Abstract
We study alternating automata with qualitative semantics over infinite binary trees: Alternation means that two opposing players construct a decoration of the input tree called a run, and the qualitative semantics says that a run of the automaton is accepting if almost all branches of the run are accepting. In this article, we prove a positive and a negative result for the emptiness problem of alternating automata with qualitative semantics.
The positive result is the decidability of the emptiness problem for the case of Büchi acceptance condition. An interesting aspect of our approach is that we do not extend the classical solution for solving the emptiness problem of alternating automata, which first constructs an equivalent non-deterministic automaton. Instead, we directly construct an emptiness game making use of imperfect information.
The negative result is the undecidability of the emptiness problem for the case of co-Büchi acceptance condition. This result has two direct consequences: the undecidability of monadic second-order logic extended with the qualitative path-measure quantifier and the undecidability of the emptiness problem for alternating tree automata with non-zero semantics, a recently introduced probabilistic model of alternating tree automata.
- Christel Baier, Marcus Größer, and Nathalie Bertrand. 2012. Probabilistic ω-automata. J. ACM 59, 1 (2012), 1.Google ScholarDigital Library
- Vince Bárány, Łukasz Kaiser, and Alex Rabinovich. 2010. Expressing cardinality quantifiers in monadic second-order logic over trees. Fundam. Inf. 100, 1--4 (2010), 1--17.Google ScholarDigital Library
- Raphaël Berthon, Emmanuel Filiot, Shibashis Guha, Bastien Maubert, Nello Murano, Laureline Pinault, Jean-François Raskin, and Sasha Rubin. 2019. Monadic second-order logic with path-measure quantifier is undecidable. CoRR abs/1901.04349.Google Scholar
- Nathalie Bertrand, Blaise Genest, and Hugo Gimbert. 2009. Qualitative determinacy and decidability of stochastic games with signals. In Proceedings of the 24th Annual IEEE Symposium on Logic in Computer Science. IEEE, 319--328.Google ScholarDigital Library
- Mikołaj Bojańczyk. 2016. Thin MSO with a probabilistic path quantifier. In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (LIPIcs), Vol. 55. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 96:1--96:13.Google Scholar
- Mikołaj Bojańczyk, Hugo Gimbert, and Edon Kelmendi. 2017. Emptiness of zero automata is decidable. In Proceedings of the 44th International Colloquium on Automata, Languages, and Programming (LIPIcs), Vol. 80. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 106:1--106:13.Google Scholar
- Mikołaj Bojańczyk, Edon Kelmendi, and Michal Skrzypczak. 2019. MSO+∇ is undecidable. In Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science. IEEE, 1--13.Google ScholarDigital Library
- Arnaud Carayol, Axel Haddad, and Olivier Serre. 2014. Randomization in automata on infinite trees. ACM Trans. Comput. Logic 15, 3 (2014), 24:1--24:33.Google ScholarDigital Library
- Arnaud Carayol, Christof Löding, and Olivier Serre. 2018. Pure strategies in imperfect information stochastic games. Fundam. Inf. 160, 4 (2018), 361--384.Google ScholarDigital Library
- Krishnendu Chatterjee and Laurent Doyen. 2014. Partial-observation stochastic games: How to win when belief fails. ACM Trans. Comput. Logic 15, 2 (2014), 16:1--16:44.Google ScholarDigital Library
- Krishnendu Chatterjee, Laurent Doyen, Thomas A. Henzinger, and Jean-François Raskin. 2007. Algorithms for omega-regular games with imperfect information. Logic. Methods Comput. Sci. 3, 3 (2007).Google Scholar
- Costas Courcoubetis and Mihalis Yannakakis. 1990. Markov decision processes and regular events (extended abstract). In Proceedings of the 17th International Colloquium on Automata, Languages, and Programming (ICALP'90), Lecture Notes in Computer Science, Vol. 443. Springer, 336--349.Google Scholar
- Luca de Alfaro. 1999. The verification of probabilistic systems under memoryless partial-information policies is hard. In Proceedings of the 2nd International Workshop on Probabilistic Methods in Verification. 19--32.Google Scholar
