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

Unification in the Description Logic EL

The Description Logic EL has recently drawn considerable attention since, on the one hand, important inference problems such as the subsumption problem are polynomial. On the other hand, EL is used to define large biomedical ontologies. Unification in Description Logics has been proposed as a novel inference service that can, for example, be used to detect redundancies in ontologies. The main result of this paper is that unification in EL is decidable. More precisely, EL-unification is NP-complete, and thus has the same complexity as EL-matching. We also show that, w.r.t. the unification type, EL is less well-behaved: it is of type zero, which in particular implies that there are unification problems that have no finite complete set of unifiers.Comment: 31page

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

Almost structural completeness; an algebraic approach

A deductive system is structurally complete if its admissible inference rules are derivable. For several important systems, like modal logic S5, failure of structural completeness is caused only by the underivability of passive rules, i.e. rules that can not be applied to theorems of the system. Neglecting passive rules leads to the notion of almost structural completeness, that means, derivablity of admissible non-passive rules. Almost structural completeness for quasivarieties and varieties of general algebras is investigated here by purely algebraic means. The results apply to all algebraizable deductive systems. Firstly, various characterizations of almost structurally complete quasivarieties are presented. Two of them are general: expressed with finitely presented algebras, and with subdirectly irreducible algebras. One is restricted to quasivarieties with finite model property and equationally definable principal relative congruences, where the condition is verifiable on finite subdirectly irreducible algebras. Secondly, examples of almost structurally complete varieties are provided Particular emphasis is put on varieties of closure algebras, that are known to constitute adequate semantics for normal extensions of S4 modal logic. A certain infinite family of such almost structurally complete, but not structurally complete, varieties is constructed. Every variety from this family has a finitely presented unifiable algebra which does not embed into any free algebra for this variety. Hence unification in it is not unitary. This shows that almost structural completeness is strictly weaker than projective unification for varieties of closure algebras

About the type of modal logics for the unification problem

Dans cette thèse, nous étudierons le problème de l'unification dans les logiques modales ordinaires, les fusions de deux logiques modales et les logiques épistémiques multi-modales. Relativement à une logique propositionnelle L, étant donnée une formule A, nous devons trouver des substitutions s telle que s(A) est dans L. Lorsqu'elles existent, ces substitutions sont appelées unifieurs de A dans L. Nous étudions différentes méthodes pour construire des ensembles minimaux complets d'unifieurs d'une formule donnée A et, en fonction de la cardinalité des ces ensembles minimaux complets, nous discutons du type de l'unification de A. Enfin, nous déterminons les types de l'unification de plusieurs logiques propositionnelles.In this thesis, we shall investigate on the unification problem in ordinary modal logics, fusions of two modal logics and multi-modal epistemic logics. With respect to a propositional logic L, given a formula A, we have to find substitutions s such that s(A) is in L. When they exist, these substitutions are called unifiers of A in L. We study different methods for the construction of minimal complete sets of unifiers of a given formula A and according to the cardinality of these minimal complete sets, we shall discuss on the unification type of A. Then, we determine the unification types of several propositional logics

Undecidability of the unification and admissibility problems for modal and description logics

We show that the unification problem `is there a substitution instance of a given formula that is provable in a given logic?' is undecidable for basic modal logics K and K4 extended with the universal modality. It follows that the admissibility problem for inference rules is undecidable for these logics as well. These are the first examples of standard decidable modal logics for which the unification and admissibility problems are undecidable. We also prove undecidability of the unification and admissibility problems for K and K4 with at least two modal operators and nominals (instead of the universal modality), thereby showing that these problems are undecidable for basic hybrid logics. Recently, unification has been introduced as an important reasoning service for description logics. The undecidability proof for K with nominals can be used to show the undecidability of unification for boolean description logics with nominals (such as ALCO and SHIQO). The undecidability proof for K with the universal modality can be used to show that the unification problem relative to role boxes is undecidable for Boolean description logic with transitive roles, inverse roles, and role hierarchies (such as SHI and SHIQ)

Unification and Finite Model Property for Linear Step-Like Temporal Multi-Agent Logic with the Universal Modality

This paper proposes a semantic description of the linear step-like temporal multi-agent logic with the universal modality $\mathcal{LTK}.sl_U$ based on the idea of non-reflexive non-transitive nature of time. We proved a finite model property and projective unification for this logic