Advertisement

Abstract

Higher-order recursive schemes are abstract forms of programs where the meaning of built-in constructs is not specified. The semantics of a scheme is an infinite tree labeled with built-in constructs. The research on recursive schemes spans over more than forty years. Still, central problems like the equality problem, and more recently, the model checking problem for schemes remain very intriguing. Even though recursive schemes were originally though of as a syntactic simplification of a fragment of the lambda calculus, we propose to go back to lambda calculus to study schemes. In particular, for the model checking problem we propose to use standard finitary models for the simply-typed lambda calculus.

Keywords

Model Check Operational Semantic Program Scheme Recursive Scheme Lambda Calculus 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aehlig, K.: A finite semantics of simply-typed lambda terms for infinite runs of automata. Logical Methods in Computer Science 3(1), 1–23 (2007)MathSciNetGoogle Scholar
  2. 2.
    Aehlig, K., de Miranda, J.G., Ong, C.-H.L.: The Monadic Second Order Theory of Trees Given by Arbitrary Level-Two Recursion Schemes Is Decidable. In: Urzyczyn, P. (ed.) TLCA 2005. LNCS, vol. 3461, pp. 39–54. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    Aho, A.V.: Indexed grammars – an extension of context-free grammars. J. ACM 15(4), 647–671 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Barendregt, H.: The type free lambda calculus. In: Handbook of Mathematical Logic, ch. D.7, pp. 1091–1132. North-Holland (1977)Google Scholar
  5. 5.
    Barendregt, H., Klop, J.W.: Applications of infinitary lambda calculus. Inf. Comput. 207(5), 559–582 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cachat, T., Walukiewicz, I.: The Complexity of Games on Higher Order Pushdown Automata. Internal reportGoogle Scholar
  7. 7.
    Carayol, A., Wöhrle, S.: The Caucal Hierarchy of Infinite Graphs in Terms of Logic and Higher-Order Pushdown Automata. In: Pandya, P.K., Radhakrishnan, J. (eds.) FSTTCS 2003. LNCS, vol. 2914, pp. 112–123. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  8. 8.
    Caucal, D.: On Infinite Terms Having a Decidable Monadic Theory. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 165–176. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Courcelle, B.: A representation of trees by languages I. Theor. Comput. Sci. 6, 255–279 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Courcelle, B.: Recursive applicative program schemes. In: Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics (B), pp. 459–492. Elesvier (1990)Google Scholar
  11. 11.
    Courcelle, B., Nivat, M.: Algebraic families of interpretations. In: FOCS (1976)Google Scholar
  12. 12.
    Courcelle, B., Walukiewicz, I.: Monadic second-order logic, graphs and unfoldings of transition systems. Annals of Pure and Applied Logic 92, 35–62 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Damm, W.: The IO– and OI–hierarchies. Theoretical Computer Science 20(2), 95–208 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Dezani-Ciancaglini, M., Giovannetti, E., de’ Liguoro, U.: Intersection Types, Lambda-models and Böhm Trees. In: MSJ-Memoir “Theories of Types and Proofs”, vol. 2, pp. 45–97. Mathematical Society of Japan (1998)Google Scholar
  15. 15.
    Engelfriet, J.: Iterated push-down automata and complexity classes. In: 15th STOC 1983, pp. 365–373 (1983)Google Scholar
  16. 16.
    Engelfriet, J., Schmidt, E.: IO and OI. Journal of Computer and System Sciences 15(3), 328–353 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Hague, M., Murawski, A.S., Ong, C.-H.L., Serre, O.: Collapsible pushdown automata and recursion schemes. In: LICS 2008, pp. 452–461. IEEE Computer Society (2008)Google Scholar
  18. 18.
    Ianov, Y.: The logical schemes of algorithms. In: Problems of Cybernetics I, pp. 82–140. Pergamon, Oxford (1969)Google Scholar
