Advertisement

A Finite Semantics of Simply-Typed Lambda Terms for Infinite Runs of Automata

  • Klaus Aehlig
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4207)

Abstract

Model checking properties are often described by means of finite automata. Any particular such automaton divides the set of infinite trees into finitely many classes, according to which state has an infinite run. Building the full type hierarchy upon this interpretation of the base type gives a finite semantics for simply-typed lambda-trees.

A calculus based on this semantics is proven sound and complete. In particular, for regular infinite lambda-trees it is decidable whether a given automaton has a run or not. As regular lambda-trees are precisely recursion schemes, this decidability result holds for arbitrary recursion schemes of arbitrary level, without any syntactical restriction. This partially solves an open problem of Knapik, Niwinski and Urzyczyn.

Keywords

Normal Form Model Check Ground Type Recursion 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., Joachimski, F.: On continuous normalization. In: Bradfield, J.C. (ed.) CSL 2002 and EACSL 2002. LNCS, vol. 2471, pp. 59–73. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    Aehlig, K., Miranda, J.G.d., 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.
    Hyland, J.M.E., Ong, C.-H.L.: On full abstraction for PCF. Information and Computation 163(2), 285–408 (2000)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Knapik, T., Niwiński, D., Urzyczyn, P.: Deciding monadic theories of hyperalgebraic trees. In: Abramsky, S. (ed.) TLCA 2001. LNCS, vol. 2044, pp. 253–267. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  5. 5.
    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
  6. 6.
    Knapik, T., Niwiński, D., Urzyczyn, P., Walukiewicz, I.: Unsafe grammars, panic automata, and decidability. 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
  7. 7.
    Kreisel, G., Mints, G.E., Simpson, S.G.: The use of abstract language in elementary metamathematics: Some pedagogic examples. In: Parikh, R. (ed.)Logic Colloquium. Lecture Notes in Mathematics, vol. 453, pp. 38–131. Springer, Heidelberg (1975)CrossRefGoogle Scholar
  8. 8.
    Mints, G.E.: Finite investigations of transfinite derivations. Journal of Soviet Mathematics, 10, 548–596 (1978); Translated from: Zap. Nauchn. Semin. LOMI 49 (1975); Cited after Grigori Mints. Selected papers in Proof Theory. Studies in Proof Theory. Bibliopolis (1992)Google Scholar
  9. 9.
    Ong, C.-H.L.: On model-checking trees generated by higher-order recursion schemes. In: Proceedings of the Twenty First Annual IEEE Symposium on Logic in Computer Science (LICS 2006) (to appear, 2006)Google Scholar
  10. 10.
    Rabin, M.O.: Decidability of second-order theories and automata on infinite trees. Transactions of the American Mathematical Society 141, 1–35 (1969)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Tait, W.W.: Intensional interpretations of functionals of finite type. The Journal of Symbolic Logic 32(2), 198–212 (1967)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Klaus Aehlig
    • 1
  1. 1.Mathematisches Institut, Ludwig-Maximilians-Universität MünchenMünchenGermany

Personalised recommendations