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Feasible Functions over Co-inductive Data

  • Conference paper
Logic, Language, Information and Computation (WoLLIC 2010)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 6188))

Abstract

Proof theoretic characterizations of complexity classes are of considerable interest because they link levels of conceptual abstraction to computational complexity. We consider here the provability of functions over co-inductive data in a highly expressive, yet proof-theoretically weak, variant of second order logic \({\rm L}^{+}_{*}\), which we believe captures the notion of feasibility more broadly than previously considered pure-logic formalisms.

Our main technical result is that every basic feasible functional (i.e. functional in the class BFF, believed to be the most adequate definition of feasibility for second-order functions) is provable in \({\rm L}^{+}_{*}\).

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References

  1. Aehlig, K.: Parameter-free polymorphic types. Ann. Pure Appl. Logic 156(1), 3–12 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Beckmann, A., Weiermann, A.: Characterizing the elementary recursive functions by a fragment of gödel’s t. Arch. Math. Log. 39(7), 475–491 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bertot, Y., Komendantskaya, E.: Inductive and coinductive components of corecursive functions in coq. Electron. Notes Theor. Comput. Sci. 203(5), 25–47 (2008)

    Article  MathSciNet  Google Scholar 

  4. Burrell, M.J., Cockett, R., Redmond, B.F.: Pola: a language for PTIME programming. Logic Computational Complexity, Logic in Computer Science (2009)

    Google Scholar 

  5. Buss, S.: Bounded Arithmetic. Bibliopolis, Naples (1986)

    Google Scholar 

  6. Buss, S.: The polynomial hierarchy and intuitionistic bounded arithmetic. In: Wiedermann, J., Gruska, J., Rovan, B. (eds.) MFCS 1986. LNCS, vol. 233, pp. 77–103. Springer, Heidelberg (1986)

    Google Scholar 

  7. Clote, P.: A note on the relation between polynomial time functionals and constable’s class k. In: Kleine-Büning, H. (ed.) CSL 1995. LNCS, vol. 1092, pp. 145–160. Springer, Heidelberg (1996)

    Google Scholar 

  8. Cobham, A.: The intrinsic computational difficulty of functions. In: Bar-Hillel, Y. (ed.) Proceedings of the International Conference on Logic, Methodology, and Philosophy of Science, pp. 24–30. North-Holland, Amsterdam (1962)

    Google Scholar 

  9. Constable, R.: Type 2 computational complexity. In: Fifth Annual ACM Symposium on Theory of Computing, pp. 108–121. ACM, New York (1973)

    Chapter  Google Scholar 

  10. Cook, S.: Computability and complexity of higher type functions. In: Moschovakis, Y. (ed.) Logic from Computer Science, pp. 51–72. Springer, New York (1991)

    Google Scholar 

  11. Cook, S.A.: Computability and complexity of higher type functions. In: Chern, Singer, Kaplansky, Mooreand, Moschovakis (eds.) Logic from Computer Science, Springer, New York (1989)

    Google Scholar 

  12. Cook, S.A., Kapron, B.M.: Characterizations of the basic feasible functionals of finite type. In: FOCS, pp. 154–159. IEEE, Los Alamitos (1989)

    Google Scholar 

  13. Cook, S.A., Kapron, B.M.: Characterizations of the Basic Feasible Functionals of Finite Type. In: Feasible Mathematics: A Mathematical Sciences Institute Workshop, pp. 71–95 (1990)

    Google Scholar 

  14. Cook, S.A., Urquhart, A.: Functional interpretations of feasible constructive arithemtic (extended abstract). In: Proceedings of the 21st ACM Symposium on Theory of Computing, pp. 107–112 (1989)

    Google Scholar 

  15. Cook, S.A., Urquhart, A.: Functional interpretations of feasible constructive arithemtic. Annals of Pure and Applied Logic 63, 103–200 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gödel, K.: Über eine bisher noch nicht benutzte erweiterung des finiten standpunktes. Dialectica 12, 280–287 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kapron, B.M., Cook, S.A.: A new characerization of type-2 feasibility. SIAM Journal of Computing 25, 117–132 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kapron, B., Cook, S.: Characterizations of the basic feasible functionals of finite type. In: Buss, S., Scott, P. (eds.) Feasible Mathematics, pp. 71–95. Birkhauser, Boston (1990)

