A semantic approach to conservativity

The aim of this paper is to describe from a semantic perspective the problem of conservativity of classical first-order theories over their intuitionistic counterparts. In particular, we describe a class of formulae for which such conservativity results can be proven in case of any intuitionistic theory T which is complete with respect to a class of T-normal Kripke models. We also prove conservativity results for intuitionistic theories which are closed under the Friedman translation and complete with respect to a class of conversely well-founded Kripke models. The results can be applied to a wide class of intuitionistic theories and can be viewed as generalization of the results obtained by syntactic methods

Bisimulation reducts and submodels of intuitionistic first-order Kripke models

We consider elementary submodels of a given intuitionistic Kripke model K meant as models that share the same theory with K and result in restricting the frame of K and/or replacing some of its worlds with their elementary substructures. We introduce the notion of bisimulation reduct of the Kripke model wich allows us to construct elementary submodels of given Kripke models in the sense of the definition above. As it was observed by A. Visser in [6], the notion of submodel can be desined for intuitionistic sirst-order Kripke models in several different ways. We can either consider models on the same frame, where the worlds of submodels are substructures of the worlds of the original model, or we can define a submodel to be the result of restricting the frame of the given model, or we can combine both of these operations. All of these possibilities were considered in the literature, see [1], [6] and [2] respectively, however it seems that we should accept the third notion as the correct one. The reason for that is, that not only such defined notion of submodel coincides with the classical notion of substructure in the case of the simplest Kripke model, but also because the well-known classical Tarski- łoś preservation theorem concerning substructures becomes a particular case of the result proven in [2]; i.e. the class of the formulas that are preserved under Kripke submodels is the class of an intuitionistic variant of universal formulas

Submodel Enumeration for CTL Is Hard

Expressing system specifications using Computation Tree Logic (CTL) formulas, formalising programs using Kripke structures, and then model checking the system is an established workflow in program verification and has wide applications in AI. In this paper, we consider the task of model enumeration, which asks for a uniform stream of output systems that satisfy the given specification. We show that, given a CTL formula and a system (potentially falsified by the formula), enumerating satisfying submodels is always hard for CTL - regardless of which subset of CTL operators is considered. As a silver lining on the horizon, we present fragments via restrictions on the allowed Boolean functions that still allow for fast enumeration.Comment: To be published in AAAI2

A note on bisimulations of finite Kripke models

In our paper we consider the notion of bounded bisimulation of Kripke models for intuitionistic first-order theories. As it is already known, in this case, the existence of bisimulation between given two Kripke models implies their logical equivalence. We present a new result which states that, under some additional conditions, for every two first-order Kripke models that are equivalent, there is a bisimulation between them

Bisymulacje modeli Kripkego dla teorii intuicjonistycznych pierwszego rzędu

Rozprawa poświęcona bisymulacjom modeli Kripkego dla teorii intuicjonistycznych pierwszego rzędu

Submodels of Kripke Models

A Kripke model K is a submodel of another Kripke modelMif K is obtained by restricting the set of nodes of M. In this paper we showthat the class of formulas of Intuitionistic Predicate Logic that is preserved under taking submodels of Kripke models is precisely the class of semipositive formulas. This result is an analogue of the Los-Tarski theorem for the Classical Predicate Calculus. In appendix A we provethat for theories with decidable identity we can take as the embeddings between domains in Kripke models of the theory, the identical embeddings. This is a well known fact, but we know of no correct proof in the literature. In appendix B we answer, negatively, a question posed by Sam Buss: whether there is a classical theory T, such that HT is HA. HereHT is the theory of all Kripke modelsMsuch that the structures assigned to the nodes of M all satisfy T in the sense o

Submodels of Kripke Models

A Kripke model K is a submodel of another Kripke model M if K is obtained by restricting the set of nodes of M. In this paper we show that the class of formulas of Intuitionistic Predicate Logic that is preserved under taking submodels of Kripke models is precisely the class of semipositive formulas. This result is an analogue of the Lós-Tarski theorem for the Classical Predicate Calculus. In appendix A we prove that for theories with decidable identity we can take as the embeddings between domains in Kripke models of the theory, the identical embeddings. This is a well known fact, but we know of no correct proof in the literature. In appendix B we answer, negatively, a question posed by Sam Buss: whether there is a classical theory T, such that HT is HA. Here HT is the theory of all Kripke models M such that the structures assigned to the nodes of M all satisfy T in the sense of classical model theory