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We study a variant ATL FB of the alternating-time temporal logic ATL with a non-standard, ‘finitely bounded’ semantics (FBS). FBS was originally defined as a game-theoretic semantics where players must commit to time limits when attempting to verify eventuality (respectively, to falsify safety) formulae. It turns out that FBS has a natural corresponding compositional semantics that essentially evaluates formulae only on finite initial segments of paths and imposes uniform bounds on all plays for the fulfilment of eventualities. The resulting version ATL FB differs in some essential features from the standard ATL, as it no longer has the finite model property, though the two logics are equivalent on finite models. We develop two tableau systems for ATL FB . The first one deals with infinite sets of formulae and may run in a transfinite sequence of steps, whereas the second one deals only with finite sets of formulae in an extended language allowing explicit symbolic indication of time limits in formulae. We prove soundness and completeness of the infinitary tableau system and prove that it is equivalent to the finitary one. We also show that the finitary tableau system provides an exponential-time decision procedure for the satisfiability problem of ATL FB and thus establishes its EXPTIME-completeness. Furthermore, we present an infinitary axiomatization for ATL FB and prove its soundness and completeness.

We investigate the question of how much hybrid machinery can be added to the interval neighbourhood logic PNL and its metric extension MPNL without losing the decidability of their satisfiability problem in N. In particular, we consider the natural hybrid extension of MPNL obtained by adding binders on integer variables ranging over lengths of intervals, thus enabling storage of the length of the current interval undecidable, which is somewhat surprising, being in contrast with the decidability of MPNL, which can be seen as a hybrid language with length constraints only involving constants over interval lengths. These results show that MPNL itself is, in this sense, a maximal decidable (weakly) hybrid extension of PNL.

In (CON 06b) we introduced the algorithmSQEMA for computing first-order equiv- alents and proving canonicity of modal formulae, and thus es tablished a very general cor- respondence and canonical completeness result. SQEMA is based on transformation rules, the most important of which employs a modal version of a resul t by Ackermann that enables elimination of an existentially quantified predicate variable in a formula, provided a certain negative polarity condition on that variable is satisfied. I this paper we develop several exten- sions of SQEMA where that syntactic condition is replaced by a semantic one , viz. downward monotonicity. For the first, and most general, extensionSemSQEMA we prove correctness for a large class of modal formulae containing an extension o the Sahlqvist formulae, defined by replacing polarity with monotonicity. By employing a spe cial modal version of Lyndon's monotonicity theorem and imposing additional requirement s on the Ackermann rule we obtain restricted versions ofSemSQEMA which guarantee canonicity, too.

We introduce special pseudo-models for the interval logic of proper subintervals over dense linear orderings. We prove finite model property with respect to such pseudo-models, and using that result we develop a decision procedure based on a sound, complete, and terminating tableau for that logic. The case of proper subintervals is essentially more complicated than the case of strict subintervals, for which we developed a similar tableau-based decision procedure in a recent work.

We generalize and extend the class of Sahlqvist formulae in arbitrary polyadic modal languages, to the class of so called inductive formulae . To introduce them we use a representation of modal polyadic languages in a combinatorial style and thus, in particular, develop what we believe to be a better syntactic approach to elementary canonical formulae altogether. By generalizing the method of minimal valuations a la Sahlqvist–van Benthem and the topological approach of Sambin and Vaccaro we prove that all inductive formulae are elementary canonical and thus extend Sahlqvist’s theorem over them. In particular, we give a simple example of an inductive formula which is not frame-equivalent to any Sahlqvist formula. Then, after a deeper analysis of the inductive formulae as set-theoretic operators in descriptive and Kripke frames, we establish a somewhat stronger model-theoretic characterization of these formulae in terms of a suitable equivalence to syntactically simpler formulae (‘primitive regular formulae’) in the extension of the language with reversive modalities. Lastly, we study and characterize the elementary canonical formulae in reversive languages with nominals, where the relevant notion of persistence is with respect to discrete frames.

Alternating-time Temporal Logic (ATL), introduced by Alur, Henzinger and Kupferman, is a logical formalism for the specification and verification of open systems involving multiple autonomous players (agents, components). In particular, this logic allows for the explicit expression of coalition abilities in such systems, modelled as infinite transition games between the coalition and its complement.Formally, ATL is a non-normal multi-modal extension of CTL (regarded as a one-player fragment of ATL) with temporal operators indexed by coalitions of players, and thus expressing selective quantification over those paths which can be effected as outcomes of infinite transition games between the coalition and its complement.We present a sound and complete axiomatization of the logic ATL, based on Pauly's axiomatization of his Coalition Logic, augmented with axioms and rules for fixed point formulae characterizing the temporal operators. The completeness proof is by construction of a bounded branching tree model for each ATL-consistent formula. These models can be folded into finite models, thus rendering the finite model property for ATL.We also describe an automata-based decision procedure for ATL by translating the satisfiability problem to the nonemptiness problem for alternating automata on infinite trees. When considering formulae over a fixed finite set of players the decidability problem is shown to be EXPTIME-complete.