La génération de conséquences en logique modale

La notion clé de la génération de conséquences est celle de l'impliqué premier, qui désigne une clause qui est impliquée par une formule et telle qu'il n'existe pas de clause logiquement plus forte impliquée par la formule. La notion d'impliqué premier s'est montrée très utile en intelligence artificielle, notamment pour la compilation de connaissances et le raisonnement abductif. Dans cette thèse nous étudions comment cette notion, jusqu'à présent étudiée en logique propositionnelle, peut être étendue à la logique modale Kn. Nous commençons par comparer plusieurs definitions plausibles d'impliqués premiers dans Kn, avant d'en selectionner une. Ensuite, nous proposons des algorithmes pour générer et reconnaître des impliqués premiers, et nous étudions la complexité de ces deux tâches. Puis, nous utilisons notre notion d'impliqué premier pour élaborer une forme normale pour Kn qui est dotée de propriétés intéressantes du point de vue de la compilation de connaissances.The key notion in consequence finding is that of prime implicates, which are defined to be the logically strongest clausal consequences of a formula. Prime implicates have proven useful in artificial intelligence, especially in knowledge compilation and abductive reasoning. In this thesis, we extend the investigation of prime implicates from propositional logic to the basic multi-modal logic Kn. We begin by comparing the properties of several plausible definitions of prime implicates in Kn in order to isolate the most suitable definition. We next study the computational aspects of the selected definition. Specifically, we provide algorithms for prime implicate generation and recognition, and we study the complexity of these tasks. Finally, we show how our notion of prime implicates can be used to define a normal form for Kn with interesting knowledge compilation properties

Heinrich Behmann's Contributions to Second-Order Quantifier Elimination from the View of Computational Logic

For relational monadic formulas (the L\"owenheim class) second-order quantifier elimination, which is closely related to computation of uniform interpolants, projection and forgetting - operations that currently receive much attention in knowledge processing - always succeeds. The decidability proof for this class by Heinrich Behmann from 1922 explicitly proceeds by elimination with equivalence preserving formula rewriting. Here we reconstruct the results from Behmann's publication in detail and discuss related issues that are relevant in the context of modern approaches to second-order quantifier elimination in computational logic. In addition, an extensive documentation of the letters and manuscripts in Behmann's bequest that concern second-order quantifier elimination is given, including a commented register and English abstracts of the German sources with focus on technical material. In the late 1920s Behmann attempted to develop an elimination-based decision method for formulas with predicates whose arity is larger than one. His manuscripts and the correspondence with Wilhelm Ackermann show technical aspects that are still of interest today and give insight into the genesis of Ackermann's landmark paper "Untersuchungen \"uber das Eliminationsproblem der mathematischen Logik" from 1935, which laid the foundation of the two prevailing modern approaches to second-order quantifier elimination