Certifying Inexpressibility

Orna Kupferman; Salomon Sickert

doi:10.1007/978-3-030-71995-1_20

. 2021 Mar 23;12650:385–405. doi: 10.1007/978-3-030-71995-1_20

Certifying Inexpressibility

Orna Kupferman ¹⁰, Salomon Sickert ^10,^11,^✉

Editors: Stefan Kiefer⁸, Christine Tasson⁹

PMCID: PMC7984136

Abstract

Different classes of automata on infinite words have different expressive power. Deciding whether a given language $L \subseteq Σ^{ω}$ can be expressed by an automaton of a desired class can be reduced to deciding a game between Prover and Refuter: in each turn of the game, Refuter provides a letter in $Σ$ , and Prover responds with an annotation of the current state of the run (for example, in the case of Büchi automata, whether the state is accepting or rejecting, and in the case of parity automata, what the color of the state is). Prover wins if the sequence of annotations she generates is correct: it is an accepting run iff the word generated by Refuter is in L. We show how a winning strategy for Refuter can serve as a simple and easy-to-understand certificate to inexpressibility, and how it induces additional forms of certificates. Our framework handles all classes of deterministic automata, including ones with structural restrictions like weak automata. In addition, it can be used for refuting separation of two languages by an automaton of the desired class, and for finding automata that approximate L and belong to the desired class.

Keywords: Automata on infinite words, Expressive power, Games

Footnotes

The full version of this article is available from [27]. Orna Kupferman is supported in part by the Israel Science Foundation, grant No. 2357/19. Salomon Sickert is supported in part by the Deutsche Forschungsgemeinschaft (DFG) under project numbers 436811179 and 317422601 (“Verified Model Checkers”), and in part funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 787367 (PaVeS).

Contributor Information

Stefan Kiefer, Email: stekie@cs.ox.ac.uk.

Christine Tasson, Email: christine.tasson@lip6.fr.

Orna Kupferman, Email: orna@cs.huji.ac.il.

Salomon Sickert, Email: salomon.sickert@mail.huji.ac.il, Email: s.sickert@tum.de.

