Advertisement

Dependent choice as a termination principle

  • Thomas PowellEmail author
Article
  • 3 Downloads

Abstract

We introduce a new formulation of the axiom of dependent choice, which can be viewed as an abstract termination principle that in particular generalises recursive path orderings, the latter being fundamental tools used to establish termination of rewrite systems. We consider several variants of our termination principle, and relate them to general termination theorems in the literature.

Keywords

Dependent choice Arithmetic in all finite types Termination Path orderings Higher order reverse mathematics 

Mathematics Subject Classification

03F03 03B15 03B70 

Notes

Acknowledgements

I am grateful to the anonymous referees for their many comments and corrections. I am also indebted to Georg Moser for suggesting the proof theoretic study of abstract path orderings, and to Sam Sanders for further comments on an earlier draft of this work.

References

  1. 1.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  2. 2.
    Berger, U.: A computational interpretation of open induction. In: Proceedings of LICS 2004, pp. 326–334. IEEE Computer Society (2004)Google Scholar
  3. 3.
    Buchholz, W.: Proof-theoretic analysis of termination proofs. Ann. Pure Appl. Logic 75, 57–65 (1995)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Coquand, T.: Constructive topology and combinatorics. Construct. Comput. Sci., LNCS 613, 159–164 (1991)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dershowitz, N.: Orderings for term rewriting systems. Theoret. Comput. Sci. 17(3), 279–301 (1982)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dershowitz, N.: Termination of rewriting. J. Symbolic Comput. 3, 69–116 (1987)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ferreira, M.C.F., Zantema, H.: Well-foundedness of term orderings. In: N. Dershowitz (Ed.) Conditional Term Rewriting Systems (CTRS ’94), LNCS, Vol. 968, pp. 106–123CrossRefGoogle Scholar
  8. 8.
    Goubault-Larrecq, J.: Well-founded recursive relations. In: Computer Science Logic (CSL’01), LNCS, Vol. 2142, pp. 484–498 (2001)CrossRefGoogle Scholar
  9. 9.
    Knuth, D.E., Bendix, P.: Simple word problems in universal algebras. In: Automation of Reasoning. Symbolic Computation, pp. 236–297. Springer (1970)Google Scholar
  10. 10.
    Kohlenbach, U.: Higher order reverse mathematics. In: Reverse Mathematics 2001, Lecture Notes in Logic, Vol. 21, pp. 281–295. ASL (2005)Google Scholar
  11. 11.
    Melliès, P.A.: On a duality between Kruskal and Dershowitz theorem. Proc. ICALP, LNCS 1443, 518–529 (1998)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Moser, G., Powell, T.: On the computational content of termination proofs. In: Proceedings of Computability in Europe (CiE 2015), LNCS, Vol. 9136, pp. 276–285 (2015)CrossRefGoogle Scholar
  13. 13.
    Powell, T.: On bar recursive interpretations of analysis. Ph.D. thesis, Queen Mary University of London (2013)Google Scholar
  14. 14.
    Powell, T.: The equivalence of bar recursion and open recursion. Ann. Pure Appl. Logic 165(11), 1727–1754 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Powell, T.: A proof theoretic study of abstract termination principles (2019). To appear in: Journal of Logic and ComputationGoogle Scholar
  16. 16.
    Powell, T.: Well quasi-orders and the functional interpretation. In: Schuster, P., Seisenberger, M., Weiermann, A. (eds.) Well-Quasi Orders in Computation, Logic, Language and Reasoning. Trends in Logic, vol. 53. Springer, pp. 221–269 (2020)Google Scholar
  17. 17.
    Seisenberger, M.: On the constructive content of proofs. Ph.D. thesis, Ludwig Maximilians Universität München (2003)Google Scholar
  18. 18.
    Sternagel, C.: A mechanized proof of Higman’s lemma by open induction. In: Schuster, P., Seisenberger, M., Weiermann, A. (eds.) Well-Quasi Orders in Computation, Logic, Language and Reasoning. Trends in Logic, vol. 53. Springer, pp. 339–350 (2020)Google Scholar
  19. 19.
    Troelstra, A.S.: Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Lecture Notes in Mathematics, vol. 344. Springer, Berlin (1973)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Fachbereich Mathematik, TU DarmstadtDarmstadtGermany

Personalised recommendations