We present axiom systems, and provide soundness and strong completeness theorems, for classes of Kripke models with restricted extension rules among the node structures of the model. As examples we present an axiom system for the class of co5nal extension Kripke models, and an axiom system for the class of end-extension Kripke models. We also show that Heyting arithmetic (HA) is strongly comple...

We investigate the complexity of primal logic with disjunction according to the Kripke semantics as defined in [1] and the quasi-boolean semantics as defined in [2]. We show that the validity problem is coNP-complete, even for variable-free sequents. For quasi-boolean semantics, the satisfiability problem is shown to be NPcomplete (even for variable-free sequents), whereas for Kripke semantics ...

This paper presents a novel theoretical framework for the state space reduction of Kripke structures. We define two equivalence relations, Kripke minimization equivalence (KME) and weak Kripke minimization equivalence (WKME). We define the quotient system under these relations and show that these relations are strictly coarser than strong (bi)simulation and divergence-sensitive stutter (bi)simu...

We define a variant of the standard Kripke semantics for intuitionistic logic, motivated by the connection between constructive logic and the Medvedev lattice. We show that while the new semantics is still complete, it gives a simple and direct correspondence between Kripke models and algebraic structures such as factors of the Medvedev lattice.

A class P ∗ of formulas was defined in [4] which whenever satisfied in a classical structure associated with a node of a Kripke model must also be forced at that node. Here we define a dual class R of formulas which whenever forced at a node of a Kripke model must be satisfied in the classical structure associated with that node.

1 Introduction: Instead of considering propositional S4 (say) as a theory built on top of classical propositional logic, we can consider it as the classical first order theory of its Kripke models. This first order theory can be formulated in any of the ways first order theories usually are: tableau, Gentzen system, natural deduction, conventional axiom system, ε-calculus. If the first order th...

From classical, Fräıssé-homogeneous, (≤ ω)-categorical theories over finite relational languages (which we refer to as JRS theories), we construct intuitionistic theories that are complete, prove negations of classical tautologies, and admit quantifier elimination. The technique we use considers Kripke models as functors from a small category to the category of L-structures with morphisms, rath...

It is shown that every normal intuitionistic modal logic L over MIPC has the nite model property if there exists a universal rstorder sentence such that (1) L is characterized by the class of Kripke frames satisfying and (2) every Kripke frame that validates L satis es . Here, MIPC is a well-known intuitionistic modal logic introduced by Prior (1957).

Kripke’s schema with parameters turns out to be equivalent to each of the following two statements from metric topology: every open subspace of a separable metric space is separable; every open subset of a separable metric space is a countable union of open balls. Thus Kripke’s schema serves as a point of reference for classifying theorems of classical mathematics within Bishop-style constructi...