References

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2.Alpern, B., Schneider, F.: Recognizing safety and liveness. Distributed computing 2, 117–126 (1987)
3.Baier, C., de Alfaro, L., Forejt, V., Kwiatkowska, M.: Model checking probabilistic systems. In: Handbook of Model Checking., pp. 963–999. Springer (2018)
4.Baumeister, T., Finkbeiner, B., Torfah, H.: Explainable reactive synthesis. In: 18th Int. Symp. on Automated Technology for Verification and Analysis (2020). 10.1007/978-3-030-59152-6_23
5.Bloem, R., Chatterjee, K., Jobstmann, B.: Graph games and reactive synthesis. In: Handbook of Model Checking., pp. 921–962. Springer (2018)
6.Boigelot, B., Jodogne, S., Wolper, P.: On the use of weak automata for deciding linear arithmetic with integer and real variables. In: Proc. Int. Joint Conf. on Automated Reasoning. Lecture Notes in Computer Science, vol. 2083, pp. 611–625. Springer (2001)
7.Boker, U., Kupferman, O.: Co-ing Büchi made tight and useful. In: Proc. 24th IEEE Symp. on Logic in Computer Science. pp. 245–254 (2009)
8.Büchi, J.: On a decision method in restricted second order arithmetic. In: Proc. Int. Congress on Logic, Method, and Philosophy of Science. 1960. pp. 1–12. Stanford University Press (1962)
9.Büchi, J., Landweber, L.: Solving sequential conditions by finite-state strategies. Trans. AMS 138, 295–311 (1969)
10.Clarke, E.M., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement for symbolic model checking. Journal of the ACM 50(5), 752–794 (2003)
11.Czerwinski, W., Lasota, S., Meyer, R., Muskalla, S., Kumar, K., Saivasan, P.: Regular separability of well-structured transition systems. In: Proc. 29th Int. Conf. on Concurrency Theory. LIPIcs, vol. 118, pp. 35:1–35:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
12.Czerwinski, W., Martens, W., Masopust, T.: Efficient separability of regular languages by subsequences and suffixes. In: Proc. 40th Int. Colloq. on Automata, Languages, and Programming. Lecture Notes in Computer Science, vol. 7966, pp. 150–161. Springer (2013)
13.Dimitrova, R., Finkbeiner, B., Torfah, H.: Approximate automata for omega-regular languages. In: 17th Int. Symp. on Automated Technology for Verification and Analysis. Lecture Notes in Computer Science, vol. 11781, pp. 334–349. Springer (2019)
14.Eisner, C., Fisman, D.: A Practical Introduction to PSL. Springer (2006)
15.Emerson, E., Jutla, C.: The complexity of tree automata and logics of programs. In: Proc. 29th IEEE Symp. on Foundations of Computer Science. pp. 328–337 (1988)
16.Emerson, E., Jutla, C.: Tree automata, $μ$ -calculus and determinacy. In: Proc. 32nd IEEE Symp. on Foundations of Computer Science. pp. 368–377 (1991)
17.Emerson, E., Lei, C.L.: Modalities for model checking: Branching time logic strikes back. Science of Computer Programming 8, 275–306 (1987)
18.Gange, G., Ganty, P., Stuckey, P.: Fixing the state budget: Approximation of regular languages with small dfas. In: 15th Int. Symp. on Automated Technology for Verification and Analysis. Lecture Notes in Computer Science, vol. 10482, pp. 67–83. Springer (2017)
19.Krishnan, S., Puri, A., Brayton, R.: Deterministic $ω$ -automata vis-a-vis deterministic Büchi automata. In: Algorithms and Computations. Lecture Notes in Computer Science, vol. 834, pp. 378–386. Springer (1994)
20.Kupferman, O.: Automata theory and model checking. In: Handbook of Model Checking, pp. 107–151. Springer (2018)
21.Kupferman, O., Morgenstern, G., Murano, A.: Typeness for $ω$ -regular automata. International Journal on the Foundations of Computer Science 17(4), 869–884 (2006)
22.Kupferman, O., Sheinvald-Faragy, S.: Finding shortest witnesses to the nonemptiness of automata on infinite words. In: Proc. 17th Int. Conf. on Concurrency Theory. Lecture Notes in Computer Science, vol. 4137, pp. 492–508. Springer (2006)
23.Kupferman, O., Vardi, M.: On bounded specifications. In: Proc. 8th Int. Conf. on Logic for Programming Artificial Intelligence and Reasoning. Lecture Notes in Computer Science, vol. 2250, pp. 24–38. Springer (2001)
24.Kupferman, O., Vardi, M.: From complementation to certification. Theoretical Computer Science 305, 591–606 (2005)
25.Kupferman, O., Vardi, M.: From linear time to branching time. ACM Transactions on Computational Logic 6(2), 273–294 (2005)
26.Kupferman, O., Vardi, M.: Safraless decision procedures. In: Proc. 46th IEEE Symp. on Foundations of Computer Science. pp. 531–540 (2005)
27.Kupferman, O., Sickert, S.: Certifying inexpressibility (2021), https://arxiv.org/abs/2101.08756, (Full Version)
28.Kurshan, R.: Computer Aided Verification of Coordinating Processes. Princeton Univ. Press (1994)
29.Landweber, L.: Decision problems for $ω$ –automata. Mathematical Systems Theory 3, 376–384 (1969)
30.Leshkowitz, O., Kupferman, O.: On repetition languages. In: 45th Int. Symp. on Mathematical Foundations of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs) (2020)
31.Löding, C.: Methods for the transformation of automata: Complexity and connection to second order logic (1999), M.Sc. Thesis, Christian-Albrechts-University of Kiel
32.Löding, C.: Efficient minimization of deterministic weak $ω$ -automata. Information Processing Letters 79(3), 105–109 (2001)
33.McNaughton, R.: Testing and generating infinite sequences by a finite automaton. Information and Control 9, 521–530 (1966)
34.Meyer, A., Stockmeyer, L.: The equivalence problem for regular expressions with squaring requires exponential space. In: Proc. 13th IEEE Symp. on Switching and Automata Theory. pp. 125–129 (1972)
35.Mostowski, A.: Regular expressions for infinite trees and a standard form of automata. In: Computation Theory. Lecture Notes in Computer Science, vol. 208, pp. 157–168. Springer (1984)
36.Muller, D., Saoudi, A., Schupp, P.: Alternating automata, the weak monadic theory of the tree and its complexity. In: Proc. 13th Int. Colloq. on Automata, Languages, and Programming. Lecture Notes in Computer Science, vol. 226, pp. 275 – 283. Springer (1986)
37.Myhill, J.: Finite automata and the representation of events. Tech. Rep. WADD TR-57-624, pages 112–137, Wright Patterson AFB, Ohio (1957)
38.Nerode, A.: Linear automaton transformations. Proceedings of the American Mathematical Society 9(4), 541–544 (1958)
39.Perrin, D., Pin, J.E.: Infinite words - automata, semigroups, logic and games, Pure and applied mathematics series, vol. 141. Elsevier Morgan Kaufmann (2004)
40.Place, T., Zeitoun, M.: Separating regular languages with first-order logic. Log. Methods Comput. Sci. 12(1) (2016)
41.Rabin, M.: Decidability of second order theories and automata on infinite trees. Transaction of the AMS 141, 1–35 (1969)
42.Safra, S.: On the complexity of $ω$ -automata. In: Proc. 29th IEEE Symp. on Foundations of Computer Science. pp. 319–327 (1988)
43.Safra, S.: Exponential determinization for $ω$ -automata with strong-fairness acceptance condition. In: Proc. 24th ACM Symp. on Theory of Computing (1992)
44.S.Almagor, Chistikov, D., Ouaknine, J., Worrell, J.: O-minimal invariants for linear loops. In: Proc. 45th Int. Colloq. on Automata, Languages, and Programming. LIPIcs, vol. 107, pp. 114:1–114:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
45.Schewe, S.: Büchi complementation made tight. In: Proc. 26th Symp. on Theoretical Aspects of Computer Science. LIPIcs, vol. 3, pp. 661–672. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany (2009)
46.Schewe, S.: Beyond Hyper-Minimisation—Minimising DBAs and DPAs is NP-Complete. In: Proc. 30th Conf. on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), vol. 8, pp. 400–411 (2010)
47.di Stasio, A., Murano, A., Vardi, M.: Solving parity games: Explicit vs symbolic. In: 23rd International Conference on Implementation and Application of Automata. Lecture Notes in Computer Science, vol. 10977, pp. 159–172. Springer (2018)
48.Thomas, W.: Automata on infinite objects. Handbook of Theoretical Computer Science pp. 133–191 (1990).
49.Vardi, M., Wolper, P.: Reasoning about infinite computations. Information and Computation 115(1), 1–37 (1994)
50.Wagner, K.: On $ω$ -regular sets. Information and Control 43, 123–177 (1979)

