Towards a Model Theory of Ordered Logics: Expressivity and Interpolation (Extended version)

TL;DR

This paper explores the model theory of ordered logics, establishing their expressive power, bisimulation notions, and interpolation properties, thereby deepening understanding of their logical and computational characteristics.

Contribution

It introduces bisimulation concepts for ordered logics, characterizes their expressivity as FO fragments, and proves which fragments satisfy the Craig Interpolation Property.

Findings

Ordered logics' bisimulation notions are defined and analyzed.

Ordered and guarded ordered logics satisfy the Craig Interpolation Property.

The fluted and forward fragments do not enjoy the Craig Interpolation Property.

Abstract

We consider the family of guarded and unguarded ordered logics, that constitute a recently rediscovered family of decidable fragments of first-order logic (FO), in which the order of quantification of variables coincides with the order in which those variables appear as arguments of predicates. While the complexities of their satisfiability problems are now well-established, their model theory, however, is poorly understood. Our paper aims to provide some insight into it. We start by providing suitable notions of bisimulation for ordered logics. We next employ bisimulations to compare the relative expressive power of ordered logics, and to characterise our logics as bisimulation-invariant fragments of FO a la van Benthem. Afterwards, we study the Craig Interpolation Property~(CIP). We refute yet another claim from the infamous work by Purdy, by showing that the fluted and forward fragments do not enjoy CIP. We complement this result by showing that the ordered fragment and the guarded ordered logics enjoy CIP. These positive results rely on novel and quite intricate model constructions, which take full advantage of the "forwardness" of our logics.