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The satisfiability and finite satisfiability problems for the two-variable fragment of firstorder logic with counting quantifiers are both in NEXPTIME, even when counting quantifiers are coded succinctly.Key words: two-variable fragment, counting quantifiers, logic, complexity BackgroundThe two-variable fragment with counting quantifiers, here denoted C2, is the set of function-free, first-order formulas containing at most two variables, but with the counting quantifiers 3c and 3=c (for every C > 0) allowed. The satisfiability problem, Sat-C2, is the problem of deciding, for a given formula of C2, whether has a model; the finite satisfiability problem, Fin-Sat-C2, is the problem of deciding, for a given formula of C2, whether

We consider the satisfiability and finite satisfiability problems for extensions of the two-variable fragment of firstorder logic in which an equivalence closure operator can be applied to a fixed number of binary predicates. We show that the satisfiability problem for two-variable, first-order logic with equivalence closure applied to two binary predicates is in 2NEXPTIME, and we obtain a matching lower bound by showing that the satisfiability problem for two-variable first-order logic in the presence of two equivalence relations is 2NEXPTIME-hard. The logics in question lack the finite model property; however, we show that the same complexity bounds hold for the corresponding finite satisfiability problems. We further show that the satisfiability (=finite satisfiability) problem for the twovariable fragment of first-order logic with equivalence closure applied to a single binary predicate is NEXPTIME-complete.

Graded modal logic is the formal language obtained from ordinary modal logic by endowing its modal operators with cardinality constraints. Under the familiar possibleworlds semantics, these augmented modal operators receive interpretations such as "It is true at no fewer than 15 accessible worlds that . . . ", or "It is true at no more than 2 accessible worlds that . . . ". We investigate the complexity of satisfiability for this language over some familiar classes of frames. This problem is more challenging than its ordinary modal logic counterpart-especially in the case of transitive frames, where graded modal logic lacks the tree-model property. We obtain tight complexity bounds for the problem of determining the satisfiability of a given graded modal logic formula over the classes of frames characterized by any combination of reflexivity, seriality, symmetry, transitivity and the Euclidean property.

The Aristotelian syllogistic cannot account for the validity of many inferences involving relational facts. In this paper, we investigate the prospects for providing a relational syllogistic. We identify several fragments based on (a) whether negation is permitted on all nouns, including those in the subject of a sentence; and (b) whether the subject noun phrase may contain a relative clause. The logics we present are extensions of the classical syllogistic, and we pay special attention to the question of whether reductio ad absurdum is needed. Thus our main goal is to derive results on the existence (or non-existence) of syllogistic proof systems for relational fragments. We also determine the computational complexity of all our fragments.