Decidability Issues for Two-Variable Logics with Several Linear Orders

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, that the finite satisfiability problem for FO2 with two linear orders and with no additional non-unary relations is decidable. We observe that our proof can be adapted to some interesting fragments of FO2, in particular it works for the two-variable guarded fragment, GF2, even if the order relations are used only as guards. Finally, we show that GF2 with an arbitrary number of linear orders which can be used only as guards becomes decidable if except linear orders only unary relations are allowed

Satisfiability for two-variable logic with two successor relations on finite linear orders

We study the finitary 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

Two-variable Logic with Counting and a Linear Order

We study the finite satisfiability problem for the two-variable fragment of first-order logic extended with counting quantifiers (C2) 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 relation. Surprisingly, the complexity of the problem explodes when we add one binary symbol more: C2 with one linear order and in the presence of other binary predicate symbols is equivalent, under elementary reductions, to the emptiness problem for multicounter automata

Adding Transitivity and Counting to the Fluted Fragment

We study the impact of adding both counting quantifiers and a single transitive relation to the fluted fragment - a fragment of first-order logic originating in the work of W.V.O. Quine. The resulting formalism can be viewed as a multi-variable, non-guarded extension of certain systems of description logic featuring number restrictions and transitive roles, but lacking role-inverses. We establish the finite model property for our logic, and show that the satisfiability problem for its k-variable sub-fragment is in (k+1)-NExpTime. We also derive ExpSpace-hardness of the satisfiability problem for the two-variable, fluted fragment with one transitive relation (but without counting quantifiers), and prove that, when a second transitive relation is allowed, both the satisfiability and the finite satisfiability problems for the two-variable fluted fragment with counting quantifiers become undecidable

Register Automata with Extrema Constraints, and an Application to Two-Variable Logic

We introduce a model of register automata over infinite trees with extrema constraints. Such an automaton can store elements of a linearly ordered domain in its registers, and can compare those values to the suprema and infima of register values in subtrees. We show that the emptiness problem for these automata is decidable. As an application, we prove decidability of the countable satisfiability problem for two-variable logic in the presence of a tree order, a linear order, and arbitrary atoms that are MSO definable from the tree order. As a consequence, the satisfiability problem for two-variable logic with arbitrary predicates, two of them interpreted by linear orders, is decidable