Dynamic Polyhedral Logic

2026-08-26

AI Summary

本文通过引入基于多面体语义和路径空间可达性算子的时空多面体可达性逻辑，解决了动态拓扑逻辑中的问题，并证明了其在可逆动力系统中的健全性和完备性。

Abstract

We introduce spatio-temporal polyhedral reachability logics, extending dynamic topological logic with polyhedral semantics and a path-based spatial reachability operator. Formulas are interpreted over polyhedra, with admissible valuations ranging over polyhedral subsets; the spatial modality is interpreted as interior, the binary operator $γ(\varphi,ψ)$ expresses reachability of a $ψ$-point through a $\varphi$-region, and the temporal modalities are interpreted by a PL-homeomorphism and its inverse. We define h-dynamic reachability spaces and axiomatize the corresponding h-dynamic extensions of the known reachability logics of topological, finite, Alexandroff, and polyhedral spaces. The main result is soundness and completeness for the intended classes of invertible dynamical systems.

Problem

• dynamic polyhedral logic
• spatio-temporal reachability
• path-based spatial reachability

Innovation

• spatio-temporal polyhedral reachability logics
• polyhedral semantics
• path-based spatial reachability operator
• h-dynamic reachability spaces