1Relational ProofsComputational Logic Lecture 8Michael Genesereth Autumn 20072Logical EntailmentA set of premises logically entails a conclusion if andonly if every interpretation that satisfies the premisesalso satisfies the conclusion.23Propositional Interpretationsp q r0 0 00 0 10 1 00 0 11 0 01 0 11 1 01 1 1For a language with n constants, there are 2n interpretations.4Relational Interpretations |i| a b r{,} {}{,} {}{,} {}{,} {, }{,} {}{,} {}{,} {}{,} {, }{,} {}{,} {}{,} {}{,} {, } . . .Infinitely many interpretations.35Logical Entailment and ProvabilityGood News: If Δ logically entails ϕ, then there is afinite proof of ϕ from Δ. And vice versa.More Good News: If Δ logically entails ϕ, it ispossible to find such a proof in finite time.Sad News: If Δ does not logically entail ϕ, theprocess of finding a proof may run forever.Not So Bad News: In many cases, the process canbe stopped after finitely many steps.6Formal ProofsA formal proof of ϕ from Δ is a sequence ofsentences terminating in ϕ in which each item iseither:1. a premise (a member of Δ)2. an instance of an axiom schema3. the result of applying a rule of inference toearlier items in the sequence.47Old Rules of InferenceModus Ponens (MP) Modus Tolens (MT)And Introduction (AI) And Elimination (AE)ϕ⇒ψϕψϕ⇒ψ¬ψ¬ϕϕ∧ψϕψϕψϕ∧ψ8Universal GeneralizationRule of InferenceExamples: p(x) p(x) ⇒ q(x) ∀x.p(x) ∀x.(p(x) ⇒ q(x))€ ϕ∀ν.ϕ59Existential GeneralizationRule of InferenceExamples: p(a) p(a) ∨ q(a) ∃x.p(x) ∃x.(p(x) ∨ q(x))€ ϕ[τ]∃ν.ϕ[ν]10Idea for Universal Instantiation€ ∀ν.ϕϕ[ν←τ]Warning: This is not quite right.611Examples∀y.hates(jane,y)hates(jane,jill) y←jillhates(jane,mother(jane)) y←mother(jane)hates(jane,y) y←yhates(jane,z) y←z∀x.∃y.hates(x,y)∃y.hates(jane,y) x←jane∃y.hates(y,y) x←y Wrong!!12BoundingA term τ is bound by ν in ϕ if and only if τ contains avariable µ and there is some free occurrence of ν in ϕand that occurrence lies in the scope of a quantifier of µ.mother(x) is bound by y in ∃x.hates(x,y).Why? The term mother(x) contains a variable x. There is a free occurrence of y in ∃x.hates(x,y). That occurrence of y lies in scope of quantifier of x.713SubstitutabilityA term τ is substitutable for ν in ϕ if and only if it isnot bound by ν in ϕ.Some texts say “x is free for y in ϕ” instead of “x issubstitutable for y in ϕ”.mother(jane) is free for y in hates(jane,y).mother(x) is free for y in hates(jane,y).mother(x) is free for y in ∃z.hates(z,y).mother(x) is not free for y in ∃x.hates(x,y).mother(x) is free for y in (∀x.∀y.l(x,y) ∧ ∃z.h(z,y)).14InappropriatenessAn occurrence of a term τ is inappropriate for avariable ν in ϕ if and only if τ contains a variable µand there is some free occurrence of ν in ϕ that liesin the scope of a quantifier of µ.mother(x) is inappropriate for y in ∃x.hates(x,y).815SubstitutabilityA term τ is substitutable for ν in ϕ if and only if it isnot inappropriate with ν in ϕ.Some texts say “x is free for y in ϕ” instead of “x issubstitutable for y in ϕ”.mother(jane) is free for y in hates(jane,y).mother(x) is free for y in hates(jane,y).mother(x) is free for y in ∃z.hates(z,y).mother(x) is not free for y in ∃x.hates(x,y).mother(x) is free for y in (∀x.