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A small reflection principle for bounded arithmetic

Published online by Cambridge University Press:  12 March 2014

Rineke Verbrugge  and
Albert Visser
Rineke Verbrugge*
Affiliation:
Department of Mathematics and Computer Science, University of Amsterdam, 1018 Tv Amsterdam, The Netherlands
Albert Visser
Affiliation:
Department of Philosophy, University of Gothenburg, S-412 G8 Gothenburg, Sweden, E-mail: Albert.Visser@phil.RUU.nl
*
Department of Philosophy, University of Gothenburg, S-412 G8 Gothenburg, Sweden, E-mail: Rineke.Verbrugge@phil.gu.se
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Abstract

We investigate the theory IΔ0+Ω1 and strengthen [Bu86, Theorem 8.6] to the following: if NP ≠ co-NP, then Σ-completeness for witness comparison foumulas is not provable in bounded arithmetic. i.e.,

Next we study a “small reflection principle” in bounded arithmetic. We prove that for all sentences φ

The proof hinges on the use of definable cuts and partial satisfaction predicates akin to those introduced by Pudlák in [Pu86].

Finally, we give some applications of the small reflection principle, showing that the principle can sometimes be invoked in order to circumvent the use of provable Σ-completeness for witness comparison formulas.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 3 , September 1994 , pp. 785 - 812
Copyright
Copyright © Association for Symbolic Logic 1994

