Interpolation for the two-way modal mu-calculus
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915438901854208 |
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| author | Kloibhofer, Johannes Venema, Yde |
| author_facet | Kloibhofer, Johannes Venema, Yde |
| contents | The two-way modal mu-calculus is the extension of the (standard) one-way mu-calculus with converse (backward-looking) modalities. For this logic we introduce two new sequent-style proof calculi: a non-wellfounded system admitting infinite branches and a finitary, cyclic version of this that employs annotations. As is common in sequent systems for two-way modal logics, our calculi feature an analytic cut rule. What distinguishes our approach is the use of so-called trace atoms, which serve to apply Vardi's two-way automata in a proof-theoretic setting. We prove soundness and completeness for both systems and subsequently use the cyclic calculus to show that the two-way mu-calculus has the (local) Craig interpolation property, with respect to both propositions and modalities. Our proof uses a version of Maehara's method adapted to cyclic proof systems. As a corollary we prove that the two-way mu-calculus also enjoys Beth's definability property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12899 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Interpolation for the two-way modal mu-calculus Kloibhofer, Johannes Venema, Yde Logic in Computer Science Logic The two-way modal mu-calculus is the extension of the (standard) one-way mu-calculus with converse (backward-looking) modalities. For this logic we introduce two new sequent-style proof calculi: a non-wellfounded system admitting infinite branches and a finitary, cyclic version of this that employs annotations. As is common in sequent systems for two-way modal logics, our calculi feature an analytic cut rule. What distinguishes our approach is the use of so-called trace atoms, which serve to apply Vardi's two-way automata in a proof-theoretic setting. We prove soundness and completeness for both systems and subsequently use the cyclic calculus to show that the two-way mu-calculus has the (local) Craig interpolation property, with respect to both propositions and modalities. Our proof uses a version of Maehara's method adapted to cyclic proof systems. As a corollary we prove that the two-way mu-calculus also enjoys Beth's definability property. |
| title | Interpolation for the two-way modal mu-calculus |
| topic | Logic in Computer Science Logic |
| url | https://arxiv.org/abs/2505.12899 |