On feasible numbers

  • Vladimir Yu. Sazonov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 960)


A formal approach to feasible numbers, as well as to middle and small numbers, is introduced, based on ideas of Parikh (1971) and improving his formalization. The “vague” set F of feasible numbers intuitively satisfies the axioms 0 ∈ F, F+1⊑F and 21000F, where the latter is stronger than a condition considered by Parikh, and seems to be treated rigorously here for the first time. Our technical considerations, though quite simple, have some unusual consequences. A discussion of methodological questions and of relevance to the foundations of mathematics and of computer science is an essential part of the paper.


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  1. 1.
    Buss, S.R. (1986) Bounded Arithmetic, Bibliopolis, Napoli, 1986.Google Scholar
  2. 2.
    Buss, S.R. (1986) The polynomial hierarchy and intuitionistic bounded arithmetic, in: Structure in Complexity Theory, Lecture Notes in Computer Science 223 (Springer, Berlin) 125–143.Google Scholar
  3. 3.
    Cook, S.A. (1975) Feasibly constructive proofs and the prepositional calculus, in: Proceedings 7th ACM Symposium on the Theory of Computation, 83–97.Google Scholar
  4. 4.
    Cook, S.A. and Urquhart, A. (1993) Functional interpretations of feasibly constructive arithmetic, Annals of Pure and Applied Logic 63, 103–200.CrossRefGoogle Scholar
  5. 5.
    Dragalin, A.G. (1985) Correctness of inconsistent theories with notions of feasibility, in: Lecture Notes in Computer Science, 208, Springer-Verlag, 58–79.Google Scholar
  6. 6.
    Gandy, R.O. (1982) Limitations to mathematical knowledge, in: D. van Dalen, D.Laskar, J. Smiley eds., Logic Colloquium'80, North-Holland, Amsterdam, 129–146.Google Scholar
  7. 7.
    Gurevich, Y. (1983) Algebras of feasible functions, in: FOCS'83, pp.210–214.Google Scholar
  8. 8.
    Hájek, P., and Pudlák, P. (1993) Metamathematics of First-order Arithmetic, Perspectives of Mathematical Logic, 460 pp. Springer-Verlag.Google Scholar
  9. 9.
    Immerman, N. (1982) Relational queries computable in polynomial time, in: 14th STOC, pp. 147–152.Google Scholar
  10. 10.
    Kolmogorov, A.N.(1979) Automata and life (in Russian), Kibernetika — neogranichennye vozmozhnosti i vozmozhnye ogranichenija. Itogi razvitija. Moskwa, Nauka, 10–29.Google Scholar
  11. 11.
    Krajiček, J., (1995) Bounded Arithmetic, Propositional Logic and Complexity Theory, to appear in Cambridge University Press.Google Scholar
  12. 12.
    Nelson, E. (1986) Predicative arithmetic, Princeton University Press, Princeton, New Jersey.Google Scholar
  13. 13.
    Orevkov, V.P. (1979) The lower bounds of complexity the deductions increasing after cut elimination. In: Zapiski nauchnych seminarov LOMI AN SSSR, 88,137–162. (In Russian)Google Scholar
  14. 14.
    Parikh, R. (1971) Existence and feasibility in arithmetic, JSL, 36, (3), 494–508.Google Scholar
  15. 15.
    Prawitz, D. (1965) Natural Deduction, Stockholm.Google Scholar
  16. 16.
    Sazonov, V.Yu. (1980) Polynomial computability and recursivity in finite domains. Elektronische Informationsverarbeitung und Kybernetik, 16, (7), 319–323.Google Scholar
  17. 17.
    Sazonov, V.Yu. (1980a) A logical approach to the problem “P=NP?”, in: Lecture Notes in Computer Science, 88, Springer, New York, 562–575. (An important correction to this paper is given in [Lecture Notes in Computer Science, 118, Springer, New York,1981, p.490.])Google Scholar
  18. 18.
    Sazonov, V.Yu. (1987) Bounded set theory and polynomial computability, FCT'87, Lecture Notes in Computer Science, 278, p.391–397.Google Scholar
  19. 19.
    Sazonov, V.Yu. (1989) An equivalence between polynomial constructivity of Markov's principle and the equality P=NP (in Russian), in: Trudy instituta matematiki, Sibirskoje otdelenie akademii nauk SSSR, “Matematicheskaja logika i algoritmicheskije problemy”, Novosibirsk, “Nauka”, Sibirskoje otdelenije, 138–165. (See also shorter English version with the same title in P.Petkov ed., Mathematical Logic, Proceedings ffof the Heyting's conference, sept., 1988, Varna, Plenum Press, New York, 1990, 351–360.).Google Scholar
  20. 20.
    Sazonov, V.Yu. (1992) On feasible numbers, Abstracts of papers of European Summer Meeting of the Association for Symbolic Logic, Logic Colloquium'89, Berlin, JSL, 57 (1) 331.Google Scholar
  21. 21.
    Statman, R. (1978) Bounds for proof-search and speed-up of the predicate calculus. Ann. Math. Logic, 15 (3), 225–287.Google Scholar
  22. 22.
    Statman, R. (1979) Lower bounds on Herbrand's theorem, Proc. of the AMS, 75 (1).Google Scholar
  23. 23.
    Troelstra, A.S. (1990) Remarks on intuitionism and the philosophy of mathematics (revised version), ITLI Prepublication Series X-90-01, University of Amsterdam, 18 pp.Google Scholar
  24. 24.
    Troelstra, A.S. and van Dalen, D. (1988) Constructivism in Mathematics. An introduction, Vol. I, II, North-Holland, Amsterdam.Google Scholar
  25. 25.
    Vardi, M.Y. (1982) The complexity of relational query languages, STOC'82, pp. 137–146.Google Scholar
  26. 26.
    Vopenka, P. (1979) Mathematics in the Alternative Set Theory, Leipzig.Google Scholar
  27. 27.
    Yesenin-Volpin, A.S., (1959) Analysis of the potential feasibility, in: Logicheskije issledovanija, Moskwa, AN SSSR, 218–262. (In Russian).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Vladimir Yu. Sazonov
    • 1
  1. 1.Program Systems Institute of Russian Academy of SciencesRussia

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