Separating Regular Languages over Infinite Words with Respect to the Wagner Hierarchy
We investigate the separation problem for regular ω-languages with respect to the Wagner hierarchy where the input languages are given as deterministic Muller automata (DMA). We show that a minimal separating DMA can be computed in exponential time and that some languages require separators of exponential size. Further, we show that in this setting it can be decided in polynomial time whether a separator exists on a certain level of the Wagner hierarchy and that emptiness of the intersection of two languages given by DMAs can be decided in polynomial time. Finally, we show that separation can also be decided in polynomial time if the input languages are given as deterministic parity automata.
Separation
Regular
Wagner Hierarchy
Muller Automata
Parity Automata
Product Automata
Membership
Theory of computation~Automata over infinite objects
Theory of computation~Regular languages
46:1-46:13
Regular Paper
I want to thank Christof Löding who supervised my Master thesis on which this paper is based and Dietrich Kuske for clarifying discussions. Further, I want to thank the reviewers of former versions of this paper for their helpful suggestions.
Christopher
Hugenroth
Christopher Hugenroth
TU Ilmenau, Germany
10.4230/LIPIcs.FSTTCS.2021.46
Udi Boker. On the (in) succinctness of Muller automata. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017.
Udi Boker. Why these automata types? In LPAR, volume 18, pages 143-163, 2018.
Olivier Carton and Ramón Maceiras. Computing the Rabin index of a parity automaton. RAIRO-Theoretical Informatics and Applications, 33(6):495-505, 1999.
Thomas Colcombet and Christof Löding. The nesting-depth of disjunctive μ-calculus for tree languages and the limitedness problem. In International Workshop on Computer Science Logic, pages 416-430. Springer, 2008.
Damian Niwiński and Igor Walukiewicz. Relating hierarchies of word and tree automata. In Annual Symposium on Theoretical Aspects of Computer Science, pages 320-331. Springer, 1998.
Damian Niwiński and Igor Walukiewicz. Deciding nondeterministic hierarchy of deterministic tree automata. Electronic Notes in Theoretical Computer Science, 123:195-208, 2005.
Dominique Perrin and Jean-Éric Pin. Infinite words: automata, semigroups, logic and games. Academic Press, 2004.
Thomas Place and Marc Zeitoun. Separating regular languages with first-order logic. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 1-10, 2014.
Thomas Place and Marc Zeitoun. The tale of the quantifier alternation hierarchy of first-order logic over words. ACM SIGLOG News, 2(3):4-17, 2015.
Shmuel Safra. Complexity of automata on infinite objects. PhD thesis, The Weizmann Institute of Science, 1989.
Marcel Paul Schützenberger. On finite monoids having only trivial subgroups. Inf. Control., 8(2):190-194, 1965.
Klaus Wagner. On ω-regular sets. Information and control, 43(2):123-177, 1979.
Christopher Hugenroth
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode