We show that the satisfiability and the finite satisfiability problems for two-variable logic, FO2, over the class of structures with three linear orders, are undecidable. This sharpens an earlier result that FO2 with eight linear orders is undecidable. The theorem holds for a restricted case in which linear orders are the only non-unary relations. Recently, a contrasting result has been shown,...

We study the finite satisfiability problem for first order logic with two variables and two binary relations, corresponding to the induced successor relations of two finite linear orders. We show that the problem is decidable in NExpTime.

We consider two-variable first-order logic on finite words with a fixed number of quantifier alternations. We show that all languages with a neutral letter definable using the order and finite-degree predicates are also definable with the order predicate only. From this result we derive the separation of the alternation hierarchy of two-variable logic on this signature.

We consider a modified version of the situation calculus built using a two-variable fragment of the first-order logic extended with counting quantifiers. We mention several additional groups of axioms that need to be introduced to capture taxonomic reasoning. We show that the regression operator in this framework can be defined similarly to regression in the Reiter’s version of the situation ca...

Following recent works connecting two-variable logic to circuits and monoids, we establish, for numerical predicate sets P satisfying a certain closure property, a one-to-one correspondence between FO[<, P]-uniform linear circuits, two-variable formulas with P predicates, and weak block products of monoids. In particular, we consider the case of linear TC, majority quantifiers, and finitely typ...

The finite satisfiability problem of two-variable logic extended by a linear order successor and a preorder successor is shown to be undecidable. The decidability of the finite satisfiability problem of two-variable logic extended by orders and preorders as well as their corresponding successor relations has been currently investigated by several papers [2, 5, 3]. This short note extends [3] wh...

Fragments of first-order logic over words can often be characterized in terms of finite monoids, and identities of omega-terms are an effective mechanism for specifying classes of monoids. Huschenbett and the first author have shown how to use infinite Ehrenfeucht-Fra¨ıssé games on linear orders for showing that some given fragment satisfies an identity of omega-terms (STACS 2014). After revisi...

We study the finite satisfiability problem for the two-variable fragment of first-order logic extended with counting quantifiers (C) and interpreted over linearly ordered structures. We show that the problem is undecidable in the case of two linear orders (in the presence of two other binary symbols). In the case of one linear order it is NExpTime-complete, even in the presence of the successor...

The data-complexity of both satisfiability and finite satisfiability for the two-variable fragment with counting is NP-complete; the data-complexity of both query-answering and finite query-answering for the two-variable guarded fragment with counting is co-NP-complete.

We investigate extensions of dependence logic with generalized quantifiers. We also introduce and investigate the notion of a generalized atom. We define a system of semantics that can accommodate variants of dependence logic, possibly extended with generalized quantifiers and generalized atoms, under the same umbrella framework. The semantics is based on pairs of teams, or double teams. We als...