Polyhedral semantics of modal logic
P. Maurice Dekker

Abstract:
Polyhedral semantics is a way of interpreting modal formulas on polyhedra. This semantics has recently been introduced by N. Bezhanishvili, D. Gabelaia, S. Adam-Day, V. Marra and others. We introduce a new variation on this semantics, and derive metamathematical properties of both semantics.
We show that the polyhedral logic of a piecewise linear manifold-with-boundary is determined by its dimension, and that there are 2^{\aleph_0} polyhedrally-complete logics. We prove that p-morphisms between simplicial complexes factor through subdivisions. Using this, we show that the problem of comparing the logics of two polyhedra reduces to the problem of checking the validity of a formula on a polyhedron. We establish decidability of these problems for polyhedra embeddable in R^3. Moreover, we demonstrate the difficulty of checking the validity of a formula on a polyhedron in R^4, and leave the decidability of this as an open problem.