈¬p,q〉Premise4. 〈q,¬p〉Premise5. 〈¬q〉 Goal6. 〈p〉 2,57. 〈¬p〉 4,58. 〈 〉 6,714Horn Clauses and Horn ChainsA Horn clause is a clause containing at most one positiveliteral.A Horn chain is a chain containing at most one positiveliteral.Example: 〈r,¬p,¬q〉Example: 〈¬p,¬q,¬r〉Example: 〈p〉Non-Example: 〈q,r,¬p〉In Horn chains, positive literals are usually written first.15CompletenessMetatheorem: If ∆ consists entirely of Horn chains and allchains are ordered with positive literals first, then there isa resolution refutation of ∆ if and only if there is a set ofsupport refutation using ordered resolution.16Example1. 〈m〉Premise2. 〈p,¬m〉Premise3. 〈q,¬m〉Premise4. 〈r,¬p,¬q〉Premise5. 〈¬r〉 Goal6. 〈¬p,¬q〉 4,57. 〈¬m,¬q〉 2,68. 〈¬q〉 1,79. 〈¬m〉 3,810. 〈 〉 1,917Input Resolution1. {p,q}Premise2. {p,¬q}Premise3. {¬p,q}Premise4. {¬p,¬q} Goal5. {p} 1,26. {q} 1,37. {p,¬p} 2,38. {q,¬q} 2,39. {¬q} 2,410. {¬p} 3,418Linear Resolution1. 〈p,q〉Premise2. 〈q, p〉Premise3. 〈¬p,q〉Premise4. 〈q,¬p〉Premise5. 〈¬q〉 Goal6. 〈p〉 2,57. 〈¬p〉 4,58. 〈 〉 6,71. 〈p,q〉Premise2. 〈q, p〉Premise3. 〈¬p,q〉Premise4. 〈q,¬p〉Premise5. 〈¬q〉 Goal6. 〈p〉 2,57. 〈q〉 3,68. 〈 〉 5,7Nonlinear Linear19Model EliminationModel Elimination is a variant of Ordered Resolution thatincorporates the Linearity Restriction in the definition of therules of inference.Using Model Elimination alone, it is possible to build atheorem prover that is sound and complete for all of RelationalLogic.Moreover, it works with the Set of Support strategy and theInput Restriction.20Normal and Reduced LiteralsNormal Literals:p¬qReduced Literals:[p][¬q]Chains:〈p,¬q,[p],r〉21Model Elimination RulesReductionCancellationDropping〈ϕ,ϕ1,...,ϕm〉〈ψ,ψ1,...,ψn〉〈ϕ1,...,ϕm,[ψ],ψ1,...,ψn〉σwhere σ = mgu(¬ϕ,ψ)〈[ϕ],ϕ1,...,ϕm〉〈ϕ1,...,ϕm〉〈ϕ,ϕ1,...,ϕm,[ψ],ψ1,...,ψn〉〈ϕ1,...,ϕm,[ψ],ψ1,...,ψn〉σwhere σ = mgu(¬ϕ,ψ)22Example1. 〈p〉Premise2. 〈q〉Premise3. 〈r,¬p,¬q〉Premise4. 〈¬r〉 Goal5. 〈¬p,¬q,[¬r]〉 Reduction :3,46. 〈[¬p],¬q,[¬r]〉 Reduction :1,57. 〈¬q,[¬r]〉 Dropping: 68. 〈[¬q],[¬r]〉 Reduction :2,79. 〈[¬r]〉 Dropping :810. 〈 〉 Dropping : 923Example1. 〈 p,q〉 Premise2. 〈 p,¬q〉 Premise3. 〈¬p,q〉 Premise4. 〈¬p,¬q〉 Goal5. 〈q,[¬p],¬q〉 Reduction :1,46. 〈 p,[q],[¬p],¬q〉 Reduction :2,57. 〈[q], [¬p],¬q〉Cancellation: 68. 〈[¬p],¬q〉 Dropping :79. 〈¬q〉 Dropping :810. 〈 p,[¬q]〉 Reduction :1,911. 〈q,[p],[¬q]〉 Reduction :3,1012. 〈[p],[¬q]〉Cancellation:1113. 〈[¬q]〉 Dropping :1214. 〈 〉 Dropping :1324No Forward Cancellation1. 〈p,q〉Premise2. 〈¬q,¬r〉Premise3. 〈r〉Premise4. 〈¬p,¬q〉 Goal5. 〈q,[¬p],¬q〉 Reduction :1,46. 〈¬r,[q],[¬p],¬q〉 Reduction :2,57. 〈[q],[¬p],¬q〉 Reduction :3,68. 〈[q],[¬p]〉 Cancellation :7 Wrong!!9. 〈[¬p]〉 Dropping :810. 〈 〉 Dropping : 925SummaryOrdered Resolution with FactoringComplete without strategiesIncompatible with Set of SupportIncompatible with Input RestrictionSemi-Ordered Resolution and/or ContrapositivesCompleteWorks with Set of SupportIncompatible with Input RestrictionModel EliminationCompleteWorks with Set of SupportWorks with Input
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