Dynamic algebras which are not Kripke structures

Abstract

The first example of a dynamic algebra which is not isomorphic to any Kripke structure was given by Kozen [8]. We analyze properties of dynamic algebras to get general arguments making it possible to construct a lot of other examples.

Dynamic algebras were introduced by Kozen [6] and Pratt [12] to provide models of propositional dynamic logic (PDL); see [12] for motivation, examples and relation to computer science problems. Dynamic algebras include (standard) Kripke structures, the traditional models of PDL. They are axiomatized in a purely algebraic way in [12] and in a nearly algebraic way in [6], in contrary to Kripke structures which are a kind of relational structures. A relation between abstract dynamic algebras and Kripke structures was investigated by Kozen; he proved

The aim of the current paper is to present other examples of that kind. In fact, we show that many Boolean algebras are Boolean parts of separable dynamic algebras which are not isomorphic to any Kripke structure.

The paper has five sections. Preliminaries are contained in sections 1 and 2. In section 3 we prove that every Kripke structure fulfils a condition, called relative $\mathfrak{S}$-continuity. By means of this result, we show that every Boolean algebra, in which the zero is an infimum of a strictly decreasing sequence, is a Boolean part of a dynamic algebra which is not relatively $\mathfrak{S}$-continuous (and hence not isomorphic to any Kripke structure) and the regular part of which has one generator. We also bring examples showing that the Kozen's axiomatization [6] of dynamic algebras is actually stronger than that of Pratt [12]. In section 4, we show that, on the other hand, the $\mathfrak{S}$-continuity (and even complete continuity) of a dynamic algebra is not sufficient for it to be isomorphic to a Kripke structure. Section 5 contains remarks concerning dynamic algebras with reversion.