[CR1] 1.Almagor, S., Lahijanian, M.: Explainable multi agent path finding. In: Proc. 19th International Conference on Autonomous Agents and Multiagent Systems. pp. 34–42 (2020)

[CR2] 2.Alpern, B., Schneider, F.: Recognizing safety and liveness. Distributed computing 2, 117–126 (1987)

[CR3] 3.Baier, C., de Alfaro, L., Forejt, V., Kwiatkowska, M.: Model checking probabilistic systems. In: Handbook of Model Checking., pp. 963–999. Springer (2018)

[CR4] 4.Baumeister, T., Finkbeiner, B., Torfah, H.: Explainable reactive synthesis. In: 18th Int. Symp. on Automated Technology for Verification and Analysis (2020). 10.1007/978-3-030-59152-6_23

[CR5] 5.Bloem, R., Chatterjee, K., Jobstmann, B.: Graph games and reactive synthesis. In: Handbook of Model Checking., pp. 921–962. Springer (2018)

[CR6] 6.Boigelot, B., Jodogne, S., Wolper, P.: On the use of weak automata for deciding linear arithmetic with integer and real variables. In: Proc. Int. Joint Conf. on Automated Reasoning. Lecture Notes in Computer Science, vol. 2083, pp. 611–625. Springer (2001)

[CR7] 7.Boker, U., Kupferman, O.: Co-ing Büchi made tight and useful. In: Proc. 24th IEEE Symp. on Logic in Computer Science. pp. 245–254 (2009)

[CR8] 8.Büchi, J.: On a decision method in restricted second order arithmetic. In: Proc. Int. Congress on Logic, Method, and Philosophy of Science. 1960. pp. 1–12. Stanford University Press (1962)

