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Proving consistency of equational theories in bounded arithmetic

Published online by Cambridge University Press:  12 March 2014

Arnold Beckmann†
Arnold Beckmann†*
Affiliation:
Institut Für Mathematische Logik und Grundlagenforschung, Westfälische Wilhelms-Universität, Einsteinstr. 62, 48149 Münster, Germany, E-mail: [email protected] Mathematical Institute, University of Oxford, 24-29 St. Giles', Oxford OX1 3LB, UK
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Abstract

We consider equational theories for functions denned via recursion involving equations between closed terms with natural rules based on recursive definitions of the function symbols. We show that consistency of such equational theories can be proved in the weak fragment of arithmetic S21. In particular this solves an open problem formulated by Takeuti (c.f. [5, p.5 problem 9.]).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 1 , March 2002 , pp. 279 - 296
Copyright
Copyright © Association for Symbolic Logic 2002

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References

REFERENCES

[1]Beckmann, Arnold and Weiermann, Andreas, A term rewriting characterization of the polytime functions and related complexity classes, Archive of Mathematical Logic, vol. 36 (1996), pp. 11–30.CrossRefGoogle Scholar
[2]Buss, Samuel R., Bounded arithmetic, Studies in proof theory. Lecture notes, 3, Bibliopolis, Naples, 1986.Google Scholar
[3]Buss, Samuel R. and Ignjatović, Aleksandar, Unprovability of consistency statements in fragments of bounded arithmetic, Annals of Pure and Applied Logic, vol. 74 (1995), no. 3, pp. 221–244.CrossRefGoogle Scholar
[4]Cichon, E. A. and Weiermann, A., Term rewriting theory for the primitive recursive functions, Annals of Pure and Applied Logic, vol. 83 (1997), no. 3, pp. 199–223.CrossRefGoogle Scholar
[5]Clote, Peter and Kxajíček, Jan, Open problems, Arithmetic, proof theory, and computational complexity. Papers from the conference held in Prague, July 2–5, 1991 (Clote, Peter and Krajíček, Jan, editors), Oxford Logic Guides, no. 23, The Clarendon Press, Oxford University Press, New York, 1993, pp. 1–9.Google Scholar
[6]Cook, Stephen A., Feasibly constructive proofs and the propositional calculus, Seventh annual ACM symposium on theory of computing (Albuquerque, NM, 1975), Association for Computing Machinery, New York, 1975, pp. 83–97.CrossRefGoogle Scholar
[7]Dershowitz, Nachum and Jouannaud, Jean-Pierre, Rewrite systems, Handbook of theoretical computer science. Volume B (van Leeuwen, Jan, editor), Elsevier Science Publishers, Amsterdam, 1990, pp. 243–320.Google Scholar
[8]Pudlák, Pavel, A note on bounded arithmetic, Fundamenta Mathematical, vol. 136 (1990), no. 2, pp. 85–89.CrossRefGoogle Scholar
[9]Weiermann, Andreas, Termination proofs for term rewriting systems with lexicographic path orderings imply multiply recursive derivation lengths, Theoretical Computer Science, vol. 139 (1995), pp. 355–362.CrossRefGoogle Scholar
[10]Wilkie, Alex J. and Paris, Jeff B., On the scheme of induction for bounded arithmetic formulas, Annals of Pure and Applied Logic, (1987), pp. 261–302.Google Scholar