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Abstract Interpretation of Indexed Grammars

Part of the Lecture Notes in Computer Science book series (LNPSE,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 which is not context-free, but they cannot recognize 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.

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References

  1. Adams, J., Freden, E., Mishna, M.: From indexed grammars to generating functions. RAIRO Theor. Inform. Appl. 47(4), 325–350 (2013)

    MathSciNet  CrossRef  Google Scholar 

  2. Aho, A.V.: Indexed grammars - an extension of context-free grammars. J. ACM 15(4), 647–671 (1968)

    MathSciNet  CrossRef  Google Scholar 

  3. Aho, A.V.: Nested stack automata. J. ACM 16(3), 383–406 (1969)

    MathSciNet  CrossRef  Google Scholar 

  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. Bertsch, E.: On the relationship between indexed grammars and logic programs. J. Log. Program. 18(1), 81–98 (1994)

    MathSciNet  CrossRef  Google Scholar 

  6. Chomsky, N.: On certain formal properties of grammars. Inf. Control. 2(2), 137–167 (1959)

    MathSciNet  CrossRef  Google Scholar 

  7. Clarke, E.M., Grumberg, O., Jha, S.: Verifying parameterized networks. ACM Trans. Program. Lang. Syst. 19(5), 726–750 (1997)

    CrossRef  Google Scholar 

  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. 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)

    MathSciNet  CrossRef  Google Scholar 

  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. 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. Cousot, P., Cousot, R.: Abstract interpretation frameworks. J. Log. Comput. 2(4), 511–547 (1992)

    MathSciNet  CrossRef  Google Scholar 

  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_58

    CrossRef  MATH  Google Scholar 

  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. Cousot, P., Cousot, R.: Grammar semantics, analysis and parsing by abstract interpretation. Theor. Comput. Sci. 412(44), 6135–6192 (2011)

    MathSciNet  CrossRef  Google Scholar 

  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. 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_14

    CrossRef  Google Scholar 

  18. Dalla Preda, M., Giacobazzi, R., Debray, S.K.: Unveiling metamorphism by abstract interpretation of code properties. Theor. Comput. Sci. 577, 74–97 (2015)

    MathSciNet  CrossRef  Google Scholar 

  19. Deutsch, A.: Interprocedural may-alias analysis for pointers: beyond k-limiting. SIGPLAN Not. 29(6), 230–241 (1994)

    CrossRef  Google Scholar 

  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_3

    CrossRef  Google Scholar 

  21. Giacobazzi, R., Ranzato, F., Scozzari, F.: Making abstract interpretation complete. J. ACM. 47(2), 361–416 (2000)

    MathSciNet  CrossRef  Google Scholar 

  22. Ginsburg, S.: The Mathematical Theory of Context Free Languages. McGraw-Hill Book Company, New York (1966)

    MATH  Google Scholar 

  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)

    CrossRef  Google Scholar 

  24. Partee, B.B., ter Meulen, A.G., Wall, R.: Mathematical Methods in Linguistics, vol. 30. Springer, Dordrecht (2012). https://doi.org/10.1007/978-94-009-2213-6

    CrossRef  MATH  Google Scholar 

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Campion, M., Dalla Preda, M., Giacobazzi, R. (2019). Abstract Interpretation of Indexed Grammars. In: Chang, BY. (eds) Static Analysis. SAS 2019. Lecture Notes in Computer Science(), vol 11822. Springer, Cham. https://doi.org/10.1007/978-3-030-32304-2_7

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  • DOI: https://doi.org/10.1007/978-3-030-32304-2_7

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