Polyhedral Completeness of Intermediate Logics: The Nerve Criterion

Abstract

We investigate a recently devised polyhedral semantics for intermediate logics, in which formulas are interpreted in n-dimensional polyhedra. An intermediate logic is polyhedrally complete if it is complete with respect to some class of polyhedra. The first main result of this paper is a necessary and sufficient condition for the polyhedral completeness of a logic. This condition, which we call the Nerve Criterion, is expressed in terms of Alexandrov’s notion of the nerve of a poset. It affords a purely combinatorial characterisation of polyhedrally complete logics. Using the Nerve Criterion we show, easily, that there are continuum many intermediate logics that are not polyhedrally complete but which have the finite model property. We also provide, at considerable combinatorial labour, a countably infinite class of logics axiomatised by the Jankov–Fine formulas of ‘starlike trees’ all of which are polyhedrally complete. The polyhedral completeness theorem for these ‘starlike logics’ is the second main result of this paper.

Geometric Model Checking of Continuous Space

“…Triangulation is a standard technique of piecewise linear geometry in which each polyhedron is decomposed in simplexes. That triangulations play an important role in the logical analysis of polyhedra has already been observed in [BMMP18,AD19,ABGM21]. However, here we show this also for the language enriched with the reachability modality γ.…”