[CR9] 9.Büchi, J., Landweber, L.: Solving sequential conditions by finite-state strategies. Trans. AMS 138, 295–311 (1969)

[CR10] 10.Clarke, E.M., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement for symbolic model checking. Journal of the ACM 50(5), 752–794 (2003)

[CR11] 11.Czerwinski, W., Lasota, S., Meyer, R., Muskalla, S., Kumar, K., Saivasan, P.: Regular separability of well-structured transition systems. In: Proc. 29th Int. Conf. on Concurrency Theory. LIPIcs, vol. 118, pp. 35:1–35:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

[CR12] 12.Czerwinski, W., Martens, W., Masopust, T.: Efficient separability of regular languages by subsequences and suffixes. In: Proc. 40th Int. Colloq. on Automata, Languages, and Programming. Lecture Notes in Computer Science, vol. 7966, pp. 150–161. Springer (2013)

[CR13] 13.Dimitrova, R., Finkbeiner, B., Torfah, H.: Approximate automata for omega-regular languages. In: 17th Int. Symp. on Automated Technology for Verification and Analysis. Lecture Notes in Computer Science, vol. 11781, pp. 334–349. Springer (2019)

[CR14] 14.Eisner, C., Fisman, D.: A Practical Introduction to PSL. Springer (2006)

[CR15] 15.Emerson, E., Jutla, C.: The complexity of tree automata and logics of programs. In: Proc. 29th IEEE Symp. on Foundations of Computer Science. pp. 328–337 (1988)

[CR16] 16.Emerson, E., Jutla, C.: Tree automata, $μ$ -calculus and determinacy. In: Proc. 32nd IEEE Symp. on Foundations of Computer Science. pp. 368–377 (1991)

[CR17] 17.Emerson, E., Lei, C.L.: Modalities for model checking: Branching time logic strikes back. Science of Computer Programming 8, 275–306 (1987)

[CR18] 18.Gange, G., Ganty, P., Stuckey, P.: Fixing the state budget: Approximation of regular languages with small dfas. In: 15th Int. Symp. on Automated Technology for Verification and Analysis. Lecture Notes in Computer Science, vol. 10482, pp. 67–83. Springer (2017)

[CR19] 19.Krishnan, S., Puri, A., Brayton, R.: Deterministic $ω$ -automata vis-a-vis deterministic Büchi automata. In: Algorithms and Computations. Lecture Notes in Computer Science, vol. 834, pp. 378–386. Springer (1994)

[CR20] 20.Kupferman, O.: Automata theory and model checking. In: Handbook of Model Checking, pp. 107–151. Springer (2018)

[CR21] 21.Kupferman, O., Morgenstern, G., Murano, A.: Typeness for $ω$ -regular automata. International Journal on the Foundations of Computer Science 17(4), 869–884 (2006)

[CR22] 22.Kupferman, O., Sheinvald-Faragy, S.: Finding shortest witnesses to the nonemptiness of automata on infinite words. In: Proc. 17th Int. Conf. on Concurrency Theory. Lecture Notes in Computer Science, vol. 4137, pp. 492–508. Springer (2006)

[CR23] 23.Kupferman, O., Vardi, M.: On bounded specifications. In: Proc. 8th Int. Conf. on Logic for Programming Artificial Intelligence and Reasoning. Lecture Notes in Computer Science, vol. 2250, pp. 24–38. Springer (2001)

[CR24] 24.Kupferman, O., Vardi, M.: From complementation to certification. Theoretical Computer Science 305, 591–606 (2005)

[CR25] 25.Kupferman, O., Vardi, M.: From linear time to branching time. ACM Transactions on Computational Logic 6(2), 273–294 (2005)

[CR26] 26.Kupferman, O., Vardi, M.: Safraless decision procedures. In: Proc. 46th IEEE Symp. on Foundations of Computer Science. pp. 531–540 (2005)

[CR27] 27.Kupferman, O., Sickert, S.: Certifying inexpressibility (2021), https://arxiv.org/abs/2101.08756, (Full Version)

[CR28] 28.Kurshan, R.: Computer Aided Verification of Coordinating Processes. Princeton Univ. Press (1994)

