Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-18T21:21:55.576Z Has data issue: false hasContentIssue false
  • English
  • Français

Natural deduction via graphs: formal definition and computation rules

Published online by Cambridge University Press:  01 June 2007

HERMAN GEUVERS  and
IRIS LOEB
HERMAN GEUVERS
Affiliation:
Radboud University Nijmegen, Institute for Computing and Information Sciences, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands Email: H.Geuvers@cs.ru.nl; I.Loeb@cs.ru.nl
IRIS LOEB
Affiliation:
Radboud University Nijmegen, Institute for Computing and Information Sciences, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands Email: H.Geuvers@cs.ru.nl; I.Loeb@cs.ru.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we introduce the formalism of deduction graphs as a generalisation of both Gentzen–Prawitz style natural deduction and Fitch style flag deduction. The advantage of this formalism is that, as with flag deductions (but not natural deduction), subproofs can be shared, but the linearisation used in flag deductions is avoided. Our deduction graphs have both nodes and boxes, which are collections of nodes that also form a node themselves. This is reminiscent of the bigraphs of Milner, where the link graph describes the nodes and edges and the place graph describes the nesting of nodes. We give a precise definition of deduction graphs, together with some illustrative examples. Furthermore, we analyse their computational behaviour by studying the process of cut-elimination and by defining translations from deduction graphs to simply typed lambda terms. From a slight variation of this translation, we conclude that the process of cut-elimination is strongly normalising. The translation to simple type theory removes quite a lot of structure, so we also propose a translation to a context calculus with lets that faithfully captures the structure of deduction graphs. The proof nets of linear logic also offer a graph-like presentation of natural deduction, and we point out some similarities between the two formalisms.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 17 , Issue 3 , June 2007 , pp. 485 - 526
Copyright
Copyright © Cambridge University Press 2007

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

Asperti, A. and Guerrini, S. (1998) The Optimal Implementation of Functional Programming Languages. Cambridge Tracts in Theoretical Computer Science 45, Cambridge University Press.Google Scholar
Cosmo, R. D. and Kesner, D. (1997) Strong normalization of explicit substitutions via cut-elimination in proof nets. In: Twelfth Annual IEEE Symposium on Logic in Computer Science (LICS), IEEE Computer Society Press 35–46.CrossRefGoogle Scholar
Fitch, F. B. (1952) Symbolic Logic, Ronald Press Company, New York.Google Scholar
Gentzen, G. (1969) Collected Works (edited by Szabo, M. E.), North-Holland.Google Scholar
Geuvers, H. and Nederpelt, R. (2004) Rewriting for Fitch style natural deductions. In: van Oostrom, V. (ed.) Proceedings of RTA 2004. Springer-Verlag Lecture Notes in Computer Science 391 134–154.CrossRefGoogle Scholar
Girard, J.-Y. (1987) Linear Logic. Theoretical Computer Science 50 (1)1–101, 1987.CrossRefGoogle Scholar
Klop, J. W. (1980) Combinatory Reduction Systems, Ph.D. thesis, University of Utrecht. (Mathematical Centre Tracts 127, CWI, Amsterdam.)Google Scholar
Kesner, D. and Lengrand, S. (2005) Extending the Explicit Substitution Paradigm. In: Proceedings of the 16th International Conference on Rewriting Techniques and Applications (RTA), Nara, Japan. Springer-Verlag Lecture Notes in Computer Science 3467 407–422.CrossRefGoogle Scholar
Lafont, Y. (1990) Interaction nets. In: Proceedings of the 17th ACM Symposium on Principles of Programming Languages (POPL'90), ACM Press 95–108.Google Scholar
Maraist, J., Odersky, M., Turner, D. N. and Wadler, P. (1998) Call-by-name, call-by-value, call-by-need, and the linear lambda calculus. Theoretical Computer Science 228 (1-2)175–210.CrossRefGoogle Scholar
Milner, R. (2004) Axioms for bigraphical structure. Technical Report UCAM-CL-TR-581, Computer Laboratory, University of Cambridge.Google Scholar
Nederpelt, R. P. (1973) Strong normalization in a typed lambda calculus with lambda structured types, Ph.D. thesis, Technical University Eindhoven.Google Scholar
Prawitz, D. (1965) Natural Deduction, Almquist and Wiksell, Stockholm.Google Scholar
Prawitz, D. (1971) Ideas and results in Proof Theory. In: Fenstad, J. E. (ed.) Proc. of the second Scandinavian Logic Symposium 235–307.CrossRefGoogle Scholar
Sørensen, M. H. (1997) Normalization in Lambda-Calculus and Type Theory, Ph.D. thesis, Department of Computer Science, University of Copenhagen.Google Scholar