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Büchi Automata Can Have Smaller Quotients

  • Lorenzo Clemente
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6756)

Abstract

We study novel simulation-like preorders for quotienting nondeterministic Büchi automata. We define fixed-word delayed simulation, a new preorder coarser than delayed simulation. We argue that fixed-word simulation is the coarsest forward simulation-like preorder which can be used for quotienting Büchi automata, thus improving our understanding of the limits of quotienting. Also, we show that computing fixed-word simulation is PSPACE-complete.

On the practical side, we introduce proxy simulations, which are novel polynomial-time computable preorders sound for quotienting. In particular, delayed proxy simulation induce quotients that can be smaller by an arbitrarily large factor than direct backward simulation. We derive proxy simulations as the product of a theory of refinement transformers: A refinement transformer maps preorders nondecreasingly, preserving certain properties. We study under which general conditions refinement transformers are sound for quotienting.

Keywords

Direct Simulation Forward Simulation Input Word Simulation Game Deterministic Automaton 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Abdulla, P., Chen, Y.-F., Holik, L., Vojnar, T.: Mediating for Reduction. In: FSTTCS, pp. 1–12. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik (2009)Google Scholar
  2. 2.
    Aziz, A., Singhal, V., Swamy, G.M., Brayton, R.K.: Minimizing Interacting Finite State Machines. Technical Report UCB/ERL M93/68, UoC, Berkeley (1993)Google Scholar
  3. 3.
    Clemente, L.: Büchi Automata can have Smaller Quotients. Technical Report EDI-INF-RR-1399, LFCS, University of Edinburgh (April 2011)Google Scholar
  4. 4.
    Clemente, L., Mayr, R.: Multipebble Simulations for Alternating Automata. In: Gastin, P., Laroussinie, F. (eds.) CONCUR 2010. LNCS, vol. 6269, pp. 297–312. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  5. 5.
    Dill, D.L., Hu, A.J., Wont-Toi, H.: Checking for Language Inclusion Using Simulation Preorders. In: Larsen, K.G., Skou, A. (eds.) CAV 1991. LNCS, vol. 575, Springer, Heidelberg (1992)CrossRefGoogle Scholar
  6. 6.
    Etessami, K.: A Hierarchy of Polynomial-Time Computable Simulations for Automata. In: Brim, L., Jančar, P., Křetínský, M., Kučera, A. (eds.) CONCUR 2002. LNCS, vol. 2421, pp. 131–144. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. 7.
    Etessami, K., Wilke, T., Schuller, R.A.: Fair Simulation Relations, Parity Games, and State Space Reduction for Büchi Automata. SIAM J. Comput. 34(5), 1159–1175 (2005)CrossRefzbMATHGoogle Scholar
  8. 8.
    Fritz, C., Wilke, T.: Simulation Relations for Alternating Büchi Automata. Theor. Comput. Sci. 338(1-3), 275–314 (2005)CrossRefzbMATHGoogle Scholar
  9. 9.
    Gramlich, G., Schnitger, G.: Minimizing NFA’s and Regular Expressions. Journal of Computer and System Sciences 73(6), 908–923 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Gurumurthy, S., Bloem, R., Somenzi, F.: Fair Simulation Minimization. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 610–624. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  11. 11.
    Henzinger, T.A., Kupferman, O., Rajamani, S.K.: Fair Simulation. Information and Computation 173, 64–81 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Henzinger, T.A., Rajamani, S.K.: Fair Bisimulation. In: Graf, S. (ed.) TACAS 2000. LNCS, vol. 1785, pp. 299–314. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  13. 13.
    Jiang, T., Ravikumar, B.: Minimal NFA Problems are Hard. In: Leach Albert, J., Monien, B., Rodríguez-Artalejo, M. (eds.) ICALP 1991. LNCS, vol. 510, pp. 629–640. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  14. 14.
    Juvekar, S., Piterman, N.: Minimizing Generalized Büchi Automata. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 45–58. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Kupferman, O., Vardi, M.: Verification of Fair Transition Systems. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 372–382. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  16. 16.
    Kupferman, O., Vardi, M.: Weak Alternating Automata Are Not That Weak. ACM Trans. Comput. Logic 2, 408–429 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Lynch, N.A., Vaandrager, F.W.: Forward and Backward Simulations. Part I: Untimed Systems. Information and Computation 121(2), 214–233 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Milner, R.: Communication and Concurrency. Prentice-Hall, Englewood Cliffs (1989)zbMATHGoogle Scholar
  19. 19.
    Raimi, R.S.: Environment Modeling and Efficient State Reachability Checking. PhD thesis, The University of Texas at Austin (1999)Google Scholar
  20. 20.
    Schewe, S.: Beyond Hyper-Minimisation—Minimising DBAs and DPAs is NP-Complete. In: FSTTCS. LIPIcs, vol. 8, pp. 400–411. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2010)Google Scholar
  21. 21.
    Somenzi, F., Bloem, R.: Efficient Büchi Automata from LTL Formulae. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 248–263. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  22. 22.
    Vardi, M.: Alternating Automata and Program Verification. In: van Leeuwen, J. (ed.) Computer Science Today. LNCS, vol. 1000, pp. 471–485. Springer, Heidelberg (1995)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Lorenzo Clemente
    • 1
  1. 1.LFCS, School of InformaticsUniversity of EdinburghUK

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