Advertisement

Induction rules in bounded arithmetic

  • Emil JeřábekEmail author
Article
  • 18 Downloads

Abstract

We study variants of Buss’s theories of bounded arithmetic axiomatized by induction schemes disallowing the use of parameters, and closely related induction inference rules. We put particular emphasis on \(\hat{\varPi }^{b}_i\) induction schemes, which were so far neglected in the literature. We present inclusions and conservation results between the systems (including a witnessing theorem for \(T^i_2\) and \(S^i_2\) of a new form), results on numbers of instances of the axioms or rules, connections to reflection principles for quantified propositional calculi, and separations between the systems.

Keywords

Bounded arithmetic Parameter-free induction Induction rule Partial conservativity Reflection principle 

Mathematics Subject Classification

03F30 03F20 

Notes

Acknowledgements

I would like to thank Andrés Cordón-Franco and Félix Lara-Martín for stimulating discussions of the topic, and for drawing my attention to [28]. I am grateful to the anonymous reviewer for many helpful suggestions.

References

  1. 1.
    Adamowicz, Z., Bigorajska, T.: Functions provably total in \(I^-\varSigma _1\). Fundamenta Mathematicae 132, 189–194 (1989)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aehlig, K., Cook, S., Nguyen, P.: Relativizing small complexity classes and their theories. Comput. Complex. 25(1), 177–215 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Beklemishev, L.D.: Induction rules, reflection principles, and provably recursive functions. Ann. Pure Appl. Log. 85(3), 193–242 (1997)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Beklemishev, L.D.: Parameter free induction and provably total computable functions. Theor. Comput. Sci. 224, 13–33 (1999)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bigorajska, T.: On \({\varSigma }_{1}\)-definable functions provably total in \({I\varPi }_{1}^{-}\). Math. Log. Q. 41, 135–137 (1995)Google Scholar
  6. 6.
    Bloch, S.A.: Divide and conquer in parallel complexity and proof theory. Ph.D. thesis, University of California, San Diego (1992)Google Scholar
  7. 7.
    Buss, S.R.: Bounded Arithmetic. Bibliopolis, Naples (1986). Revision of 1985 Princeton University Ph.D. thesisGoogle Scholar
  8. 8.
    Buss, S.R.: Relating the bounded arithmetic and polynomial time hierarchies. Ann. Pure Appl. Log. 75(1–2), 67–77 (1995)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Buss, S.R., Kołodziejczyk, L.A., Zdanowski, K.: Collapsing modular counting in bounded arithmetic and constant depth propositional proofs. Trans. Am. Math. Soc. 367(11), 7517–7563 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Buss, S.R., Krajíček, J.: An application of boolean complexity to separation problems in bounded arithmetic. Proc. Lond. Math. Soc. 69(3), 1–21 (1994)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chang, R., Kadin, J.: The Boolean hierarchy and the polynomial hierarchy: a closer connection. SIAM J. Comput. 25(2), 340–354 (1996)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chiari, M., Krajíček, J.: Witnessing functions in bounded arithmetic and search problems. J. Symb. Log. 63(3), 1095–1115 (1998)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chiari, M., Krajíček, J.: Lifting independence results in bounded arithmetic. Arch. Math. Log. 38(2), 123–138 (1999)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Clote, P., Takeuti, G.: First order bounded arithmetic and small boolean circuit complexity classes. In: Clote, P., Remmel, J.B. (eds.) Feasible Mathematics II, Progress in Computer Science and Applied Logic, vol. 13, pp. 154–218. Birkhäuser, Boston (1995)zbMATHGoogle Scholar
  15. 15.
    Cook, S.A.: Feasibly constructive proofs and the propositional calculus. In: Proceedings of the 7th Annual ACM Symposium on Theory of Computing, pp. 83–97 (1975)Google Scholar
  16. 16.
    Cook, S.A., Krajíček, J.: Consequences of the provability of \(\mathbf{NP}\subseteq \mathbf{P}/\mathbf{poly}\). J. Symb. Log. 72(4), 1353–1371 (2007)CrossRefGoogle Scholar
  17. 17.
    Cook, S.A., Nguyen, P.: Logical Foundations of Proof Complexity. Perspectives in Logic. Cambridge University Press, New York (2010)CrossRefGoogle Scholar
