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arXiv · 2511.02597

The mu-calculus' Alternation Hierarchy is Strict over Non-Trivial Fusion Logics

Leonardo Pacheco
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Abstract

The modal mu-calculus is obtained by adding least and greatest fixed-point operators to modal logic. Its alternation hierarchy classifies the mu-formulas by their alternation depth: a measure of the codependence of their least and greatest fixed-point operators. The mu-calculus' alternation hierarchy is strict over the class of all Kripke frames: for all n, there is a mu-formula with alternation depth n+1 which is not equivalent to any formula with alternation depth n. This does not always happen if we restrict the semantics. For example, every mu-formula is equivalent to a formula without fixed-point operators over S5 frames. We show that the multimodal mu-calculus' alternation hierarchy is strict over non-trivial fusions of modal logics. We also comment on two examples of multimodal logics where the mu-calculus collapses to modal logic.

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Leonardo Pacheco. 2025-11-04. The mu-calculus' Alternation Hierarchy is Strict over Non-Trivial Fusion Logics. https://doi.org/10.4204/eptcs.435.8

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