Valentin Goranko

Modal logic with names

We investigate an enrichment of the propositional modal languageℒ with a “universal” modality ▪ having semanticsx ⊧ ▪ϕ iff āy(y ⊧ ϕ), and a countable set of “names” — a special kind of propositional variables ranging over singleton sets of worlds. The obtained language ℒc proves to have a great expressive power. It is equivalent with respect to modal definability to another enrichment ℒ() ofℒ, where is an additional modality with the semanticsx ⊧ϕ iff āy(y⊧ x → y ⊧ϕ). Model-theoretic characterizations of modal definability in these languages are obtained. Further we consider deductive systems in ℒc. Strong completeness of the normal ℒc-logics is proved with respect to models in which all worlds are named. Every ℒc-logic axiomatized by formulae containing only names (but not propositional variables) is proved to be strongly frame-complete. Problems concerning transfer of properties ([in]completeness, filtration, finite model property etc.) fromℒ to ℒcare discussed. Finally, further perspectives for names in multimodal environment are briefly sketched.

A Road Map of Interval Temporal Logics and Duration Calculi

We survey main developments, results, and open problems on interval temporal logics and duration calculi. We present various formal systems studied in the literature and discuss their distinctive features, emphasizing on expressiveness, axiomatic systems, and (un)decidability results.

Complete axiomatization and decidability of alternating-time temporal logic

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 Paulys 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.

Propositional interval neighborhood logics: Expressiveness, decidability, and undecidable extensions

Abstract In this paper, we investigate the expressiveness of the variety of propositional interval neighborhood logics (PNL), we establish their decidability on linearly ordered domains and some important subclasses, and we prove the undecidability of a number of extensions of PNL with additional modalities over interval relations. All together, we show that PNL form a quite expressive and nearly maximal decidable fragment of Halpern–Shoham’s interval logic HS.

Algorithmic correspondence and completeness in modal logic. I. The core algorithm SQEMA

Modal formulae express monadic second-order properties on Kripke frames, but in many important cases these have first-order equivalents. Computing such equiva- lents is important for both logical and computational reasons. On the other hand, canon- icity of modal formulae is important, too, because it implies frame-completeness of logics axiomatized with canonical formulae. Computing a first-order equivalent of a modal formula amounts to elimination of second- order quantifiers. Two algorithms have been developed for second-order quantifier elim- ination: SCAN, based on constraint resolution, and DLS, based on a logical equivalence established by Ackermann. In this paper we introduce a new algorithm, SQEMA, for computing first-order equiva- lents (using a modal version of Ackermanns lemma) and, moreover, for proving canonicity of modal formulae. Unlike SCAN and DLS, it works directly on modal formulae, thus avoiding Skolemization and the subsequent problem of unskolemization. We present the core algorithm and illustrate it with some examples. We then prove its correctness and the canonicity of all formulae on which the algorithm succeeds. We show that it succeeds not only on all Sahlqvist formulae, but also on the larger class of inductive formulae, in- troduced in our earlier papers. Thus, we develop a purely algorithmic approach to proving canonical completeness in modal logic and, in particular, establish one of the most general completeness results in modal logic so far.

Comparing Semantics of Logics for Multi-Agent Systems

We draw parallels between several closely related logics that combine — in different proportions — elements of game theory, computation tree logics, and epistemic logics to reason about agents and their abilities. These are: the coalition game logics CL and ECL introduced by Pauly 2000, the alternating-time temporal logic ATL developed by Alur, Henzinger and Kupferman between 1997 and 2002, and the alternating-time temporal epistemic logic ATEL by van der Hoek and Wooldridge (2002). In particular, we establish some subsumption and equivalence results for their semantics, as well as interpretation of the alternating-time temporal epistemic logic into ATL.The focus in this paper is on models: alternating transition systems, multi-player game models (alias concurrent game structures) and coalition effectivity models turn out to be intimately related, while alternating epistemic transition systems share much of their philosophical and formal apparatus. Our approach is constructive: we present ways to transform between different types of models and languages.

Elementary canonical formulae: extending Sahlqvist's theorem

Abstract 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.

Hierarchies of modal and temporal logics with reference pointers

We introduce and study hierarchies of extensions of the propositional modal and temporal languages with pairs of new syntactic devices: “point of reference-reference pointer” which enable semantic references to be made within a formula. We propose three different but equivalent semantics for the extended languages, discuss and compare their expressiveness. The languages with reference pointers are shown to have great expressive power (especially when their frugal syntax is taken into account), perspicuous semantics, and simple deductive systems. For instance, Kamps and Stavis temporal operators, as well as nominals (names, clock variables), are definable in them. Universal validity in these languages is proved undecidable. The basic modal and temporal logics with reference pointers are uniformly axiomatized and a strong completeness theorem is proved for them and extended to some classes of their extensions.

Decidable and Undecidable Fragments of Halpern and Shoham's Interval Temporal Logic: Towards a Complete Classification

Interval temporal logics are based on temporal structures where time intervals, rather than time instants, are the primitive ontological entities. They employ modal operators corresponding to various relations between intervals, known as Allens relations. Technically, validity in interval temporal logics translates to dyadic second-order logic, thus explaining their complex computational behavior. The full modal logic of Allens relations, called HS, has been proved to be undecidable by Halpern and Shoham under very weak assumptions on the class of interval structures, and this result was discouraging attempts for practical applications and further research in the field. A renewed interest has been recently stimulated by the discovery of interesting decidable fragments of HS. This paper contributes to the characterization of the boundary between decidability and undecidability of HS fragments. It summarizes known positive and negative results, it describes the main techniques applied so far in both directions, and it establishes a number of new undecidability results for relatively small fragments of HS.

Tableaux for Logics of Subinterval Structures over Dense Orderings

In this article, we develop tableau-based decision procedures for the logics of subinterval structures over dense linear orderings. In particular, we consider the two difficult cases: the relation of strict subintervals (with both endpoints strictly inside the current interval) and the relation of proper subintervals (that can share one endpoint with the current interval). For each of these logics, we establish a small pseudo-model property and construct a sound, complete and terminating tableau that searches systematically for existence of such a pseudo-model satisfying the input formulas. Both constructions are non-trivial, but the latter is substantially more complicated because of the presence of beginning and ending subintervals which require special treatment. We prove PSPACE completeness for both procedures and implement them in the generic tableau-based theorem prover Lotrec.