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A Characterisation of Definable NP Search Problems in Peano Arithmetic

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Logic, Language, Information and Computation (WoLLIC 2009)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 5514))

Abstract

The complexity class of ≺-bounded local search problems with goals is introduced for well-orderings ≺, and is used to give a characterisation of definable NP search problems in Peano Arithmetic.

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References

  1. Aehlig, K., Beckmann, A.: On the computational complexity of cut-reduction. Submitted to APAL (2008)

    Google Scholar 

  2. Beckmann, A., Buss, S.R.: Polynomial local search in the polynomial hierarchy and witnessing in fragments of bounded arithmetic. Technical Report CSR15-2008, Department of Computer Science, Swansea University (December 2008)

    Google Scholar 

  3. Beckmann, A., Buss, S.R.: Characterising definable search problems in bounded arithmetic via proof notations. Technical report, Department of Computer Science, Swansea University (January 2009)

    Google Scholar 

  4. Beckmann, A., Buss, S.R., Pollett, C.: Ordinal notations and well-orderings in bounded arithmetic. Ann. Pure Appl. Logic 120(1-3), 197–223 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Buss, S.R., Krajíček, J.: An application of Boolean complexity to separation problems in bounded arithmetic. Proc. London Math. Soc. (3) 69(1), 1–21 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Buchholz, W.: Notation systems for infinitary derivations. Archive for Mathematical Logic 30, 277–296 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. Buchholz, W.: Explaining Gentzen’s consistency proof within infinitary proof theory. In: Gottlob, G., Leitsch, A., Mundici, D. (eds.) KGC 1997. LNCS, vol. 1289, pp. 4–17. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  8. Buss, S.R.: Bounded arithmetic. Studies in Proof Theory. Lecture Notes, vol. 3. Bibliopolis, Naples (1986)

    MATH  Google Scholar 

  9. Gentzen, G.: Die Widerspruchsfreiheit der reinen Zahlentheorie. Math. Ann. 112, 493–565 (1936)

    Article  MathSciNet  MATH  Google Scholar 

  10. Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: How easy is local search? J. Comput. System Sci. 37(1), 79–100 (1988); 26th IEEE Conference on Foundations of Computer Science (Portland, OR, 1985)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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)

    Chapter  Google Scholar 

  12. Krajíček, J.: Fragments of bounded arithmetic and bounded query classes. Trans. Amer. Math. Soc. 338(2), 587–598 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Krajíček, J., Skelley, A., Thapen, N.: NP search problems in low fragments of bounded arithmetic. J. Symbolic Logic 72(2), 649–672 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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)

    Article  MATH  Google Scholar 

  15. Pohlers, W.: Proof theory. The first step into impredicativity. Universitext. Springer, Berlin (2009)

    MATH  Google Scholar 

  16. Pollett, C.: Structure and definability in general bounded arithmetic theories. Ann. Pure Appl. Logic 100(1-3), 189–245 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pudlák, P.: Consistency and games—in search of new combinatorial principles. In: Logic Colloquium 2003. Lect. Notes Log., vol. 24, pp. 244–281. Assoc. Symbol. Logic, La Jolla (2006)

    Google Scholar 

  18. Paris, J., Wilkie, A.: Counting problems in bounded arithmetic. In: Dold, A., Eckmann, B. (eds.) Methods in Mathematical Logic (Proceedings Caracas 1983). Lecture Notes in Mathematics, vol. 1130, pp. 317–340. Springer, Heidelberg (1985)

    Chapter  Google Scholar 

  19. Skelley, A., Thapen, N.: The provable total search problems of bounded arithmetic (2007); Typeset manuscript

    Google Scholar 

  20. Tait, W.W.: Normal derivability in classical logic. In: Barwise, J. (ed.) The Syntax and Semantics of Infinitatry Languages. Lecture Notes in Mathematics, vol. 72, pp. 204–236. Springer, Heidelberg (1968)

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, Swansea University, Swansea, SA2 8PP, UK

    Arnold Beckmann

Authors
  1. Arnold Beckmann
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Editor information

Editors and Affiliations

  1. Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, 923-1292, Ishikawa, Japan

    Hiroakira Ono

  2. National Institute of Informatics, 2–1–2 Hitotsubashi, Chiyoda-ku, 101–8430, Tokyo, Japan

    Makoto Kanazawa

  3. Centro de Informática, Universidade Federal de Pernambuco, Recife, PE, Brazil

    Ruy de Queiroz

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Beckmann, A. (2009). A Characterisation of Definable NP Search Problems in Peano Arithmetic. In: Ono, H., Kanazawa, M., de Queiroz, R. (eds) Logic, Language, Information and Computation. WoLLIC 2009. Lecture Notes in Computer Science(), vol 5514. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02261-6_1

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  • DOI: https://doi.org/10.1007/978-3-642-02261-6_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02260-9

  • Online ISBN: 978-3-642-02261-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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