    Google Scholar 

  19. Kapron, B., Cook, S.A.: A new characterization of type-2 feasibility. SIAM J. on Computing 25(1), 117–132 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  20. Leivant, D.: Finitely stratified polymorphism. Inf. Comput. 93(1), 93–113 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  21. Leivant, D.: A foundational delineation of poly-time. Information and Computation 110, 391–420 (1994); (Special issue of selected papers from LICS’91, edited by Kahn, G.). Preminary report: A foundational delineation of computational feasibility, In: Proceedings of the Sixth IEEE Conference on Logic in Computer Science. IEEE Computer Society Press, Los Alamitos (1991)

    Google Scholar 

  22. Leivant, D.: Intrinsic theories and computational complexity. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 177–194. Springer, Heidelberg (1995)

    Google Scholar 

  23. Leivant, D.: Intrinsic reasoning about functional programs I: First order theories. Annals of Pure and Applied Logic 114, 117–153 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  24. Leivant, D.: Intrinsic reasoning about functional programs ii: unipolar induction and primitive-recursion. Theor. Comput. Sci. 318(1-2), 181–196 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Leivant, D., Ramyaa, R.: Implicit complexity for coinductive data: a proof-theoretic characterization of primitive corecursion. In: Conference submission (2010), see current draft as http://www.cs.indiana.edu/~leivant/corecursion.pdf

  26. Mehlhorn, K.: Polynomial and abstract subrecursive classes. J. Comput. Syst. Sci. 12(2), 147–178 (1976)

    MATH  MathSciNet  Google Scholar 

  27. Moschovakis, Y.N.: The formal language of recursion. J. Symb. Log. 54(4), 1216–1252 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  28. Parsons, C.: On a number-theoretic choice schema and its relation to induction. In: Kino, A., Myhill, J., Vesley, R. (eds.) Intuitionism and Proof Theory, pp. 459–473. North-Holland, Amsterdam (1970)

    Google Scholar 

  29. Paulson, L.C.: Mechanizing coinduction and corecursion in higher-order logic. Journal of Logic and Computation 7 (1997)

    Google Scholar 

  30. Pezzoli, E.: On the computational complexity of type 2 functionals. In: Nielsen, M., Thomas, W. (eds.) CSL 1997. LNCS, vol. 1414, pp. 373–388. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  31. Prawitz, D.: Natural Deduction. Almqvist and Wiksell, Uppsala (1965)

    MATH  Google Scholar 

  32. Seth, A.: Some desirable conditions for feasible functionals of type~2. In: LICS, pp. 320–331 (1993)

    Google Scholar 

  33. Seth, A.: Complexity theory of higher type functionals. PhD thesis, University of Bombay, PhD Thesis (1994)

    Google Scholar 

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Authors and Affiliations

  1. Indiana University,  

    Ramyaa Ramyaa & Daniel Leivant

Authors
  1. Ramyaa Ramyaa
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  2. Daniel Leivant
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Editor information

Editors and Affiliations

  1. Computer Laboratory, University of Cambridge, J.J. Thomson Avenue, CB3 0FD, Cambridge, UK

    Anuj Dawar

  2. Centro de Informática, Universidade Federal de Pernambuco, Avenida Prof Luis Freire s/n, Cidade Universitária, 50740-540, Recife, PE, Brazil

    Ruy de Queiroz

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Ramyaa, R., Leivant, D. (2010). Feasible Functions over Co-inductive Data. In: Dawar, A., de Queiroz, R. (eds) Logic, Language, Information and Computation. WoLLIC 2010. Lecture Notes in Computer Science(), vol 6188. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13824-9_16

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  • DOI: https://doi.org/10.1007/978-3-642-13824-9_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13823-2

  • Online ISBN: 978-3-642-13824-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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