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A Categorical Approach to Coalgebraic Fixpoint Logic (2405.00237v1)
Published 30 Apr 2024 in cs.LO
Abstract: We define a framework for incorporating alternation-free fixpoint logics into the dual-adjunction setup for coalgebraic modal logics. We achieve this by using order-enriched categories. We give a least-solution semantics as well as an initial algebra semantics, and prove they are equivalent. We also show how to place the alternation-free coalgebraic $\mu$-calculus in this framework, as well as PDL and a logic with a probabilistic dynamic modality.
- Positive fragments of coalgebraic logics. In CALCO, volume 8089 of Lecture Notes in Computer Science, pages 51–65. Springer, 2013.
- Coalgebra learning via duality. In FoSSaCS, volume 11425 of Lecture Notes in Computer Science, pages 62–79. Springer, 2019.
- Janusz A. Brzozowski. Derivatives of regular expressions. J. ACM, 11(4):481–494, 1964.
- EXPTIME tableaux for the coalgebraic mu-calculus. Log. Methods Comput. Sci., 7(3), 2011.
- Modal logics are coalgebraic. In BCS Int. Acad. Conf, pages 128–140. British Computer Society, 2008.
- Constructive versions of Tarski’s fixed point theorems. Pacific Journal of Mathematics, 82(1):43 – 57, 1979.
- Fredrik Dahlqvist. Coalgebraic completeness-via-canonicity - principles and applications. In CMCS, volume 9608 of Lecture Notes in Computer Science, pages 174–194. Springer, 2016.
- A characterization theorem for the alternation-free fragment of the modal μ𝜇\mathrm{\mu}italic_μ-calculus. In LICS, pages 478–487. IEEE Computer Society, 2013.
- Non-iterative modal logics are coalgebraic. In AiML, pages 229–248. College Publications, 2020.
- Hennessy-Milner results for probabilistic PDL. In MFPS, volume 352 of Electronic Notes in Theoretical Computer Science, pages 283–304. Elsevier, 2020.
- Weak completeness of coalgebraic dynamic logics. In FICS, volume 191 of EPTCS, pages 90–104, 2015.
- Generic model checking for modal fixpoint logics in COOL-MC. In VMCAI (1), volume 14499 of Lecture Notes in Computer Science, pages 171–185. Springer, 2024.
- Global caching for the alternation-free μ𝜇\muitalic_μ-calculus. In CONCUR, volume 59 of LIPIcs, pages 34:1–34:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016.
- On observing nondeterminism and concurrency. In ICALP, volume 85 of Lecture Notes in Computer Science, pages 299–309. Springer, 1980.
- Max Kelly. The basic concepts of enriched category theory. Reprints in Theory and Applications of Categories [electronic only], 2005, 01 2005.
- Bartek Klin. Coalgebraic modal logic beyond sets. In MFPS, volume 173 of Electronic Notes in Theoretical Computer Science, pages 177–201. Elsevier, 2007.
- Dexter Kozen. A probabilistic PDL. J. Comput. Syst. Sci., 30(2):162–178, 1985.
- An elementary proof of the completness of PDL. Theor. Comput. Sci., 14:113–118, 1981.
- Algebraic semantics for coalgebraic logics. In CMCS, volume 106 of Electronic Notes in Theoretical Computer Science, pages 219–241. Elsevier, 2004.
- Coalgebraic semantics of modal logics: An overview. Theor. Comput. Sci., 412(38):5070–5094, 2011.
- Enriched logical connections. Appl. Categorical Struct., 21(4):349–377, 2013.
- A focus system for the alternation-free μ𝜇\muitalic_μ-calculus. In TABLEAUX, volume 12842 of Lecture Notes in Computer Science, pages 371–388. Springer, 2021.
- Testing semantics: Connecting processes and process logics. In AMAST, volume 4019 of Lecture Notes in Computer Science, pages 308–322. Springer, 2006.
- Structural congruence for bialgebraic semantics. J. Log. Algebraic Methods Program., 85(6):1268–1291, 2016.
- Steps and traces. J. Log. Comput., 31(6):1482–1525, 2021.
- Jan J. M. M. Rutten. Universal coalgebra: a theory of systems. Theor. Comput. Sci., 249(1):3–80, 2000.
- PSPACE bounds for rank-1 modal logics. ACM Trans. Comput. Log., 10(2):13:1–13:33, 2009.
- Flat coalgebraic fixed point logics. In CONCUR, volume 6269 of Lecture Notes in Computer Science, pages 524–538. Springer, 2010.
- Yde Venema. Automata and fixed point logic: A coalgebraic perspective. Inf. Comput., 204(4):637–678, 2006.