Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-18T03:06:33.376Z Has data issue: false hasContentIssue false
  • English
  • Français

A proof-theoretic treatment of λ-reduction with cut-elimination: λ-calculus as a logic programming language

Published online by Cambridge University Press:  12 March 2014

Michael Gabbay
Michael Gabbay*
Affiliation:
Department of Philosophy, Kings College London, Strand, WC2R 2LS, UK, E-mail: michael.gabbay@kcl.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We build on an existing a term-sequent logic for the λ-calculus. We formulate a general sequent system that fully integrates αβη-reductions between untyped λ-terms into first order logic.

We prove a cut-elimination result and then offer an application of cut-elimination by giving a notion of uniform proof for λ-terms. We suggest how this allows us to view the calculus of untyped αβ-reductions as a logic programming language (as well as a functional programming language, as it is traditionally seen).

Keywords

term-sequentcut-eliminationlambda-calculuslogic programming
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 2 , June 2011 , pp. 673 - 699
Copyright
Copyright © Association for Symbolic Logic 2011

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

REFERENCES

[1]Baader, Franz and Nipkow, Tobias, Term rewriting and all that, Cambridge University Press, Great Britain, 1998.CrossRefGoogle Scholar
[2]Beeson, Michael, Lambda logic, Second international joint conference on automated reasoning (IJCAR 2004), Lecture Notes in Computer Science, vol. 3097, Springer, Berlin, 2004, pp. 460–474.Google Scholar
[3]Deplagne, Eric and Kirchner, Claude, Deduction versus computation: the case of induction, AISC and Calculemos joint international conferences, Lecture Notes in Artificial Intelligence, vol. 2385, Springer, Berlin, 2002, pp. 4–6.Google Scholar
[4]Dowek, Gilles and Werner, Benjamin, Arithmetic as a theory modulo, Term rewriting and applications, Lecture Notes in Comput. Sci., vol. 3467, Springer, Berlin, 2005, pp. 423–437.CrossRefGoogle Scholar
[5]Dummett, Michael, Elements of intuitionism, 1 ed., Clarendon Press, Oxford, 1977.Google Scholar
[6]Gabbay, Michael and Gabbay, Murdoch J., Term sequent logic, Proceedings of the 17th international workshop on functional and (constraint) logic programming (WFLP 2008), Electronic Notes in Theoretical Computer Science, vol. 246, 08 2009, pp. 87–106.Google Scholar
[7]Gabbay, Murdoch J. and Gabbay, Michael J., A simple class of Kripke-style models in which logic and computation have equal standing, International conference on logic for programming artificial intelligence and reasoning (LPAR 2010) (Clarke, Edmund M. and Voronkov, Andrei, editors), Lecture Notes in Computer Science, vol. 6355, Springer, 2010, pp. 231–254.Google Scholar
[8]Gabbay, Murdoch J. and Gabbay, Michael J., a-logic with arrows, Proceedings of the 16th international workshop on functional and (constraint) logic programming (WFLP 2007) (Echahed, Rachid, editor), 06 2007, pp. 47–62.Google Scholar
[9]Hacking, Ian, What is logic?. The Journal of Philosophy, vol. 76 (1979), pp. 285–319.CrossRefGoogle Scholar
[10]Hindley, J. Roger and Seldin, Jonathan P., Lambda-calculus and combinators, an introduction, 2 ed., Cambridge University Press, Cambridge, 2008.CrossRefGoogle Scholar
[11]Kaplan, Stephane and Jouannaud, Jean-Pierre (editors), Conditional term rewriting systems, Lecture Notes in Computer Science, vol. 308, Berlin, Springer-Verlag, 07 1988.CrossRefGoogle Scholar
[12]Liang, Zhi An and Miller, Dale, Focusing and polarization in intuitionistic logic, Computer science logic, CSL 2007 (Duparc, Jacques and Henzinger, Thomas A., editors), Lecture Notes in Computer Science, vol. 4646, Springer, 2007, pp. 451–465.Google Scholar
[13]Martí-Oliet, N. and Meseguer, J., Rewriting logic as a logical and semantic framework, Electronic notes in theoretical computer science (Meseguer, J., editor), vol. 4, Elsevier Science Publishers, 2000.Google Scholar
[14]Miller, Dale, A logic programming language with lambda-abstraction, function variables, and simple unification, Journal of Logic and Computation, vol. 1 (1991), no. 4, pp. 497–536.CrossRefGoogle Scholar
[15]Miller, Dale, Nadathur, Gopalan, Pfenning, Frank, and Scedrov, Andre, Uniform proofs as a foundation for logic programming, Technical report, Durham, NC, USA, 1991.CrossRefGoogle Scholar
[16]O'Hearn, Peter W. and Pym, David J., The logic of bunched implications, The Bulletin of Symbolic Logic, vol. 5 (1999), no. 2, pp. 215–244.CrossRefGoogle Scholar
[17]Selinger, Peter, Order-incompleteness andfinite lambda reduction models, Theoretical Computer Science, vol. 309 (2003), no. 1, pp. 43–63.CrossRefGoogle Scholar