Finite Satisfiability of the Two-Variable Guarded Fragment with Transitive Guards and Related Variants

TL;DR

This paper investigates the finite satisfiability problem for extensions of the two-variable guarded fragment with various transitive relations, establishing complexity bounds and model construction techniques.

Contribution

It identifies which guarded fragment extensions retain the finite model property and provides tight complexity bounds using novel multidimensional grid constructions.

Findings

GF2 with equivalence guards (without equality) has the finite model property.

Finite models for other fragments are at most doubly exponential in size.

Finite satisfiability is NExpTime-complete for GF2 with equivalence guards and 2ExpTime-complete for GF2 with transitive guards.

Abstract

We consider extensions of the two-variable guarded fragment, GF2, where distinguished binary predicates that occur only in guards are required to be interpreted in a special way (as transitive relations, equivalence relations, pre-orders or partial orders). We prove that the only fragment that retains the finite (exponential) model property is GF2 with equivalence guards without equality. For remaining fragments we show that the size of a minimal finite model is at most doubly exponential. To obtain the result we invent a strategy of building finite models that are formed from a number of multidimensional grids placed over a cylindrical surface. The construction yields a 2NExpTime-upper bound on the complexity of the finite satisfiability problem for these fragments. We improve the bounds and obtain optimal ones for all the fragments considered, in particular NExpTime for GF2 with equivalence guards, and 2ExpTime for GF2 with transitive guards. To obtain our results we essentially use some results from integer programming.