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

Positive Logic with Adjoint Modalities: Proof Theory, Semantics and Reasoning about Information

We consider a simple modal logic whose non-modal part has conjunction and disjunction as connectives and whose modalities come in adjoint pairs, but are not in general closure operators. Despite absence of negation and implication, and of axioms corresponding to the characteristic axioms of (e.g.) T, S4 and S5, such logics are useful, as shown in previous work by Baltag, Coecke and the first author, for encoding and reasoning about information and misinformation in multi-agent systems. For such a logic we present an algebraic semantics, using lattices with agent-indexed families of adjoint pairs of operators, and a cut-free sequent calculus. The calculus exploits operators on sequents, in the style of "nested" or "tree-sequent" calculi; cut-admissibility is shown by constructive syntactic methods. The applicability of the logic is illustrated by reasoning about the muddy children puzzle, for which the calculus is augmented with extra rules to express the facts of the muddy children scenario.Comment: This paper is the full version of the article that is to appear in the ENTCS proceedings of the 25th conference on the Mathematical Foundations of Programming Semantics (MFPS), April 2009, University of Oxfor

A Note on Parameterised Knowledge Operations in Temporal Logic

We consider modeling the conception of knowledge in terms of temporal logic. The study of knowledge logical operations is originated around 1962 by representation of knowledge and belief using modalities. Nowadays, it is very good established area. However, we would like to look to it from a bit another point of view, our paper models knowledge in terms of linear temporal logic with {\em past}. We consider various versions of logical knowledge operations which may be defined in this framework. Technically, semantics, language and temporal knowledge logics based on our approach are constructed. Deciding algorithms are suggested, unification in terms of this approach is commented. This paper does not offer strong new technical outputs, instead we suggest new approach to conception of knowledge (in terms of time).Comment: 10 page