- Ronald Fagin, Joseph Y. Halpern, Yoram. Moses, and Moshe Y. Vardi. 1995. Reasoning about Knowledge. MIT Press.Google Scholar
- Nathanaël Fijalkow. 2017. Undecidability results for probabilistic automata. SIGLOG News 4, 4 (2017), 10--17.Google ScholarDigital Library
- Nathanaël Fijalkow, Hugo Gimbert, Edon Kelmendi, and Youssouf Oualhadj. 2015. Deciding the value 1 problem for probabilistic leaktight automata. Logic. Methods Comput. Sci. 11, 2 (2015), 1--42.Google Scholar
- Nathanaël Fijalkow, Sophie Pinchinat, and Olivier Serre. 2013. Emptiness of alternating tree automata using games with imperfect information. In Proceedings of IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (LIPIcs), Vol. 24. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 299--311.Google Scholar
- Nathanaël Fijalkow, Sophie Pinchinat, and Olivier Serre. 2013. Emptiness of Alternating Tree Automata Using Games with Imperfect Information. Retrieved from https://hal.inria.fr/hal-01260682.Google Scholar
- Paulin Fournier and Hugo Gimbert. 2018. Alternating nonzero automata. In Proceedings of the 29th International Conference on Concurrency Theory (LIPIcs), Vol. 118. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 13:1--13:16.Google Scholar
- Hugo Gimbert and Youssouf Oualhadj. 2010. Probabilistic automata on finite words: Decidable and undecidable problems. In Proceedings of the 37th International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, Vol. 6199. Springer, 527--538.Google Scholar
- Hugo Gimbert and Wiesław Zielonka. 2007. Perfect information stochastic priority games. In Proceedings of the 34th International Colloquium on Automata, Languages, and Programming, Lecture Notes in Computer Science, Vol. 4596. Springer, 850--861.Google Scholar
- Vincent Gripon and Olivier Serre. 2009. Qualitative concurrent stochastic games with imperfect information. In Proceedings of the 36th International Colloquium on Automata, Languages, and Programming, Lecture Notes in Computer Science, Vol. 5556. Springer, 200--211.Google Scholar
- Antonín Kučera. 2011. Turn-based stochastic games. In Lectures in Game Theory for Computer Scientists, Krzysztof R. Apt and Erich Grdel (Eds.). Cambridge University Press, New York, NY, Chapter 5, 146--184.Google Scholar
- Henryk Michalewski and Matteo Mio. 2016. Measure quantifier in monadic second order logic. In Proceedings of the International Symposium on Logical Foundations of Computer Science, Lecture Notes in Computer Science, Vol. 9537. Springer, 267--282.Google Scholar
- Matteo Mio, Michał Skrzypczak, and Henryk Michalewski. 2018. Monadic second order logic with measure and category quantifiers. Logic. Methods Comput. Sci. 14, 2 (2018).Google Scholar
- Azaria Paz. 1971. Introduction to Probabilistic Automata. Academic Press.Google ScholarDigital Library
- Martin L. Puterman. 1994. Markov Decision Processes: Discrete Stochastic Dynamic Programming. John Wiley 8 Sons, Inc., New York, NY.Google Scholar
- Michael O. Rabin. 1963. Probabilistic automata. Inf. Contr. 6, 3 (1963), 230--245.Google ScholarCross Ref
- Michael O. Rabin. 1969. Decidability of second-order theories and automata on infinite trees. Trans. AMS 141 (1969), 1--35. https://www.ams.org/journals/tran/1969-141-00/S0002-9947-1969-0246760-1/.Google Scholar
- J. H. Reif. 1979. Universal games of incomplete information. In Proceedings of the Annual ACM Symposium on Theory of Computing (STOC’79). ACM, 288--308.Google ScholarDigital Library
- J. H. Reif. 1984. The complexity of two-player games of incomplete information. J. Comput. Syst. Sci. 29, 2 (1984), 274--301.Google ScholarCross Ref
- Wolfgang Thomas. 1997. Languages, automata, and logic. In Handbook of Formal Language Theory, G. Rozenberg and A. Salomaa (Eds.). Vol. III. 389--455.Google Scholar
- Wiesław Zielonka. 1998. Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theor. Comput. Sci. 200, 1--2 (1998), 135--183.Google ScholarDigital Library
Index Terms
- Alternating Tree Automata with Qualitative Semantics
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