Finite Model Property and Bisimulation for LFD

Abstract

Recently, Baltag & van Benthem introduced a decidable logic of functional dependence (LFD) that extends the logic of Cylindrical Relativized Set Algebras (CRS) with atomic local dependence statements. Its semantics can be given in terms of generalised assignment models or their modal counterparts, hence the logic is both a first-order and a modal logic. We show that LFD has the finite model property (FMP) using Herwig's theorem on extending partial isomorphisms, and prove a bisimulation invariance theorem characterizing LFD as a fragment of first-order logic.

Local Dependence and Guarding

“…As a warming up, we reprove a known result, namely the finite model property for GF [7], Recall that a logic has the finite model property (FMP) if every satisfiable formula has a finite model. It has been long known that GF has the FMP [7], and FMP was only recently shown to hold for LFD [12]. Therefore, the following transfer fact does not give us a new result, but it does provide an illustration of the value of our translations.…”

“…By corollary 3.6, since ϕ is satisfiable on standard models, it follows that τ ρ (ϕ) is satisfiable on dependence models, for some ρ : free(ϕ) → V LF D (for V LF D sufficiently large). By the finite model property of LFD [12], there is a finite dependence model M satisfying τ ρ (ϕ) at some s ∈ A. We can take M to be distinguished by Proposition 2.9 (which preserves finiteness).…”

“…We can take M to be distinguished by Proposition 2.9 (which preserves finiteness). In fact, inspection of the proof in [12] shows that its construction via Herwig's theorem already ensures distinguishedness. By Theorem 3.4, then, G(M) is a finite model of ϕ.…”

“…Given its similarities with modal logic, LFD comes with a notion of dependence-bisimulations, which leaves the truth values of formulas invariant [9,12]. For each set of variables V , dependence model M, and assignment s, the dependence-closure…”

“…• We show how these results allow for transfer of important properties between LFD and GF. For instance, the finite model property for GF [7], can be derived from that for LFD, established in [12]. Conversely, the decidability of LFD follows from that of GF.…”