Translating LTL Formulas to First-Order Logic & Interpreting Relational Models in CS, Assignments of Computer Science

Download Translating LTL Formulas to First-Order Logic & Interpreting Relational Models in CS and more Assignments Computer Science in PDF only on Docsity! CS 598mp: Fall 2005: Homework #3 Due on Fri 14 Oct Hand over in class or to Colin Robertson at 3229 SC Problem 1. (LTL → FO) Consider LTL over infinite words, over the set of proposition P. Let Σ = 2P . For any LTL formula ϕ over P, show that there is a formula ψ(x) (over exactly one free variable x) in first-order logic over Σ-labelled ω-words such that: • for any word α ∈ Σω and i ∈ N, α, i |= ϕ iff α |=I ψ(x) where I(x) = i. Problem 2. (Interpretations) Let M1 = (U1, R1, . . . , Rk,=) and M2 = (U2, S1, . . . , Sl,=) be two relational models with equality, where each Ri and each Si are binary relations. Consider an interpretation of M2 in M1, 〈f, ϕf , ψ1, . . . , ψl〉 given as follows: • f : U2 −→ U1 is an injective map • ϕf (x) is an MSO formula over M1 with free variable x which evaluates to true only for elements in U1 which are in the range of f (i.e. only for elements in U1 such that there is an element in U2 which gets mapped to it by f). • Each ψi(x, y) is an MSO formula over M1 such that for any interpretation of x and y over elements in the range of f , the formula evaluates to true if and only if the corresponding elements in U2 satisfies Si. In other words, if v1, v2 ∈ U2, f(v1) = u1 and f(v2) = u2, then 〈u1, u2〉 satisfies ψi iff 〈v1, v2〉 satisfies Si. Show that there is an effective translation h that maps MSO sentences over M2 to MSO sentences over M1 such that for any MSO sentence ϕ over M2, M2 |= ϕ if and only if M1 |= h(ϕ). [Hence, if M1 has a decidable MSO theory, then so will M2.]