Skip to main content Accessibility help
×
Home
Hostname: page-component-7f7b94f6bd-98q29 Total loading time: 0.181 Render date: 2022-06-29T21:58:49.965Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

Proving consistency of equational theories in bounded arithmetic

Published online by Cambridge University Press:  12 March 2014

Arnold Beckmann†*
Affiliation:
Institut Für Mathematische Logik und Grundlagenforschung, Westfälische Wilhelms-Universität, Einsteinstr. 62, 48149 Münster, Germany, E-mail: Arnold.Beckmann@math.uni-muenster.de Mathematical Institute, University of Oxford, 24-29 St. Giles', Oxford OX1 3LB, UK

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
Copyright
Copyright © Association for Symbolic Logic 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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. 1130.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. 221244.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. 199223.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. 19.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. 8397.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. 243320.Google Scholar
[8]Pudlák, Pavel, A note on bounded arithmetic, Fundamenta Mathematical, vol. 136 (1990), no. 2, pp. 8589.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. 355362.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. 261302.Google Scholar
2
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Proving consistency of equational theories in bounded arithmetic
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Proving consistency of equational theories in bounded arithmetic
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Proving consistency of equational theories in bounded arithmetic
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *