The Alternation Hierarchy of the $$\mu $$-calculus over Weakly Transitive Frames

Pacheco, Leonardo; Tanaka, Kazuyuki

doi:10.1007/978-3-031-15298-6_13

Cited by 3 publications

(1 citation statement)

References 28 publications

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“…This raises the question of whether it is possible to define simulability in anything short of the full µ-calculus. As an upper bound, it is known that the topological µ-calculus collapses to its alternation-free fragment [20], but this fragment is substantially more complex from a syntactic perspective than the tangled fragments. Can something similar be obtained in the general topological setting?…”

Section: Expressive Completeness Of Tangled Fragmentsmentioning

confidence: 99%

Fixed Point Logics on Hemimetric Spaces

Fernández–Duque

Gougeon

2023

2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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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 wellknown 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.

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Section: Expressive Completeness Of Tangled Fragmentsmentioning

confidence: 99%

Fixed Point Logics on Hemimetric Spaces

Fernández–Duque

Gougeon

2023

2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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The Universal Tangle for Spatial Reasoning

Fernández-Duque,

Papafilippou

2023

Lecture Notes in Computer Science

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The Topological Mu-Calculus: Completeness and Decidability

Baltag,

Bezhanishvili,

Fernández-Duque

2023

J. ACM

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We study the topological μ -calculus, based on both Cantor derivative and closure modalities, proving completeness, decidability and finite model property over general topological spaces, as well as over T 0 and T D spaces. We also investigate the relational μ -calculus, providing general completeness results for all natural fragments of the μ -calculus over many different classes of relational frames. Unlike most other such proofs for μ -calculi, ours is model-theoretic, making an innovative use of a known method from modal logic (–the ‘final’ submodel of the canonical model), that has the twin advantages of great generality and essential simplicity.

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The Alternation Hierarchy of the $$\mu $$-calculus over Weakly Transitive Frames

Cited by 3 publications

References 28 publications

Fixed Point Logics on Hemimetric Spaces

Fixed Point Logics on Hemimetric Spaces

The Universal Tangle for Spatial Reasoning

The Topological Mu-Calculus: Completeness and Decidability

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