Model-theoretic characterization of intuitionistic predicate formulas

Abstract

Abstract. 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. Finally, it is proved that a first-order formula is equivalent to a standard translation of an intuitionistic predicate formula over a class of intuitionistic models (intuitionistic models with constant domain) iff it is invariant with respect to asimulations between intuitionistic models (intuitionistic models with constant domain).Van Benthem's well-known modal characterization theorem shows that expressive power of modal propositional logic as a fragment of first-order logic can be described via the notion of bisimulation invariance. Moreover, it is known that modal predicate logic, initially considered as an extension of first-order logic, can also be viewed as its fragment, although somewhat bigger than the fragment induced by propositional modal logic. Expressive power of modal predicate logic, from this vantage point, is described by the notion of world-object bisimulation which appears to be a rather direct combination of bisimulation and partial isomorphism (see, e. g. [Van Benthem 2010, p. 124, Theorem 21]).Although intuitionistic logic has been treated as a fragment of modal logic for quite a long while, results analogous to propositional and predicate version of Van Benthem's modal characterization theorem were not obtained for it until recently. 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. In this paper we do the same job for intuitionistic predicate logic without identity.The layout of the paper is as follows. Starting from some notational conventions and 1

On generalized Van Benthem-type characterizations

“…In much the same way, there seem to be no principal difficulties in obtaining a 'parametrized' version of Theorem 1 similar to [6,Theorem 2], [7,Theorem 2], or [8,Theorem 1].…”

“…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.Later we modified and extended asimulations in order to capture the expressive powers of first-order intuitionistic logic (in [7]) and some variants of basic modal intuitionistic logic (in [8]) viewed as fragments of classical first-order logic. Some other authors were also working in this direction; e.g.…”

Failure of Interpolation in Constant Domain Intuitionistic Logic

“…The version we are using here is a simplified version of the general notion; the simplifications are possible because we are operating in the context of the constant domain semantics. The second author has defined a more general version [16] that is suited to the context of Kripke's semantics for intuitionistic predicate logic, and has also proved that it serves to characterize the first-order properties expressed by formulas of propositional and predicate logic in that framework. In the present case, however, we do not need the full characterization theorems; the following Lemma is sufficient for our purpose of refuting the interpolation theorem for the logic CD.…”

On expressive power of basic modal intuitionistic logic as a fragment of classical FOL

“…Section 1 starts with notational conventions, after which we introduce the main variants of Kripke style semantics for the basic modal intuitionistic system. All in all we consider 4 different variants of semantics, of which 2 are easily discharged by the versions of clauses employed in [7] and [8] for basic intutionistic logic. However, for the other 2 systems their semantical characterization is less obvious, mainly due to their treatment of diamond modality.…”