[CR29] 29.Landweber, L.: Decision problems for $ω$ –automata. Mathematical Systems Theory 3, 376–384 (1969)

[CR30] 30.Leshkowitz, O., Kupferman, O.: On repetition languages. In: 45th Int. Symp. on Mathematical Foundations of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs) (2020)

[CR31] 31.Löding, C.: Methods for the transformation of automata: Complexity and connection to second order logic (1999), M.Sc. Thesis, Christian-Albrechts-University of Kiel

[CR32] 32.Löding, C.: Efficient minimization of deterministic weak $ω$ -automata. Information Processing Letters 79(3), 105–109 (2001)

[CR33] 33.McNaughton, R.: Testing and generating infinite sequences by a finite automaton. Information and Control 9, 521–530 (1966)

[CR34] 34.Meyer, A., Stockmeyer, L.: The equivalence problem for regular expressions with squaring requires exponential space. In: Proc. 13th IEEE Symp. on Switching and Automata Theory. pp. 125–129 (1972)

[CR35] 35.Mostowski, A.: Regular expressions for infinite trees and a standard form of automata. In: Computation Theory. Lecture Notes in Computer Science, vol. 208, pp. 157–168. Springer (1984)

[CR36] 36.Muller, D., Saoudi, A., Schupp, P.: Alternating automata, the weak monadic theory of the tree and its complexity. In: Proc. 13th Int. Colloq. on Automata, Languages, and Programming. Lecture Notes in Computer Science, vol. 226, pp. 275 – 283. Springer (1986)

[CR37] 37.Myhill, J.: Finite automata and the representation of events. Tech. Rep. WADD TR-57-624, pages 112–137, Wright Patterson AFB, Ohio (1957)

[CR38] 38.Nerode, A.: Linear automaton transformations. Proceedings of the American Mathematical Society 9(4), 541–544 (1958)

[CR39] 39.Perrin, D., Pin, J.E.: Infinite words - automata, semigroups, logic and games, Pure and applied mathematics series, vol. 141. Elsevier Morgan Kaufmann (2004)

[CR40] 40.Place, T., Zeitoun, M.: Separating regular languages with first-order logic. Log. Methods Comput. Sci. 12(1) (2016)

[CR41] 41.Rabin, M.: Decidability of second order theories and automata on infinite trees. Transaction of the AMS 141, 1–35 (1969)

[CR42] 42.Safra, S.: On the complexity of $ω$ -automata. In: Proc. 29th IEEE Symp. on Foundations of Computer Science. pp. 319–327 (1988)

[CR43] 43.Safra, S.: Exponential determinization for $ω$ -automata with strong-fairness acceptance condition. In: Proc. 24th ACM Symp. on Theory of Computing (1992)

[CR44] 44.S.Almagor, Chistikov, D., Ouaknine, J., Worrell, J.: O-minimal invariants for linear loops. In: Proc. 45th Int. Colloq. on Automata, Languages, and Programming. LIPIcs, vol. 107, pp. 114:1–114:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

[CR45] 45.Schewe, S.: Büchi complementation made tight. In: Proc. 26th Symp. on Theoretical Aspects of Computer Science. LIPIcs, vol. 3, pp. 661–672. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany (2009)

[CR46] 46.Schewe, S.: Beyond Hyper-Minimisation—Minimising DBAs and DPAs is NP-Complete. In: Proc. 30th Conf. on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), vol. 8, pp. 400–411 (2010)

[CR47] 47.di Stasio, A., Murano, A., Vardi, M.: Solving parity games: Explicit vs symbolic. In: 23rd International Conference on Implementation and Application of Automata. Lecture Notes in Computer Science, vol. 10977, pp. 159–172. Springer (2018)

[CR48] 48.Thomas, W.: Automata on infinite objects. Handbook of Theoretical Computer Science pp. 133–191 (1990).

[CR49] 49.Vardi, M., Wolper, P.: Reasoning about infinite computations. Information and Computation 115(1), 1–37 (1994)

[CR50] 50.Wagner, K.: On $ω$ -regular sets. Information and Control 43, 123–177 (1979)

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Certifying Inexpressibility

Orna Kupferman

Salomon Sickert

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Footnotes

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Certifying Inexpressibility

Orna Kupferman

Salomon Sickert

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