∀y.l(x,y) ∧ ∃z.h(z,y)).16Universal Instantiation€ ∀ν.ϕϕ[ν←τ]where τ is free for ν in ϕ917Existential Instantiation I€ ∃ν.ϕϕ[ν←σ]where ∃ν.ϕ contains no free variableswhere σ is a new object constant18Examples∃y.p(y)p(c)∃y.y*y=01*1=0 Wrong!∃y.y*y=xc*c=x Wrong!c*c=4c*c=61019Existential Instantiation II€ ∃ν.ϕϕ[ν←π(τ1,...,τn)]where τ1,...,τn are free in ∃ν.ϕwhere π is a new function constant20Examples∃y.y*y=xf(x)*f(x)=xf(4)*f(4)=4f(6)*f(6)=6∃y.y*y=xsqrt(x)*sqrt(x)=xlog(x)*log(x)=x Wrong!1121Formal ProofsA formal proof of ϕ from Δ is a sequence ofsentences terminating in ϕ in which each item iseither:1. a premise (a member of Δ)2. an instance of an axiom schema3. the result of applying a rule of inference toearlier items in the sequence.22ExampleEverybody loves somebody. Everybody loves alover. Show that Jack loves Jill.1. ∀x.∃y.loves(x, y) Premise2. ∀u.∀v.∀w.(loves(v, w) ⇒ loves(u, v)) Premise3. ∃y.loves( jill, y) UI :14. loves( jill,mike) EI : 35. ∀v.∀w.(loves(v, w) ⇒ loves( jack, v)) UI : 26. ∀w.(loves( jill, w) ⇒ loves( jack, jill)) UI : 57. loves( jill,mike) ⇒ loves( jack, jill) UI : 68. loves( jack, jill) MP : 7, 41223Harry and RalphEvery horse can outrun every dog. Some greyhounds canoutrun every rabbit. Harry is a horse. Ralph is a rabbit. CanHarry outrun Ralph?1. ∀x.∀y.( h(x) ∧ d(y) ⇒ f (x, y)) Premise2. ∃y.(g(y) ∧ ∀z.(r(z) ⇒ f (y, z))) Premise3. ∀y.( g(y) ⇒ d(y)) Premise4. ∀x.∀y.∀z.( f (x, y) ∧ f (y,z) ⇒ f (x, z)) Premise5. h(harry) Premise6. r(ralph) Premise24Harry and Ralph (continued)€ 7. g(greg)∧ ∀z.(r(z) ⇒ f (greg, z)) EI : 28. g(greg) AE : 79. ∀z.(r(z) ⇒ f (greg, z)) AE : 710. r(ralph) ⇒ f (greg,ralph) UI : 911. f (greg, ralph) MP :6,1012. g(greg) ⇒ d(greg) UI : 313. d(greg) MP : 12, 814. ∀y.(h(harry)∧ d(y) ⇒ f (harry, y)) UI :115. h(harry)∧ d(greg) ⇒ f (harry, greg) UI :1416. h(harry)∧ d(greg) AI : 5,1317. f (harry, greg) MP : 15,161325Harry and Ralph (continued)18. ∀y.∀z.( f (harry, y) ∧ f (y, z) ⇒ f (harry, z)) UI : 419. ∀z.( f (harry, greg) ∧ f (greg, z) ⇒ f (harry, z)) UI :1820. f (harry, greg) ∧ f (greg, ralph) ⇒ f (harry, ralph)) UI : 1921. f (harry, greg) ∧ f (greg, ralph) AI :17,1122. f (harry, ralph) MP : 20, 2126Mendelson LogicMendelson Logic is that subset of Relational Logic inwhich there are only two operators, viz. ¬ and ⇒, andone quantifier, viz. ∀. Fortunately, all sentences inRelational Logic can be reduced to logicallyequivalent sentences with these operators by applyingthe following rules.(ψ ⇔ ϕ) → ((ϕ ⇒ ψ) ∧ (ψ ⇒ ϕ))(ϕ ⇐ ψ) → (ψ ⇒ ϕ) (ψ ∧ ϕ) → ¬(¬ϕ ⇒ ψ) (ψ ∨ ϕ) → (¬ϕ ⇒
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