6 - 6-1Semantics of Predicate Logic(cont’d)I Evaluation...

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Unformatted text preview: 6-1Semantics of Predicate Logic (cont’d)I: Evaluation of TermsWe will define avalue functionValA:Tm(Σ)×State(A)→AWe also write[[t]]Aσ=dfValA(t, σ)∈Aso that[[t]]A:State(A)→A.The def. of [[t]]Ais bystructural induction ont:•[[x]]Aσ=σ(x)•[[c]]Aσ=cA•[[f(t1, . . . , tn)]]Aσ=fA([[t1]]Aσ, . . . ,[[tn]]Aσ)(n >0)Example. Fort≡suc(suc(0) +suc(x)),[[t]]N1σ=Note. S&A uses the notation ‘valA,σ(t)’ forValA(t, σ).Definition. ForM⊆Var,σ≈σ′(relM)orσ≈Mσ′(σagrees withσ′onM) if and only ifσ↾M=σ′↾Mi.e.,for allx∈M,σ(x) =σ′(x).Lemma.For fixedM,≈Mis an equivalence relation onState(A).JZ CAS701 F096-2Theorem 1 (Coincidence Lemma for terms).σ≈σ′(relFV(t)) =⇒[[t]]Aσ= [[t]]Aσ′.Proof:Structural induction ont.squareCorollary.Iftis closed then[[t]]Aσis independent ofσ.Notation. Hence fortclosed, write [[t]]AσastA,calledvalA(t) on p.5-4.More generally: supposeFV(t)⊆ {x1, . . . , xn}Putt≡t(x1, . . . , xn).Then [[t]]Aσdepends only on the values ofσatx1, . . . , xn.So for anya1, . . . , an∈A, writetA(a1, . . . , an)to mean: [[t]]Aσfor anyσsuch thatσ(xi) =aifori= 1, . . . , n.Theorem 2 (Substitution Lemma for terms).[[r(x/t)]]Aσ= [[r]]Aσ{x/a}wherea= [[t]]Aσ.Proof:Structural induction onr.squareExGive the proof....
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6 - 6-1Semantics of Predicate Logic(cont’d)I Evaluation...

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