  18. 18.
    Cook, S.A., Nguyen, P.: Corrections for [17] (2013). http://www.cs.toronto.edu/~sacook/homepage/corrections.pdf
  19. 19.
    Cordón-Franco, A., Fernández-Margarit, A., Lara-Martín, F.F.: Existentially closed models and conservation results in bounded arithmetic. J. Log. Comput. 19(1), 123–143 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Cordón-Franco, A., Lara-Martín, F.F.: Local induction and provably total computable functions. Ann. Pure Appl. Log. 165(9), 1429–1444 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Hájek, P., Pudlák, P.: Metamathematics of First-Order Arithmetic. Perspectives in Mathematical Logic. Springer-Verlag, Berlin (1993) (Second edition 1998)CrossRefGoogle Scholar
  22. 22.
    Håstad, J.: Almost optimal lower bounds for small depth circuits. In: Micali, S. (ed.) Randomness and Computation, Advances in Computing Research: A Research Annual, vol. 5, pp. 143–170. JAI Press, Greenwich, CT (1989)Google Scholar
  23. 23.
    Jeřábek, E.: Approximate counting by hashing in bounded arithmetic. J. Symb. Log. 74(3), 829–860 (2009)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Jeřábek, E.: On theories of bounded arithmetic for \(\mathit{NC}^1\). Ann. Pure Appl. Log. 162(4), 322–340 (2011)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Jeřábek, E., Nguyen, P.: Simulating non-prenex cuts in quantified propositional calculus. Math. Log. Q. 57(5), 524–532 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Johannsen, J., Pollett, C.: On the \(\varDelta ^b_1\)-bit-comprehension rule. In: Buss, S.R., Hájek, P., Pudlák, P. (eds.) Logic colloquium ’98: proceedings of the 1998 ASL European summer meeting held in Prague, Czech Republic, pp. 262–280. ASL (2000)Google Scholar
  27. 27.
    Kaye, R.: Parameter free induction in arithmetic. In: Proceedings of the 5th Easter Conference on Model Theory, pp. 70–81. Sektion Mathematik der Humboldt-Universität zu Berlin (1987). Seminarbericht Nr. 93Google Scholar
  28. 28.
    Kaye, R.: Axiomatizations and quantifier complexity. In: Proceedings of the 6th Easter Conference on Model Theory, pp. 65–84. Sektion Mathematik der Humboldt-Universität zu Berlin (1988). Seminarbericht Nr. 98Google Scholar
  29. 29.
    Kaye, R.: Diophantine induction. Ann. Pure Appl. Logic 46(1), 1–40 (1990)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Kaye, R., Paris, J., Dimitracopoulos, C.: On parameter free induction schemas. J. Symb. Log. 53(4), 1082–1097 (1988)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Krajíček, J.: Fragments of bounded arithmetic and bounded query classes. Trans. Am. Math. Soc. 338(2), 587–598 (1993)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Krajíček, J.: Bounded Arithmetic, Propositional Logic, and Complexity Theory, Encyclopedia of Mathematics and Its Applications, vol. 60. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
  33. 33.
    Krajíček, J., Pudlák, P.: Quantified propositional calculi and fragments of bounded arithmetic. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 36(1), 29–46 (1990)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Krajíček, J., Pudlák, P., Takeuti, G.: Bounded arithmetic and the polynomial hierarchy. Ann. Pure Appl. Log. 52(1–2), 143–153 (1991)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Paris, J.B., Wilkie, A.J.: Counting problems in bounded arithmetic. In: Di Prisco, C.A. (ed.) Methods in Mathematical Logic. Lecture Notes in Mathematics, vol. 1130, pp. 317–340. Springer-Verlag, Berlin (1985)CrossRefGoogle Scholar
  36. 36.
    Skelley, A., Thapen, N.: The provably total search problems of bounded arithmetic. Proc. Lond. Math. Soc. 103(1), 106–138 (2011)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Yao, A.C.C.: Separating the polynomial-time hierarchy by oracles. In: Tarjan, R.E. (ed.) Proceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science, pp. 1–10 (1985)Google Scholar
  38. 38.
    Zambella, D.: Notes on polynomially bounded arithmetic. J. Symb. Log. 61(3), 942–966 (1996)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of MathematicsThe Czech Academy of SciencesPrague 1Czech Republic

Personalised recommendations