Bisimulation Quantifiers and Uniform Interpolation for Guarded First Order Logic

The idea that the good model-theoretic and algorithmic properties of Modal Logics are due to the guarded nature of their quantification was put forward by Andreka, van Benthem and Nemeti in a series of papers in the 1990s, exploiting the satisfiability problem, the tree model property, and other similar properties of the Guarded Fragment of First Order Logic (GF). Since then, further work on the Guarded Fragment has been done by various authors, in some cases reinforcing this idea, in some others not. At least at first sight, Craig interpolation is on the negative side: there are implications in GF without an interpolant in GF, while Modal Logic (and even the μ-calculus, a powerful extension of Modal Logic) enjoys a much stronger form of interpolation, the uniform one, in which the interpolant of a valid implication not only exists, but only depends on the antecedent and on the common language of antecedent and consequent. However, Hoogland and Marx proved that Craig interpolation is restored in GF if we consider the modal character of GF with more attention, that is, if relations appearing on guards are viewed as “modalities” and the rest as “propositions”, and only the latter enter in the common language. In this paper we strengthen this result by showing that GF enjoys a Modal Uniform Interpolation Theorem (in the sense of Hoogland and Marx).