Advertisement

Nominal Automata with Name Binding

  • Lutz Schröder
  • Dexter Kozen
  • Stefan Milius
  • Thorsten Wißmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10203)

Abstract

Nominal sets are a convenient setting for languages over infinite alphabets, i.e. data languages. We introduce an automaton model over nominal sets, regular nondeterministic nominal automata (RNNA), which have a natural coalgebraic definition using abstraction sets to capture transitions that read a fresh letter from the input word. We prove a Kleene theorem for RNNAs w.r.t. a simple expression language that extends nominal Kleene algebra (NKA) with unscoped name binding, thus remedying the known failure of the expected Kleene theorem for NKA itself. We analyse RNNAs under two notions of freshness: global and local. Under global freshness, RNNAs turn out to be equivalent to session automata, and as such have a decidable inclusion problem. Under local freshness, RNNAs retain a decidable inclusion problem, and translate into register automata. We thus obtain decidability of inclusion for a reasonably expressive class of nondeterministic register automata, with no bound on the number of registers.

Notes

Acknowledgements

We thank Charles Paperman for useful discussions, and the anonymous reviewers of an earlier version of the paper for insightful comments that led us to discover the crucial notion of name dropping. Erwin R. Catesbeiana has commented on the empty bar language.

References

  1. 1.
    Aho, A.V., Sethi, R., Ullman, J.D.: Compilers: Principles, Techniques, and Tools. Addison-Wesley Longman Publishing Co., Inc., Boston (1986)zbMATHGoogle Scholar
  2. 2.
    Bielecki, M., Hidders, J., Paredaens, J., Tyszkiewicz, J., Bussche, J.: Navigating with a browser. In: Widmayer, P., Eidenbenz, S., Triguero, F., Morales, R., Conejo, R., Hennessy, M. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 764–775. Springer, Heidelberg (2002). doi: 10.1007/3-540-45465-9_65 CrossRefGoogle Scholar
  3. 3.
    Bojańczyk, M.: Automata for data words and data trees. In: Rewriting Techniques and Applications, RTA 2010. LIPIcs, vol. 6, pp. 1–4. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2010)Google Scholar
  4. 4.
    Bojańczyk, M.: Computation in Sets with Atoms. http://atoms.mimuw.edu.pl/wp-content/uploads/2014/03/main.pdf
  5. 5.
    Bojanczyk, M., Klin, B., Lasota, S.: Automata theory in nominal sets. Log. Methods Comput. Sci. 10, 1–44 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bollig, B., Habermehl, P., Leucker, M., Monmege, B.: A robust class of data languages and an application to learning. Log. Meth. Comput. Sci. 10, 1–23 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ciancia, V., Tuosto, E.: A novel class of automata for languages on infinite alphabets. Technical report, University of Leicester, cS-09-003 (2009)Google Scholar
  8. 8.
    Colcombet, T., Puppis, G., Skrypczak, M.: Unambiguous register automata, preprintGoogle Scholar
  9. 9.
    Colcombet, T.: Unambiguity in automata theory. In: Shallit, J., Okhotin, A. (eds.) DCFS 2015. LNCS, vol. 9118, pp. 3–18. Springer, Cham (2015). doi: 10.1007/978-3-319-19225-3_1 CrossRefGoogle Scholar
  10. 10.
    Demri, S., Lazic, R.: LTL with the freeze quantifier and register automata. ACM Trans. Comput. Log. 10, 16:1–16:30 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gabbay, M., Pitts, A.: A new approach to abstract syntax involving binders. In: Logic in Computer Science, LICS 1999, pp. 214–224. IEEE Computer Society (1999)Google Scholar
  12. 12.
    Gabbay, M.J.: Foundations of nominal techniques: logic and semantics of variables in abstract syntax. Bull. Symbolic Logic 17(2), 161–229 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gabbay, M.J., Ciancia, V.: Freshness and name-restriction in sets of traces with names. In: Hofmann, M. (ed.) FoSSaCS 2011. LNCS, vol. 6604, pp. 365–380. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-19805-2_25 CrossRefGoogle Scholar
  14. 14.
    Gabbay, M.J., Ghica, D.R., Petrisan, D.: Leaving the nest: nominal techniques for variables with interleaving scopes. In: Computer Science Logic, CSL 2015. LIPIcs, vol. 41, pp. 374–389. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2015)Google Scholar
