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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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. ...
View Full Document

Ask a homework question - tutors are online