In this paper we continue the study of Girard's Linear Logic and introduce a new Linear Logic with modalities. Our logic describes not only the consumption, but also the presence of resources. We introduce a new semantics and a new calculus for this logic. In contrast to the results of Lincoln [7] and Kanovich [4] about the NP-completeness of the problem of the construction of a proof for a given sequent in the multiplicative fragment of Girard's Linear Logic, we present (...) here a non-exponential algorithm to construct a proof for a given sequent and a given point of a given model in our Linear Logic. (shrink)

A conception of an information system has been introduced by Pawlak. The study has been continued in works of Pawlak and Orlowska and in works of Vakarelov. They had proposed some basic relations and had constructed a formal system of a modal logic that describes the relations and some of their Boolean combinations. Our work is devoted to a generalization of this approach. A class of relation systems and a complete calculus construction method for these systems are proposed. As a (...) corollary of our main result, our paper contains a solution of a Vakarelov's problem: how to construct a formal system that describes all the Boolean combinations of the basic relations. (shrink)

In this paper we introduce a new type of nets with bounded types of distributed resources . Linear Logic to describe the behaviour of BR-nets is defined. It is based on Girard's Linear Logic but captures not only consumption of resources but their presence as well. Theorem of soundness and completeness of the proposed axiomatization is proved and the complexity of the provability problem is established for the general case and some particular ones.