A Practical Linear Time Algorithm for Trivial Automata Model Checking of Higher-Order Recursion Schemes

  • Naoki Kobayashi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6604)

Abstract

The model checking of higher-order recursion schemes has been actively studied and is now becoming a basis of higher-order program verification. We propose a new algorithm for trivial automata model checking of higher-order recursion schemes. To our knowledge, this is the first practical model checking algorithm for recursion schemes that runs in time linear in the size of the higher-order recursion scheme, under the assumption that the size of trivial automata and the largest order and arity of functions are fixed. The previous linear time algorithm was impractical due to a huge constant factor, and the only practical previous algorithm suffers from the hyper-exponential worst-case time complexity, under the same assumption. The new algorithm is remarkably simple, consisting of just two fixed-point computations. We have implemented the algorithm and confirmed that it outperforms Kobayashi’s previous algorithm in a certain case.

References

  1. 1.
    Abramsky, S., McCusker, G.: Game semantics. In: Computational Logic: Proceedings of the 1997 Marktoberdorf Summer School, pp. 1–56. Springer, Heidelberg (1999)Google Scholar
  2. 2.
    Aehlig, K.: A finite semantics of simply-typed lambda terms for infinite runs of automata. Logical Methods in Computer Science 3(3) (2007)Google Scholar
  3. 3.
    Ball, T., Rajamani, S.K.: The SLAM project: Debugging system software via static analysis. In: Proc. of POPL, pp. 1–3 (2002)Google Scholar
  4. 4.
    Beyer, D., Henzinger, T.A., Jhala, R., Majumdar, R.: The software model checker Blast. International Journal on Software Tools for Technology Transfer 9(5-6), 505–525 (2007)CrossRefGoogle Scholar
  5. 5.
    Carlier, S., Polakow, J., Wells, J.B., Kfoury, A.J.: System E: Expansion variables for flexible typing with linear and non-linear types and intersection types. In: Schmidt, D. (ed.) ESOP 2004. LNCS, vol. 2986, pp. 294–309. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. 6.
    Hague, M., Murawski, A., Ong, C.-H.L., Serre, O.: Collapsible pushdown automata and recursion schemes. In: Proceedings of 23rd Annual IEEE Symposium on Logic in Computer Science, pp. 452–461. IEEE Computer Society, Los Alamitos (2008)Google Scholar
  7. 7.
    Kfoury, A.J., Wells, J.B.: Principality and type inference for intersection types using expansion variables. Theor. Comput. Sci. 311(1-3), 1–70 (2004)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Knapik, T., Niwinski, 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
  9. 9.
    Knapik, T., Niwinski, 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
  10. 10.
    Kobayashi, N.: Model-checking higher-order functions. In: Proceedings of PPDP 2009, pp. 25–36. ACM Press, New York (2009), see also [13]Google Scholar
  11. 11.
    Kobayashi, N.: TRecS (2009), http://www.kb.ecei.tohoku.ac.jp/~koba/trecs/
  12. 12.
    Kobayashi, N.: Types and higher-order recursion schemes for verification of higherorder programs. In: Proc. of POPL, pp. 416–428 (2009), see also [13]Google Scholar
  13. 13.
    Kobayashi, N.: Model checking higher-order programs. A revised and extended version of [12] and [10], available from the author’s web page (2010)Google Scholar
  14. 14.
    Kobayashi, N.: A practical linear time algorithm for trivial automata model checking of higher-order recursion schemes (2010), an extended version http://www.kb.ecei.tohoku.ac.jp/~koba/gtrecs/
  15. 15.
    Kobayashi, N., Ong, C.-H.L.: Complexity of model checking recursion schemes for fragments of the modal mu-calculus. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5556, pp. 223–234. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Kobayashi, N., Ong, C.-H.L.: A type system equivalent to the modal mu-calculus model checking of higher-order recursion schemes. In: Proceedings of LICS 2009, pp. 179–188. IEEE Computer Society Press, Los Alamitos (2009)Google Scholar
  17. 17.
    Kobayashi, N., Sato, R., Unno, H.: Predicate abstraction and cegar for higher-order model checking (July 2010) (unpublished manuscript)Google Scholar
  18. 18.
    Kobayashi, N., Tabuchi, N., Unno, H.: Higher-order multi-parameter tree transducers and recursion schemes for program verification. In: Proc. of POPL, pp. 495–508 (2010)Google Scholar
  19. 19.
    Lester, M.M., Neatherway, R.P., Ong, C.-H.L., Ramsay, S.J.: Model checking liveness properties of higher-order functional programs (2010) (unpublished manuscript)Google Scholar
  20. 20.
    Ong, C.-H.L.: On model-checking trees generated by higher-order recursion schemes. In: LICS 2006, pp. 81–90. IEEE Computer Society Press, Los Alamitos (2006)Google Scholar
  21. 21.
    Ong, C.-H.L., Ramsay, S.: Verifying higher-order programs with pattern-matching algebraic data types. In: Proceedings of POPL 2011 (to appear, 2011)Google Scholar
  22. 22.
    Rehof, J., Mogensen, T.: Tractable constraints in finite semilattices. Science of Computer Programming 35(2), 191–221 (1999)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Naoki Kobayashi
    • 1
  1. 1.Tohoku UniversityJapan

Personalised recommendations