24 research outputs found
On unification and admissible rules in Gabbay-de Jongh logics
In this paper we study the admissible rules of intermediate logics with the disjunction property. We establish some general results on extension of models and sets of formulas, and eventually specialize to provide a a basis for the admissible rules of the Gabbay-de Jongh logics and to show that that logic has finitary unification type
Multiple Conclusion Rules in Logics with the Disjunction Property
We prove that for the intermediate logics with the disjunction property any
basis of admissible rules can be reduced to a basis of admissible m-rules
(multiple-conclusion rules), and every basis of admissible m-rules can be
reduced to a basis of admissible rules. These results can be generalized to a
broad class of logics including positive logic and its extensions, Johansson
logic, normal extensions of S4, n-transitive logics and intuitionistic modal
logics
A gentle introduction to unification in modal logics
International audienceUnification in propositional logics is an active research area. In this paper, we introduce the results we have obtained within the context of modal logics and epistemic logics and we present some of the open problems whose solution will have an important impact on the future of the area.L'unification dans les logiques propositionnelles est un domaine de recherche actif. Dans cet article, nous présentons les résultats que nous avons obtenus dans le cadre des logiques modales et des logiqueś epistémiques et nous introduisons quelques uns des problèmes ouverts dont la résolution aura un impact important sur l'avenir du domaine
Inference Rules in some temporal multi-epistemic propositional logics
Multi-modal logics are among the best tools developed so far to analyse
human reasoning and agents’ interactions. Recently multi-modal
logics have found several applications in Artificial Intelligence (AI) and
Computer Science (CS) in the attempt to formalise reasoning about
the behavior of programs. Modal logics deal with sentences that are
qualified by modalities. A modality is any word that could be added to
a statement p to modify its mode of truth. Temporal logics are obtained
by joining tense operators to the classical propositional calculus, giving
rise to a language very effective to describe the flow of time. Epistemic
logics are suitable to formalize reasoning about agents possessing a
certain knowledge. Combinations of temporal and epistemic logics are
particularly effective in describing the interaction of agents through the
flow of time. Although not yet fully investigated, this approach has
found many fruitful applications. These are concerned with the development
of systems modelling reasoning about knowledge and space,
reasoning under uncertainty, multi-agent reasoning et c.
Despite their power, multi modal languages cannot handle a changing
environment. But this is exactly what is required in the case of human
reasoning, computation and multi-agent environment. For this purpose,
inference rules are a core instrument. So far, the research in this
field has investigated many modal and superintuitionistic logics. However,
for the case of multi-modal logics, not much is known concerning
admissible inference rules.
In our research we extend the investigation to some multi-modal propositional logics which combine tense and knowledge modalities. As far
as we are concerned, these systems have never been investigated before.
In particular we start by defining our systems semantically; further we
prove such systems to enjoy the effective finite model property and to
be decidable with respect to their admissible inference rules. We turn
then our attention to the syntactical side and we provide sound and
complete axiomatic systems. We conclude our dissertation by introducing
the reader to the piece of research we are currently working on.
Our original results can be found in [9, 4, 11] (see Appendix A). They
have also been presented by the author at some international conferences
and schools (see [8, 10, 5, 7, 6] and refer to Appendix B for more
details).
Our project concerns philosophy, mathematics, AI and CS. Modern
applications of logic in CS and AI often require languages able to represent
knowledge about dynamic systems. Multi-modal logics serve
these applications in a very efficient way, and we would absorb and
develop some of these techniques to represent logical consequences in
artificial intelligence and computation