  15. 15.
    Grigore, R., Tzevelekos, N.: History-register automata. Log. Meth. Comput. Sci. 12(1), 1–32 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Grumberg, O., Kupferman, O., Sheinvald, S.: Variable automata over infinite alphabets. In: Dediu, A.-H., Fernau, H., Martín-Vide, C. (eds.) LATA 2010. LNCS, vol. 6031, pp. 561–572. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-13089-2_47 CrossRefGoogle Scholar
  17. 17.
    Hennessy, M.: A fully abstract denotational semantics for the pi-calculus. Theor. Comput. Sci. 278, 53–89 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kaminski, M., Francez, N.: Finite-memory automata. Theor. Comput. Sci. 134, 329–363 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kaminski, M., Tan, T.: Regular expressions for languages over infinite alphabets. Fund. Inform. 69, 301–318 (2006)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Kaminski, M., Zeitlin, D.: Finite-memory automata with non-deterministic reassignment. Int. J. Found. Comput. Sci. 21, 741–760 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kozen, D., Mamouras, K., Petrişan, D., Silva, A.: Nominal Kleene coalgebra. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9135, pp. 286–298. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-47666-6_23 Google Scholar
  22. 22.
    Kozen, D., Mamouras, K., Silva, A.: Completeness and incompleteness in nominal Kleene algebra. In: Kahl, W., Winter, M., Oliveira, J.N. (eds.) RAMICS 2015. LNCS, vol. 9348, pp. 51–66. Springer, Cham (2015). doi: 10.1007/978-3-319-24704-5_4 CrossRefGoogle Scholar
  23. 23.
    Kürtz, K., Küsters, R., Wilke, T.: Selecting theories and nonce generation for recursive protocols. In: Formal methods in Security Engineering, FMSE 2007, pp. 61–70. ACM (2007)Google Scholar
  24. 24.
    Kurz, A., Suzuki, T., Tuosto, E.: On nominal regular languages with binders. In: Birkedal, L. (ed.) FoSSaCS 2012. LNCS, vol. 7213, pp. 255–269. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-28729-9_17 CrossRefGoogle Scholar
  25. 25.
    Libkin, L., Tan, T., Vrgoc, D.: Regular expressions for data words. J. Comput. Syst. Sci. 81, 1278–1297 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Manuel, A., Muscholl, A., Puppis, G.: Walking on data words. Theor. Comput. Sys. 59, 180–208 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Neven, F., Schwentick, T., Vianu, V.: Finite state machines for strings over infinite alphabets. ACM Trans. Comput. Log. 5, 403–435 (2004)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Pitts, A.: Nominal Sets: Names and Symmetry in Computer Science. Cambridge University Press, Cambridge (2013)CrossRefzbMATHGoogle Scholar
  29. 29.
    Rutten, J.: Universal coalgebra: a theory of systems. Theor. Comput. Sci. 249(1), 3–80 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Segoufin, L.: Automata and logics for words and trees over an infinite alphabet. In: Ésik, Z. (ed.) CSL 2006. LNCS, vol. 4207, pp. 41–57. Springer, Heidelberg (2006). doi: 10.1007/11874683_3 CrossRefGoogle Scholar
  31. 31.
    Stockhusen, C., Tantau, T.: Completeness results for parameterized space classes. In: Gutin, G., Szeider, S. (eds.) IPEC 2013. LNCS, vol. 8246, pp. 335–347. Springer, Cham (2013). doi: 10.1007/978-3-319-03898-8_28 CrossRefGoogle Scholar
  32. 32.
    Turner, D., Winskel, G.: Nominal domain theory for concurrency. In: Grädel, E., Kahle, R. (eds.) CSL 2009. LNCS, vol. 5771, pp. 546–560. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-04027-6_39 CrossRefGoogle Scholar
  33. 33.
    Tzevelekos, N.: Fresh-register automata. In: Principles of Programming Languages, POPL 2011, pp. 295–306. ACM (2011)Google Scholar
  34. 34.
    Wysocki, T.: Alternating register automata on finite words. Master’s thesis, University of Warsaw (2013). (In Polish)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Lutz Schröder
    • 1
  • Dexter Kozen
    • 2
  • Stefan Milius
    • 1
  • Thorsten Wißmann
    • 1
  1. 1.Friedrich-Alexander-Universität Erlangen-NürnbergErlangenGermany
  2. 2.Cornell UniversityIthacaUSA

Personalised recommendations