Fixed Point Logics on Hemimetric Spaces

0574237 - ÚI 2024 RIV US eng C - Conference Paper (international conference)
Fernández-Duque, David - Gougeon, Q.
Fixed Point Logics on Hemimetric Spaces.
38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) Proceedings. New York: IEEE, 2023, č. článku 190687. ISBN 979-8-3503-3588-0.
[LICS 2023: Annual ACM/IEEE Symposium on Logic in Computer Science /38./. Boston (US), 26.06.2023-29.06.2023]
R&D Projects: GA ČR(CZ) GA22-01137S
Institutional support: RVO:67985807
Keywords : Computer science * Semantics * Extraterrestrial measurements * Behavioral sciences * Proposals * Standards
OECD category: Pure mathematics
https://dx.doi.org/10.1109/LICS56636.2023.10175784

The μ-calculus can be interpreted over metric spaces and is known to enjoy, among other celebrated properties, variants of the McKinsey-Tarski completeness theorem and of Dawar and Otto's modal characterization theorem. In its topological form, this theorem states that every topological fixed point may be defined in terms of the tangled derivative, a polyadic generalization of Cantor's perfect core. However, these results fail when spaces not satisfying basic separation axioms are considered, in which case the base modal logic is not the well-known K4, but the weaker wK4.In this paper we show how these shortcomings may be overcome. First, we consider semantics over the wider class of hemimetric spaces, and obtain metric completeness results for wK4 and related logics. In this setting, the Dawar-Otto theorem still fails, but we argue that this is due to the tangled derivative not being suitably defined for general application in arbitrary topological spaces. We thus introduce the hybrid tangle, which coincides with the tangled derivative over metric spaces but is better behaved in general. We show that only the hybrid tangle suffices to define simulability of finite structures, a key 'test case' for an expressively complete fragment of the μ-calculus.
Permanent Link: https://hdl.handle.net/11104/0344576