Lecture_19_09 - Lecture 19 Lecture 19 FOL(without function...

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Unformatted text preview: Lecture 19 Lecture 19 FOL (without function symbols) FOL (without function symbols) Signature: a set R of relation symbols, a set C of constant symbols. We assume that each relation symbol is equipped with a natural number that indicates the appropriate number of arguments for that symbol. Assuming a set V of variable symbols, the first­order language over the signature is given inductively as follows: FOL (without function symbols) FOL (without function symbols) 1. 2. 3. Terms: every constant symbol is a term, and every variable symbol is a term. Formulas: If t and t’ are terms, then t ≈ t’ is a formula. If R is a relation symbol that takes n arguments, and t1, …, tn are terms, then R(t1, …, tn) is a formula. More complicated formulas are built up from the following inductive clauses: FOL (without function symbols) FOL (without function symbols) Inductive clauses (one for each connective): If ϕ ∈L, then ¬ ϕ ∈L If ϕ ,ψ∈L, then ϕ ∧ L ψ∈ If ϕ ∈L and x ∈V, then ∀x ϕ ∈L If ϕ ∈L and x ∈V, then ∃ x ϕ ∈L Interpretations Interpretations An interpretation M consists of the following ingredients: a nonempty set D, the “domain” or “universe” of M, an element cM in D for each constant symbol c, and a set RM ⊆ Dn for each relation symbol R of arity n. 1. 2. 3. Term assignments Term assignments Given interpretation M, a term assignment g is a function from the set of terms to D (the domain of M) such that g(c) = cM for every c ∈ C. If g is a term assignment and a ∈ D and x ∈ V, then g(x,a) is the term assignment that maps x to a, but agrees with g for all other terms. Semantics Semantics Atomic M ╞ (t1 ≈ tn)[g] iff g(t1) = g(tn) M ╞ P(t1,…,tn)[g] iff (g(t1),…,g(tn)) ∈ PM P(t Negation M ╞ ¬ϕ [g] iff not M ╞ ϕ [g] ¬ϕ Conjunction M ╞ (ϕ∧ )[g] iff M╞ ϕ [g] and M╞ ψ[g] ψ Semantics Semantics Universal quantification M ╞∀xϕ [g] iff M╞ ϕ [g(x,a)] for all a ∈ D iff M Existential quantification M ╞∃ xϕ [g] iff M╞ ϕ [g(x,a)] for some a ∈ D iff M Example Example Signature: H (unary), R (binary), c (constant) Interpretation: D = {0,1,2,3} HM = {1,2} RM = {(0,1),(2,3),(3,3)} cM = 3 M ╞ ∀x(R(x,c) → H(x))[g]? x(R(x,c) Example Example M ╞ ∀x(R(x,c) → H(x))[g] iff, x(R(x,c) for every a∈D, for every a M ╞ (R(x,c) → H(x))[g(x,a)] iff, M (R(x,c) for every a∈D, M ╞ R(x,c)[g(x,a)] implies M ╞ H(x)[g(x,a)] M iff, for every a ∈ D, (a,3) ∈ RM implies a ∈ HM Definability Definability If M is an interpretation and ϕ is a formula with x as its only free variable, then ϕ defines the following subset of D: {a : M ╞ ϕ [g(x,a)]} The idea extends to definable relations over the domain. ...
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