Abstract
We show how to construct uniform interpolants in the context of the modal mu-calculus. D’Agostino and Hollenberg (2000) were the first to prove that this logic has the uniform interpolation property, employing a combination of semantic and syntactic methods. This article outlines a purely proof-theoretic approach to the problem based on insights from the cyclic proof theory of mu-calculus. We argue the approach has the potential to lend itself to other temporal and fixed point logics.
Supported by the Knut and Alice Wallenberg Foundation [2015.0179] and the Swedish Research Council [2016-03502 & 2017-05111]. The authors would like to thank the anonymous referees for their valuable comments.
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Notes
- 1.
This definition depends on the choice of underlying logic. For example, in the case of polymodal logic, \(\mathsf {Voc}({\varphi })\) is the set of propositional constants and modal actions occurring in \(\varphi \).
- 2.
The case distinction based on the provability of \( { \varGamma _i , \pi \Rightarrow \varnothing }\) brings into question the computational cost of constructing uniform interpolants. Lemmas 1 and 3, however, provide that provability of sequents with empty consequent is implicit in the interpolation template.
- 3.
We assume Definition 4 is generalised to derivations with \( \mathsf {GMod}\). No additional restrictions are necessary to accommodate this rule
- 4.
A non-exhaustive list of cyclic proofs systems include: first-order logic with inductive definitions [8, 9, 11], arithmetic [7, 17, 39], linear logic [3, 4, 20], modal and dynamic logics [1, 22, 23, 28, 30, 38, 40, 41, 44], program semantics [37], automated theorem proving [10, 36, 42], higher-order logic [31] and algebras and lattices [18, 19, 32].
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Afshari, B., Leigh, G.E., Menéndez Turata, G. (2021). Uniform Interpolation from Cyclic Proofs: The Case of Modal Mu-Calculus. In: Das, A., Negri, S. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2021. Lecture Notes in Computer Science(), vol 12842. Springer, Cham. https://doi.org/10.1007/978-3-030-86059-2_20
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