Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T04:40:25.660Z Has data issue: false hasContentIssue false
  • English
  • Français

The New Normal: We Cannot Eliminate Cuts in Coinductive Calculi, But We Can Explore Them

Published online by Cambridge University Press:  22 September 2020

Ekaterina Komendantskaya Open the ORCID record for Ekaterina Komendantskaya [Opens in a new window] ,
Dmitry Rozplokhas  and
Henning Basold
Ekaterina Komendantskaya
Affiliation:
Heriot-Watt University, Edinburgh, Scotland, UK (e-mail: ek19@hw.ac.uk)
Dmitry Rozplokhas
Affiliation:
Jet Brains Research, St Petersburgh, Russia (e-mail: rozplokhas@gmail.com)
Henning Basold
Affiliation:
Leiden University, the Netherlands (e-mail: henning@basold.eu)
Get access Rights & Permissions [Opens in a new window]

Abstract

In sequent calculi, cut elimination is a property that guarantees that any provable formula can be proven analytically. For example, Gentzen’s classical and intuitionistic calculi LK and LJ enjoy cut elimination. The property is less studied in coinductive extensions of sequent calculi. In this paper, we use coinductive Horn clause theories to show that cut is not eliminable in a coinductive extension of LJ, a system we call CLJ. We derive two further practical results from this study. We show that CoLP by Gupta et al. gives rise to cut-free proofs in CLJ with fixpoint terms, and we formulate and implement a novel method of coinductive theory exploration that provides several heuristics for discovery of cut formulae in CLJ.

Keywords

Sequent CalculusHorn ClausesCoinductionCut EliminationTheory Exploration
Type
Original Article
Information
Theory and Practice of Logic Programming , Volume 20 , Issue 6: 36th International Conference on Logic Programming Special Issue II , November 2020 , pp. 990 - 1005
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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

Barendregt, H., Dekkers, W., and Statman, R. 2013. Lambda Calculus with Types. Cambridge University Press, Cambridge; New York.Google Scholar
Basold, H., Komendantskaya, E., and Li, Y. 2019. Coinduction in uniform: Foundations for corecursive proof search with horn clauses. In ESOP 2019. 783–813.Google Scholar
Blanchette, J. et al. 2017. Foundational nonuniform (co)datatypes for higher-order logic. In LICS’17. IEEE Computer Society, 1–12.Google Scholar
Boyer, R. S. and Moore, J. S. 1979. A Computational Logic . ACM Monograph Series. Academic Press.Google Scholar
Brotherston, J. and Simpson, A. 2011. Sequent calculi for induction and infinite descent. JLC 21, 6, 1177–1216.Google Scholar
Courcelle, B. 1983. Fundamental properties of infinite trees. TCS 25, 95–169.Google Scholar
Dagnino, F., Ancona, D., and E.Zucca. 2020. Flexible coinductive logic programming. TPLP.Google Scholar
Endrullis, J., Hansen, H. H., Hendriks, D., Polonsky, A., and Silva, A. 2015. A coinductive framework for infinitary rewriting and equational reasoning. In RTA’15. 143–159.Google Scholar
Fu, P. and Komendantskaya, E. 2016. Operational semantics of resolution and productivity in Horn clause logic. Formal Aspects of Computing.Google Scholar
Fu, P., Komendantskaya, E., Schrijvers, T., and Pond, A. 2016. Proof relevant corecursive resolution. In FLOPS’16. Springer, 126–143.Google Scholar
Gentzen, G. 1969. Investigations into logical deduction. In The Collected Papers of Gerhard Gentzen, M. Szabo, Ed. Studies in Logic and the Foundations of Mathematics, vol. 55. Elsevier, 68–131.Google Scholar
Gupta, G., Bansal, A., Min, R., Simon, L., and Mallya, A. 2007. Coinductive logic programming and its applications. In Logic Programming, Dahl, V. and Niemelä, I., Eds. Springer Berlin Heidelberg, Berlin, Heidelberg, 27–44.Google Scholar
Kimura, D., Nakazawa, K., Terauchi, T., and Unno, H. 2020. Failure of cut-elimination in cyclic proofs of separation logic. Computer Software 37, 39–52.Google Scholar
Komendantskaya, E. and Li, Y. 2017. Productive corecursion in logic programming. J. TPLP (ICLP’17 post-proc.) 17, 5-6, 906–923.Google Scholar
Miller, D. and Nadathur, G. 2012. Programming with Higher-order logic. Cambridge University Press.Google Scholar
Miller, D., Nadathur, G., Pfenning, F., and Scedrov, A. 1991. Uniform Proofs as a Foundation for Logic Programming . Annals of Pure and Applied Logic, vol. 51. Elsevier, 125–157.Google Scholar
Plotkin, G. D. 1970. A note on inductive generalization. Machine intelligence.Google Scholar
Saotome, K., Nakazawa, K., and Kimura, D. 2020. Restriction on cut in cyclic proof system for symbolic heaps. In FLOPS’20.Google Scholar
Schubert, A. and Urzyczyn, P. 2018. First-order answer set programming as constructive proof search. Theory Pract. Log. Program. 18, 3-4, 673–690.CrossRefGoogle Scholar
Simon, L., Mallya, A., Bansal, A., and Gupta, G. 2006. Coinductive logic programming. In ICLP. 330–345.Google Scholar
Sorensen, M. H. and Urzyczyn, P. 2006. Lectures on the Curry-Howard Isomorphism. Studies in Logic. Elsevier.Google Scholar
Sulzmann, M., Duck, G. J., Jones, S. L. P., and Stuckey, P. J. 2007. Understanding functional dependencies via constraint handling rules. J. Funct. Program. 17, 1, 83–129.Google Scholar
Troelstra, A. S. and Schwichtenberg, H. 2000. Basic Proof Theory, 2nd ed. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge.CrossRefGoogle Scholar