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

摘要

We consider extensions of the two-variable guarded fragment, GF(2), where distinguished binary predicates that occur only in guards are required to be interpreted in a special way (as transitive relations, equivalence relations, preorders, or partial orders). We prove that the only fragment that retains the finite (exponential) model property is GF(2) 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 2-NExpTime 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 GF(2) with equivalence guards, and 2-ExpTime for GF(2) with transitive guards. To obtain our results, we essentially use some results from integer programming.