Abstract A complete classification of the complexity of the local and global satisfiability problems for graded modal language over traditional classes of frames has already been established. By “traditional” classes of frames, we mean those characterized by any positive combination of reflexivity, seriality, symmetry, transitivity, and the Euclidean property. In this paper, we fill the gaps remaining in an analogous classification of the graded modal language with graded converse modalities. In particular, we show its NExpTime-completeness over the class of Euclidean frames, demonstrating this way that over this class the considered language is harder than the language without graded modalities or without converse modalities. We also consider its variation disallowing graded converse modalities, but still admitting basic converse modalities. Our most important result for this variation is confirming an earlier conjecture that it is decidable over transitive frames. This contrasts with the undecidability of the language with graded converse modalities.

We consider the satisfiability problem for the two-variable fragment of first-order logic extended with counting quantifiers, interpreted over finite words with data, denoted here with C²[≤ , succ, ∼, π_bin]. In our scenario, we allow for using arbitrary many uninterpreted binary predicates from π_bin, two navigational predicates ≤ and succ over word positions as well as a data-equality predicate ∼. We prove that the obtained logic is undecidable, which contrasts with the decidability of the logic without counting by Montanari, Pazzaglia and Sala [Angelo Montanari et al., 2016]. We supplement our results with decidability for several sub-fragments of C²[≤ , succ, ∼, π_bin], e.g. without binary predicates, without successor succ, or under the assumption that the total number of positions carrying the same data value in a data-word is bounded by an a priori given constant.

We consider an extension of two-variable, first-order logic with counting quantifiers and arbitrarily many unary and binary predicates, in which one distinguished predicate is interpreted as the mother-daughter relation in an unranked forest. We show that both the finite satisfiability and the general satisfiability problems for the extended logic are decidable in NExpTime. We also show that the decision procedure for finite satisfiability can be extended to the logic where two distinguished predicates are interpreted as the mother-daughter relations in two independent forests.

We study variants of the satisfiability problem of elementary modal logics, i.e., modal logic considered over first-order definable classes of frames. The standard semantics of modal logic allows infinite structures, but often practical applications require to restrict our attention to finite structures. A number of decidability and undecidability results for the elementary modal logics were proved separately for general satisfiability and finite satisfiability. In this paper we justify that the results for both kinds of the satisfiability problem must be shown separately – we prove that there is a universal first-order formula that defines an elementary modal logic with decidable general satisfiability problem, but undecidable finite satisfiability problem, and, the other way round, that there is a universal first-order formula that defines an elementary modal logic with decidable finite satisfiability problem, but undecidable general satisfiability problem.

We consider the two-variable logic with counting quantifiers (C 2 ) interpreted over finite structures that contain two forests of ranked trees. This logic is strictly more expressive than standard C 2 and it is no longer a fragment of first-order logic. In particular, it can express that a structure is a ranked tree, a cycle, or a connected graph of bounded degree. It is also strictly more expressive than first-order logic with two variables and two successor relations of two finite linear orders. We present a decision procedure for the satisfiability problem for this logic. The procedure runs in NE xp T ime , which is optimal since the satisfiability problem for plain C 2 is NE xp T ime -complete.

We study the finite satisfiability problem for the two-variable fragment of the 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 presence of two other binary symbols). In the case of one linear order it is NEXPTIME-complete, even in 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 its successor, in presence of other binary predicate symbols, is decidable, but it is as expressive (and as complex) as Vector Addition Systems.

We consider the two-variable logic with counting quantifiers (C2) interpreted over finite structures that contain two forests of ranked trees. This logic is strictly more expressive than standard C2and it is no longer a fragment of the first order logic. In particular, it can express that a structure is a ranked tree, a cycle or a connected graph of bounded degree. It is also strictly more expressive than the first-order logic with two variables and two successor relations of two finite linear orders. We give a decision procedure for the satisfiability problem for this logic. The procedure runs in NEXPTIME, which is optimal since the satisfiability problem for plain C2is NEXPTIME-complete.