Guillaume Hoffmann

Relation-changing modal operators

We study dynamic modal operators that can change the accessibility relation of a model during the evaluation of a formula. In particular, we extend the basic modal language with modalities that are able to delete, add or swap an edge between pairs of elements of the domain. We define a generic framework to characterize this kind of operations. First, we investigate relation-changing modal logics as fragments of classical logics. Then, we use the new framework to get a suitable notion of bisimulation for the logics introduced, and we investigate their expressive power. Finally, we show that the complexity of the model checking problem for the particular operators introduced is PSpace-complete, and we study two subproblems of model checking: formula complexity and program complexity.

Moving Arrows and Four Model Checking Results

We study dynamic modal operators that can change the model during the evaluation of a formula. In particular, we extend the basic modal language with modalities that are able to swap, delete or add pairs of related elements of the domain, while traversing an edge of the accessibility relation. We study these languages together with the sabotage modal logic, which can arbitrarily delete edges of the model. We define a suitable notion of bisimulation for the basic modal logic extended with each of the new dynamic operators and investigate their expressive power, showing that they are all uncomparable. We also show that the complexity of their model checking problems is PSpace-complete.

Modal logics with counting

We present a modal language that includes explicit operators to count the number of elements that a model might include in the extension of a formula, and we discuss how this logic has been previously investigated under different guises. We show that the language is related to graded modalities and to hybrid logics. We illustrate a possible application of the language to the treatment of plural objects and queries in natural language. We investigate the expressive power of this logic via bisimulations, discuss the complexity of its satisfiability problem, define a new reasoning task that retrieves the cardinality bound of the extension of a given input formula, and provide an algorithm to solve it.

HTab: a Terminating Tableaux System for Hybrid Logic

Hybrid logic is a formalism that is closely related to both modal logic and description logic. A variety of proof mechanisms for hybrid logic exist, but the only widely available implemented proof system, HyLoRes, is based on the resolution method. An alternative to resolution is the tableaux method, already widely used for both modal and description logics. Tableaux algorithms have also been developed for a number of hybrid logics, and the goal of the present work is to implement one of them. In this article we present the implementation of a terminating tableaux algorithm for the hybrid logic H(@,A). The performance of the tableaux algorithm is compared with the performances of HyLoRes, HyLoTab (a system based on a different tableaux algorithm) and RacerPro. HTab is written in the functional language Haskell, using the Glasgow Haskell Compiler (GHC). The code is released under the GNU GPL and can be downloaded from http://hylo.loria.fr/intohylo/htab.php.

Lightweight hybrid tableaux

Abstract We 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) [2] . 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) [6] .

Tableaux for Relation-Changing Modal Logics

We consider dynamic modal operators that can change the relation of a model during the evaluation of a formula. In this paper, we extend the basic modal language with modalities that are able to delete, add or swap pairs of related elements of the domain; and explore tableau calculi as satisfiability procedures for these logics.

Relation-Changing Logics as Fragments of Hybrid Logics.

Relation-changing modal logics are extensions of the basic modal logic that allow changes to the accessibility relation of a model during the evaluation of a formula. In particular, they are equipped with dynamic modalities that are able to delete, add, and swap edges in the model, both locally and globally. We provide translations from these logics to hybrid logic along with an implementation. In general, these logics are undecidable, but we use our translations to identify decidable fragments. We also compare the expressive power of relation-changing modal logics with hybrid logics.

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.

Satisfiability for relation-changing logics

Relation-changing modal logics are extensions of the basic modal logic with dynamic operators that modify the accessibility relation of a model during the evaluation of a formula. These languages are equipped with dynamic modalities that are able, for example, to delete, add, and swap edges in the model, both locally and globally. We study the satisfiability problem for some of these logics. We first show that they can be translated into hybrid logic. As a result, we can transfer some results from hybrid logics to relation-changing modal logics. We discuss in particular, decidability for some fragments. We then show that satisfiability is, in general, undecidable for all the languages introduced, via translations from memory logics.

Undecidability of Relation-Changing Modal Logics

Relation-changing modal logics are extensions of the basic modal logic that allow to change the accessibility relation of a model during the evaluation of a formula. In particular, they are equipped with dynamic modalities that are able to delete, add and swap edges in the model, both locally and globally. We investigate the satisfiability problem of these logics. We define satisfiability-preserving translations from an undecidable memory logic to relation-changing modal logics. This way we show that their satisfiability problems are undecidable.