Model-Theoretic Characterization of Intuitionistic Propositional Formulas

Abstract

Notions of k-asimulation and asimulation are introduced as asymmetric counterparts to k-bisimulation and bisimulation, respectively. It is proved that a first-order formula is equivalent to a standard translation of an intuitionistic propositional formula iff it is invariant with respect to k-asimulations for some k, and then that a first-order formula is equivalent to a standard translation of an intuitionistic propositional formula iff it is invariant with respect to asimulations. Finally, it is proved that a first-order formula is equivalent to a standard translation of an intuitionistic propositional formula over the class of intuitionistic Kripke models iff it is invariant with respect to asimulations between intuitionistic models. §1. Introduction.Van Benthem's well-known modal characterization theorem (Theorem 4.9 below) states that a first-order formula is equivalent to a standard translation of a modal propositional formula iff it is invariant with respect to bisimulations. There is also a weaker 'parametrized' version of this result stating that a first-order formula is equivalent to a standard translation of a modal propositional formula iff this formula is invariant with respect to k-bisimulations for some k. Although both results yield a convenient model-theoretical technique distinguishing 'modal' first-order formulas from 'nonmodal' ones, van Benthem's characterization theorem, unlike its parametrized version, also isolates a single property defining expressive powers of modal propositional logic and thus gives us an important insight into its nature when this logic is viewed as a fragment of first-order logic.Given that the view of intuitionistic logic as a fragment of modal propositional logic has a long and established tradition dating back to Tarski-Gödel translation of this logic into S4, it was natural to ask whether one can add to the bisimulation invariance some extra condition to get a criterion distinguishing 'intuitionistic propositional' first-order formulas from nonintuitionistic ones. As was shown by Visser et al. (1995), bisimulation invariance plus a form of monotonicity with respect to the accessibility relation 'almost' gives such a criterion: i. e., it tells intuitionistic propositional first-order formulas from nonintuitionistic ones over the set of intuitionistic models. It was not known, however, how to remove this restriction and provide a criterion distinguishing intuitionistic propositional first-order formulas from nonintuitionistic ones in the general case.In this paper, we take a different strategy and amend the notion of bisimulation itself rather than adding new conditions to it. In this way, we get a notion of asimulation and we show that asimulation invariance allows one to tell intuitionistic propositional first-order formulas from nonintuitionistic ones in the general case, thus providing a criterion for the

On generalized Van Benthem-type characterizations

“…The paper continues the line of [6], [7], and [8]. This results in a model-theoretic characterization of expressive powers of arbitrary finite sets of guarded connectives of degree not exceeding 1 and regular connectives of degree 2 over the language of bounded lattices.…”

“…This line started in late 2011, when we began to think about possible modifications of bisimulation relation in order to obtain the full analogue of Van Benthem modal characterization theorem for intuitionistic propositional logic. For the resulting modification, which was published in [6], we came up with a term "asimulation", since one of the differences between asimulations and bisimulations was that asimulations were not symmetrical.…”

Model-theoretic characterization of intuitionistic predicate formulas

“…However, in the predicate case differences from the corresponding notion of bisimulation are much more conspicuous than in the propositional case. Thus, if we introduced 'asimulation games' corresponding to the propositional version of asimulation defined in [Olkhovikov 2011] (the main difference from propositional case being the absence of conditions (78) and (79)) then, given the strength of condition (77) we would have these games indistinguishable from bisimulation games on the segment beginning from the first move of Duplicator. Every link between worlds established by this player would have to be symmetrical and the asymmetry of asimulation would be important only for the intial pair of worlds.…”

“…Notions of asimulation and k-asimulation introduced in [Olkhovikov 2011] are extended onto the level of predicate logic. We then prove that a firstorder formula is equivalent to a standard translation of an intuitionistic predicate formula iff it is invariant with respect to k-asimulations for some k, and then that a first-order formula is equivalent to a standard translation of an intuitionistic predicate formula iff it is invariant with respect to asimulations.…”

“…In [Olkhovikov 2011] we filled this gap for intuitionistic propositional logic. In this paper we introduced the notion of asimulation and its parametrized version, k-asimulation, and showed that they can be used to characterize expressive power of intuitionistic propositional logic in much the same way bisimulation and k-bisimulation are used to characterize modal propositional logic.…”