  19. 19.
    Indermark, K.: Schemes with Recursion on Higher Types. In: Mazurkiewicz, A. (ed.) MFCS 1976. LNCS, vol. 45, pp. 352–358. Springer, Heidelberg (1976)CrossRefGoogle Scholar
  20. 20.
    Jancar, P.: Decidability of DPDA language equivalence via first-order grammars. In: LICS 2012 (2012)Google Scholar
  21. 21.
    Knapik, T., Niwiński, D., Urzyczyn, P.: Higher-Order Pushdown Trees Are Easy. In: Nielsen, M., Engberg, U. (eds.) FOSSACS 2002. LNCS, vol. 2303, pp. 205–222. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  22. 22.
    Knapik, T., Niwiński, D., Urzyczyn, P., Walukiewicz, I.: Unsafe Grammars and Panic Automata. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1450–1461. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  23. 23.
    Kobayashi, N.: Types and higher-order recursion schemes for verification of higher-order programs. In: POPL 2009, pp. 416–428. ACM (2009)Google Scholar
  24. 24.
    Kobayashi, N.: Higher-order model checking: From theory to practice. In: LICS 2011, pp. 219–224 (2011)Google Scholar
  25. 25.
    Kobayashi, N., Ong, C.-H.L.: Complexity of model checking recursion schemes for fragments of the modal mu-calculus. Logical Methods in Computer Science 7(4) (2011)Google Scholar
  26. 26.
    Kobayashi, N., Ong, L.: A type system equivalent to modal mu-calculus model checking of recursion schemes. In: LICS 2009, pp. 179–188 (2009)Google Scholar
  27. 27.
    Manna, Z.: Mathematical Theory of Computation. McGraw-Hill (1974)Google Scholar
  28. 28.
    Maslov, A.: The hierarchy of indexed languages of an arbitrary level. Soviet. Math. Doklady 15, 1170–1174 (1974)zbMATHGoogle Scholar
  29. 29.
    Maslov, A.: Multilevel stack automata. Problems of Information Transmission 12, 38–43 (1976)Google Scholar
  30. 30.
    Milner, R.: Models of LCF. Memo AIM-186. Stanford University (1973)Google Scholar
  31. 31.
    Nivat, M.: On interpretation of recursive program schemes. In: Symposia Mathematica, vol. 15 (1975)Google Scholar
  32. 32.
    Ong, C.-H.L.: On model-checking trees generated by higher-order recursion schemes. In: LICS 2006, pp. 81–90 (2006)Google Scholar
  33. 33.
    Parys, P.: On the significance of the collapse operation. In: LICS 2012 (2012)Google Scholar
  34. 34.
    Plotkin, G.D.: LCF considered as a programming language. Theor. Comput. Sci. 5(3), 223–255 (1977)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Salvati, S.: Recognizability in the Simply Typed Lambda-Calculus. In: Ono, H., Kanazawa, M., de Queiroz, R. (eds.) WoLLIC 2009. LNCS, vol. 5514, pp. 48–60. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  36. 36.
    Salvati, S., Manzonetto, G., Gehrke, M., Barendregt, H.: Loader and Urzyczyn Are Logically Related. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part II. LNCS, vol. 7392, pp. 364–376. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  37. 37.
    Salvati, S., Walukiewicz, I.: Krivine Machines and Higher-Order Schemes. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part II. LNCS, vol. 6756, pp. 162–173. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  38. 38.
    Scott, D.: Continuous lattices. In: Proc. of Dalhousie Conference. Lecture Notes in Mathematics, vol. 188, pp. 311–366. Springer (1972)Google Scholar
  39. 39.
    Sénizergues, G.: L(A)=L(B)? Decidability results from complete formal systems. Theor. Comput. Sci. 251(1-2), 1–166 (2001)zbMATHCrossRefGoogle Scholar
  40. 40.
    Sénizergues, G.: L(A)=L(B)? A simplified decidability proof. Theor. Comput. Sci. 281(1-2), 555–608 (2002)zbMATHCrossRefGoogle Scholar
  41. 41.
    Stirling, C.: Deciding DPDA Equivalence Is Primitive Recursive. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 821–832. Springer, Heidelberg (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Igor Walukiewicz
    • 1
  1. 1.LaBRICNRS/Universit BordeauxFrance

Personalised recommendations