Complexity of finite-variable fragments of products with K

We show that products and expanding relativized products of propositional modal logics where one component is the minimal monomodal logic K are polynomial-time reducible to their single-variable fragments. Therefore, the known lower-bound complexity and undecidability results for such logics are extended to their single-variable fragments. Similar results are obtained for products where one component is a polymodal logic with a K-style modality; these include products with propositional dynamic logics.

Models of transfinite provability logic

For any ordinal Λ, we can define a polymodal logic GLPΛ, with a modality [ξ] for each ξ < Λ. These represent provability predicates of increasing strength. Although GLPΛ has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities, denoted . Later, Icard defined a topological model for which is very closely related to Ignatiev's.In this paper we show how to extend these constructions for arbitrary Λ. More generally, for each Θ, Λ we build a Kripke model and a topological model , and show that is sound for both of these structures, as well as complete, provided Θ is large enough.

A Family of Polymodal Systems and its Application to Generalized Possibility Measures and Multi-Rough Sets

Polymodal systems generally have large areas of applications to theoretical computer science including the theory of programming, while other applications are not yet fully explored. In this paper we consider a family of polymodal systems with the structure of lattices on the polymodal indices. After investigating theory of the polymodal systems such as the completeness, we study two applications. One is generalized possibility measures in which lattice-valued measures are proposed and relations with the ordinary possibility and necessity measures are uncovered. Second application is consideration of an information system as a table such as the one in the relational database. It is known that rough sets are used to discover regularities from such information tables. Applying polymodal logic concept, we generalize rough sets which are called multi-rough sets here. Our consideration is mainly to establish theoretical frameworks in these two application areas and hence no real examples are shown here.

Properties of independently axiomatizable bimodal logics

In monomodal logic there are a fair number of high-powered results on completeness covering large classes of modal systems; witness for example Fine [74], [85] and Sahlqvist [75]. Monomodal logic is therefore a well-understood subject in contrast to polymodal logic, where even the most elementary questions concerning completeness, decidability, etc. have been left unanswered. Given that in many applications of modal logic one modality is not sufficient, the lack of general results is acutely felt by the “users” of modal logics, contrary to logicians who might entertain the view that a deep understanding of one modality alone provides enough insight to be able to generalize the results to logics with several modalities. Although this view has its justification, the main results we are going to prove are certainly not of this type, for they require a fundamentally new technique. The results obtained are called transfer theorems in Fine and Schurz [91] and are of the following type. Let L ∌ ⊥ be an independently axiomatizable bimodal logic and L⎕ and L∎ its monomodal fragments. Then L has a property P iff L⎕ and L∎ have P. Properties which will be discussed are completeness, the finite model property, compactness, persistence, interpolation and Halldén-completeness. In our discussion we will prove transfer theorems for the simplest case when there are just two modal operators, but it will be clear that the proof works in the general case as well.