Advertisement

Abstract Interpretation of Indexed Grammars

  • Marco CampionEmail author
  • Mila Dalla Preda
  • Roberto Giacobazzi
Conference paper
  • 389 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11822)

Abstract

Indexed grammars are a generalization of context-free grammars and recognize a proper subset of context-sensitive languages. The class of languages recognized by indexed grammars are called indexed languages and they correspond to the languages recognized by nested stack automata. For example indexed grammars can recognize the language Open image in new window which is not context-free, but they cannot recognize Open image in new window which is context-sensitive. Indexed grammars identify a set of languages that are more expressive than context-free languages, while having decidability results that lie in between the ones of context-free and context-sensitive languages. In this work we study indexed grammars in order to formalize the relation between indexed languages and the other classes of languages in the Chomsky hierarchy. To this end, we provide a fixpoint characterization of the languages recognized by an indexed grammar and we study possible ways to abstract, in the abstract interpretation sense, these languages and their grammars into context-free and regular languages.

References

  1. 1.
    Adams, J., Freden, E., Mishna, M.: From indexed grammars to generating functions. RAIRO Theor. Inform. Appl. 47(4), 325–350 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aho, A.V.: Indexed grammars - an extension of context-free grammars. J. ACM 15(4), 647–671 (1968)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aho, A.V.: Nested stack automata. J. ACM 16(3), 383–406 (1969)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ballance, R.A., Butcher, J., Graham, S.L.: Grammatical abstraction and incremental syntax analysis in a language-based editor. In: Proceedings of the ACM SIGPLAN 1988 Conference on Programming Language Design and Implementation, PLDI 1988, pp. 185–198. ACM, New York (1988)Google Scholar
  5. 5.
    Bertsch, E.: On the relationship between indexed grammars and logic programs. J. Log. Program. 18(1), 81–98 (1994)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chomsky, N.: On certain formal properties of grammars. Inf. Control. 2(2), 137–167 (1959)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Clarke, E.M., Grumberg, O., Jha, S.: Verifying parameterized networks. ACM Trans. Program. Lang. Syst. 19(5), 726–750 (1997)CrossRefGoogle Scholar
  8. 8.
    Cousot, P.: The calculational design of a generic abstract interpreter. In: Broy, M., Steinbrüggen, R. (eds.) Calculational System Design, vol. 173, pp. 421–505. NATO Science Series, Series F: Computer and Systems Sciences. IOS Press, Amsterdam (1999)Google Scholar
  9. 9.
    Cousot, P.: Constructive design of a hierarchy of semantics of a transition system by abstract interpretation. Theor. Comput. Sci. 277(1–2), 47–103 (2002)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cousot, P., Cousot, R.: Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: Conference Record of the 4th ACM Symposium on Principles of Programming Languages (POPL 1977), pp. 238–252. ACM Press (1977)Google Scholar
  11. 11.
    Cousot, P., Cousot, R.: Systematic design of program analysis frameworks. In: Conference Record of the 6th ACM Symposium on Principles of Programming Languages (POPL 1979), pp. 269–282. ACM Press (1979)Google Scholar
  12. 12.
    Cousot, P., Cousot, R.: Abstract interpretation frameworks. J. Log. Comput. 2(4), 511–547 (1992)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Cousot, P., Cousot, R.: Compositional and inductive semantic definitions in fixpoint, equational, constraint, closure-condition, rule-based and game-theoretic form (Invited Paper). In: Wolper, P. (ed.) CAV 1995. LNCS, vol. 939, pp. 293–308. Springer, Heidelberg (1995).  https://doi.org/10.1007/3-540-60045-0_58CrossRefzbMATHGoogle Scholar
  14. 14.
    Cousot, P., Halbwachs, N.: Automatic discovery of linear restraints among variables of a program. In: POPL 1978: Proceedings of the 5th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages, pp. 84–96. ACM Press (1978)Google Scholar
  15. 15.
    Cousot, P., Cousot, R.: Grammar semantics, analysis and parsing by abstract interpretation. Theor. Comput. Sci. 412(44), 6135–6192 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Cousot, P., Cousot, R.: Abstract interpretation: past, present and future. In: Henzinger, T.A., Miller, D. (eds.) Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS 2014, Vienna, Austria, 14–18 July 2014, pp. 2:1–2:10. ACM (2014)Google Scholar
  17. 17.
    Dalla Preda, M., Giacobazzi, R., Debray, S., Coogan, K., Townsend, G.M.: Modelling metamorphism by abstract interpretation. In: Cousot, R., Martel, M. (eds.) SAS 2010. LNCS, vol. 6337, pp. 218–235. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-15769-1_14CrossRefGoogle Scholar
  18. 18.
    Dalla Preda, M., Giacobazzi, R., Debray, S.K.: Unveiling metamorphism by abstract interpretation of code properties. Theor. Comput. Sci. 577, 74–97 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Deutsch, A.: Interprocedural may-alias analysis for pointers: beyond k-limiting. SIGPLAN Not. 29(6), 230–241 (1994)CrossRefGoogle Scholar
  20. 20.
    Gazdar, G.: Applicability of indexed grammars to natural languages. In: Reyle, U., Rohrer, C. (eds.) Natural Language Parsing and Linguistic Theories, pp. 69–94. Springer, Dordrecht (1988).  https://doi.org/10.1007/978-94-009-1337-0_3CrossRefGoogle Scholar
  21. 21.
    Giacobazzi, R., Ranzato, F., Scozzari, F.: Making abstract interpretation complete. J. ACM. 47(2), 361–416 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ginsburg, S.: The Mathematical Theory of Context Free Languages. McGraw-Hill Book Company, New York (1966)zbMATHGoogle Scholar
  23. 23.
    Istrail, S.: Generalization of the Ginsburg-Rice Schützenberger fixed-point theorem for context-sensitive and recursive-enumerable languages. Theor. Comput. Sci. 18(3), 333–341 (1982)CrossRefGoogle Scholar
  24. 24.
    Partee, B.B., ter Meulen, A.G., Wall, R.: Mathematical Methods in Linguistics, vol. 30. Springer, Dordrecht (2012).  https://doi.org/10.1007/978-94-009-2213-6CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Marco Campion
    • 1
    Email author
  • Mila Dalla Preda
    • 1
  • Roberto Giacobazzi
    • 1
  1. 1.Dipartimento di InformaticaUniversity of VeronaVeronaItaly

Personalised recommendations