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References

REFERENCES

[Ad90]Adamowicz, Z., End-extending models of IΔ0 + EXP + BΣ1, Fundamenta Mathematicae, vol. 136 (1990), pp. 133–145.CrossRefGoogle Scholar
[Ad93]Adamowicz, Z., A contribution to the end-extension problem and the ∏1 conservativeness problem, Annals of Pure and Applied Logic, vol. 61 (1993). pp. 3–48.CrossRefGoogle Scholar
[BDG87]Balcázar, J. L., Díaz, J., and Gabarró, J.. Structural complexity I, Springer-Verlag, Berlin, 1987.Google Scholar
[Be89]Beklemishev, L. D., On the classification of propositional provability logics, Mathematics of the USSR Izvestiya, vol. 35 (1990), pp. 247–275.CrossRefGoogle Scholar
[Be91]Beklemishev, L. D., On bimodal provability logics for ∏1-axiomatized extensions of arithmetical theories, ITLI prepublication series, X-91-09, University of Amsterdam, Amsterdam, 1991.Google Scholar
[BV93]Berarducci, A. and Verbrugge, L. C., On the provability logic of bounded arithmetic, Annals of Pure and Applied Logic, vol. 61 (1993), pp. 75–93.CrossRefGoogle Scholar
[Bu86]Buss, S., Bounded arithmetic, Bibliopolis, Napoli, 1986.Google Scholar
[Fe60]Feferman, S., Arithmetization of metamathematics in a general setting, Fundamenta Mathematicae, vol. 49 (1960), pp. 35–92.CrossRefGoogle Scholar
[FR79]Ferrante, J. and Rackoff, C. W., The computational complexity of logical theories, Springer-Verlag, Berlin, 1979.CrossRefGoogle Scholar
[GS79]Guaspari, D. and Solovay, R. M., Rosser sentences, Annals of Mathematical Logic, vol. 16 (1979). pp. 81–99.CrossRefGoogle Scholar
[Há83]Hájek, P., On a new notion of partial conservativity, Logic colloquium ‘83 (Börger, E.et al. editors), vol. 2, Springer–Verlag, Berlin, 1983, pp. 217–232.Google Scholar
[HP93]Hájek, P. and Pudlák, P., Metamathematics of first-order arithmetic, Springer-Verlag, Berlin, 1993.CrossRefGoogle Scholar
[Jo87]de Jongh, D. H. J., A simplification of a completeness proof of Guaspari and Solovay, Studia Logica, vol. 46 (1987). pp. 187–192.CrossRefGoogle Scholar
[JM87]de Jongh, D. H. J. and Montagna, F., Generic generalized Rosser fixed points, Studia Logica, vol. 46 (1987), pp. 193–203.CrossRefGoogle Scholar
[JV88]de Jongh, D. H. J. and Veltman, F., Intensional logic, lecture notes, Philosophy Department, University of Amsterdam, Amsterdam, 1988.Google Scholar
[JMM91]de Jongh, D. H. J., Jumelet, M., and Montagna, F., On the proof of Solovay's theorem, Studia Logica, vol. 50 (1991), pp. 51–70.CrossRefGoogle Scholar
[JM91]de Jongh, D. H. J. and Montagna, F., Rosser orderings and free variables, Studia Logica, vol. 50 (1991), pp. 71–80.CrossRefGoogle Scholar
[KH82]Kent, C. F. and Hodgson, B. R., An arithmetical characterization of NP, Theoretical Computer Science, vol. 21 (1982). pp. 255–267.CrossRefGoogle Scholar
[KP89]Krajíček, J. and Pudlák, P., On the structure of initial segments of models of arithmetic, Archives of Mathematical Logic, vol. 28 (1989). pp. 91–98.CrossRefGoogle Scholar
[MA78]Manders, K. and Adleman, L., NP-complete decision problems for binary quadratics, Journal of Computer and System Sciences, vol. 16 (1978), pp. 168–184.CrossRefGoogle Scholar
[Ne86]Nelson, E., Predicative arithmetic, Mathematical Notes 32, Princeton University Press, Princeton, New Jersey, 1986.CrossRefGoogle Scholar
[Pa71]Parikh, R., Existence and feasibility in arithmetic, this Journal, vol. 36 (1971), pp. 494–508.Google Scholar
[Pu83]Pudlák, P., A definition of exponentiation by a bounded arithmetical formula, Commentationes Mathematicae Universitatis Carolinae, vol. 24 (1983). pp. 667–671.Google Scholar
[Pu85]Pudlák, P., Cuts, consistency statements and interpretations, this Journal, vol. 50 (1985), pp. 423–441.Google Scholar
[Pu86]Pudlák, P., On the length of proofs of finitistic consistency statements in first order theories, Logic colloquium ’84 (Paris, J. B.et al., editors), North-Holland, Amsterdam, 1986, pp. 165–196.Google Scholar
[Pu87]Pudlák, P., Improved bounds to the length of proofs of finitistic consistency statements, Logic and combinatorics (Simpson, S. G., editor): Contemporary Mathematics, vol. 35. American Mathematical Society, Providence, Rhode Island, 1987, pp. 309–332.Google Scholar
[Sm85]Smoryński, C., Self-reference and modal logic, Springer–Verlag, New York, 1985.CrossRefGoogle Scholar
[So76]Solovay, R. M., Provability interpretations of modal logic, Israel Journal of Mathematics, vol. 25 (1976), pp. 287–304.CrossRefGoogle Scholar
[So76b]Solovay, R. M., On interpretability in set theories, manuscript, 1976.Google Scholar
[So89]Solovay, R. M., Injecting inconsistencies into models of PA, Annals of Pure and Applied Logic, vol. 44 (1989), pp. 261–302.CrossRefGoogle Scholar
[St76]Stockmeyer, L. J., The polynomial-time hierarchy, Theoretical Computer Science, vol. 3 (1976), pp. 1–22.CrossRefGoogle Scholar
[Šv83]Švejdar, V., Modal analysis of generalized Rosser sentences, this Journal, vol. 48 (1983), pp. 986–999.Google Scholar
[Ta75]Takeuti, G., Proof theory, North-Holland, Amsterdam, 1975.Google Scholar
[Ta88]Takeuti, G., Bounded arithmetic and truth definition, Annals of Pure and Applied Logic, vol. 39 (1988). pp. 75–104.CrossRefGoogle Scholar
[Ve88]Verbrugge, L. C., Does Solovay's completeness theorem extend to bounded arithmetic?, Master's Thesis, University of Amsterdam, Amsterdam, 1988.Google Scholar
[Ve89]Verbrugge, L. C., Σ-completeness and bounded arithmetic, ITLI prepublication series for mathematical logic and foundations, ML-89-05, University of Amsterdam, Amsterdam, 1989.Google Scholar
[Vi81]Visser, A., Aspects of diagonalization & provability, Ph.D. thesis, University of Utrecht, Utrecht, 1981.Google Scholar
[Vi82]Visser, A., On the completeness principle, Annals of Mathematical Logic, vol. 22 (1982), pp. 263–295.Google Scholar
[Vi85]Visser, A., Evaluation, provably deductive equivalence in Heyting's Arithmetic of substitution instances of propositional formulas, Logic Group Prepint Series, no. 4, University of Utrecht, Utrecht, 1985.Google Scholar
[Vi90]Visser, A., Interpretabilty logic, Mathematical logic (Petkov, P. P., editor), Proceedings of the Heyting ’88 summer school. Plenum Press, New York, 1990, pp. 175–209.Google Scholar
[Vi91]Visser, A., The formalization of interpretability, Studia Logica, vol. 50 (1991), pp. 81–105.CrossRefGoogle Scholar
[WP87]Wilkie, A. J. and Paris, J. B., On the scheme of induction for bounded arithmetic formulas, Annals of Pure and Applied Logic, vol. 35 (1987), pp. 261–302.CrossRefGoogle Scholar
[WP89]Wilkie, A. J. and Paris, J. B., On the existence of end extensions of models of bounded induction, Logic, methodology and philosophy of science VIII (Fenstad, J. E.et al, editors), North-Holland, Amsterdam, 1989, pp. 143–161.Google Scholar
[Wr76]Wrathall, C., Complete sets and the polynomial time hierarchy, Theoretical Computer Science vol. 3 (1976). pp. 23–33.CrossRefGoogle Scholar