Lightweight Hybrid Tableaux

International audienceWe present a decision procedure for hybrid logic equipped with nominals, the satisfaction operator and existential, difference, converse, reflexive, symmetric and transitive modalities. This decision procedure is a prefixed tableau method based on the one introduced by Bolander and Blackburn (2007). It enhances its predecessor in terms of computational efficiency and handles more expressive logics. Its way of ensuring termination enables addition of rules for the difference modality, inspired by Kaminski and Smolka (2009)

Decision procedures for some strong hybrid logics

Hybrid logics are extensions of standard modal logics, which significantly increase the expressive power of the latter. Since most of hybrid logics are known to be decidable, decision procedures for them is a widely investigated field of research. So far, several tableau calculi for hybrid logics have been presented in the literature. In this paper we introduce a sound, complete and terminating tableau calculus TH(@,E,D,♦ −) for hybrid logics with the satisfaction operators, the universal modality, the difference modality and the inverse modality as well as the corresponding sequent calculus SH(@,E,D,♦ −). They not only uniformly cover relatively wide range of various hybrid logics but they are also conceptually simple and enable effective search for a minimal model for a satisfiable formula. The main novelty is the exploitation of the unrestricted blocking mechanism introduced as an explicit, sound tableau rule

Symmetries in Modal Logics

We generalize the notion of symmetries of propositional formulas in conjunctive normal form to modal formulas. Our framework uses the coinductive models and, hence, the results apply to a wide class of modal logics including, for example, hybrid logics. Our main result shows that the symmetries of a modal formula preserve entailment.Comment: In Proceedings LSFA 2012, arXiv:1303.713

Completeness of Tableau Calculi for Two-Dimensional Hybrid Logics

Hybrid logic is one of the extensions of modal logic. The many-dimensional product of hybrid logic is called hybrid product logic (HPL). We construct a sound and complete tableau calculus for two-dimensional HPL. Also, we made a tableau calculus for hybrid dependent product logic (HdPL), where one dimension depends on the other. In addition, we add a special rule to the tableau calculus for HdPL and show that it is still sound and complete. All of them lack termination, however.Comment: Version 2. 27 pages. 5 figures. This is a preprin

A Goal-Directed Decision Procedure for Hybrid PDL

We present the first goal-directed decision procedure for hybrid PDL. The procedure is based on a modular approach that scales from basic modal logic with eventualities to hybrid PDL. The approach is designed so that nominals and eventualities are treated orthogonally. To deal with the complex programs of PDL, the approach employs a novel disjunctive program decomposition. In arguing the correctness of our approach, we employ the novel notion of support generalizing the standard notion of Hintikka sets

Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established

Incremental decision procedures for modal logics with nominals and eventualities

This thesis contributes to the study of incremental decision procedures for modal logics with nominals and eventualities. Eventualities are constructs that allow to reason about the reflexive-transitive closure of relations. Eventualities are an essential feature of temporal logics and propositional dynamic logic (PDL). Nominals extend modal logics with the possibility to reason about state equality. Modal logics with nominals are often called hybrid logics. Incremental procedures are procedures that can potentially solve a problem by performing only the reasoning steps needed for the problem in the underlying calculus. We begin by introducing a class of syntactic models called demos and showing how demos can be used for obtaining nonincremental but worst-case optimal decision procedures for extensions of PDL with nominals, converse and difference modalities. We show that in the absence of nominals, such nonincremental procedures can be refined into incremental demo search procedures, obtaining a worst-case optimal decision procedure for modal logic with eventualities. We then develop the first incremental decision procedure for basic hybrid logic with eventualities, which we eventually extend to deal with hybrid PDL. The approach in the thesis suggests a new principled design of modular, incremental decision procedures for expressive modal logics. In particular, it yields the first incremental procedures for modal logics containing both nominals and eventualities.Diese Dissertation untersucht inkrementelle Entscheidungsverfahren für Modallogiken mit Nominalen und Eventualities. Eventualities sind Konstrukte, die erlauben, über den reflexiv-transitiven Abschluss von Relationen zu sprechen. Sie sind ein Schlüsselmerkmal von Temporallogiken und dynamischer Aussagenlogik (PDL). Nominale erweitern Modallogik um die Möglichkeit, über Gleichheit von Zuständen zu sprechen. Modallogik mit Nominalen nennt man Hybridlogik. Inkrementell ist ein Verfahren dann, wenn es ein Problem so lösen kann, dass für die Lösung nur solche Schritte in dem zugrundeliegenden Kalkül gemacht werden, die für das Problem relevant sind. Wir führen zunächst eine Klasse syntaktischer Modelle ein, die wir Demos nennen. Wir nutzen Demos um nichtinkrementelle aber laufzeitoptimale Entscheidungsverfahren für Erweiterungen von PDL zu konstruieren. Wir zeigen, dass im Fall ohne Nominale solche Verfahren durch algorithmische Verfeinerung zu inkrementellen Verfahren ausgebaut werden können. Insbesondere erhalten wir so ein optimales Verfahren für Modallogik mit Eventualities. Anschließend entwickeln wir das erste inkrementelle Verfahren für Hybridlogik mit Eventualities, welches wir schließlich auf hybrides PDL erweitern. Die Dissertation vermittelt einen neuen Ansatz zur Konstruktion modularer, inkrementeller Entscheidungsverfahren für expressive Modallogiken. Insbesondere liefert der Ansatz die ersten inkrementellen Verfahren für Modallogiken mit Nominalen und Eventualities