MoL-2023-37: Dekker, P. Maurice (2023) Polyhedral semantics of modal logic. [Report]
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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.
| Item Type: | Report |
|---|---|
| Report Nr: | MoL-2023-37 |
| Series Name: | Master of Logic Thesis (MoL) Series |
| Year: | 2023 |
| Subjects: | Logic Mathematics |
| Depositing User: | Dr Marco Vervoort |
| Date Deposited: | 26 Feb 2024 13:13 |
| Last Modified: | 26 Feb 2024 13:13 |
| URI: | https://eprints.illc.uva.nl/id/eprint/2302 |
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