Home

# Natural deduction via graphs: formal definition and computation rules

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.

References
Hide All
Asperti, A. and Guerrini, S. (1998) The Optimal Implementation of Functional Programming Languages. Cambridge Tracts in Theoretical Computer Science 45, Cambridge University Press.
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 3546.
Fitch, F. B. (1952) Symbolic Logic, Ronald Press Company, New York.
Gentzen, G. (1969) Collected Works (edited by Szabo, M. E.), North-Holland.
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.
Girard, J.-Y. (1987) Linear Logic. Theoretical Computer Science 50 (1)1101, 1987.
Klop, J. W. (1980) Combinatory Reduction Systems, Ph.D. thesis, University of Utrecht. (Mathematical Centre Tracts 127, CWI, Amsterdam.)
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 407422.
Lafont, Y. (1990) Interaction nets. In: Proceedings of the 17th ACM Symposium on Principles of Programming Languages (POPL'90), ACM Press 95108.
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)175210.
Milner, R. (2004) Axioms for bigraphical structure. Technical Report UCAM-CL-TR-581, Computer Laboratory, University of Cambridge.
Nederpelt, R. P. (1973) Strong normalization in a typed lambda calculus with lambda structured types, Ph.D. thesis, Technical University Eindhoven.
Prawitz, D. (1965) Natural Deduction, Almquist and Wiksell, Stockholm.
Prawitz, D. (1971) Ideas and results in Proof Theory. In: Fenstad, J. E. (ed.) Proc. of the second Scandinavian Logic Symposium 235307.
Sørensen, M. H. (1997) Normalization in Lambda-Calculus and Type Theory, Ph.D. thesis, Department of Computer Science, University of Copenhagen.
Recommend this journal

Mathematical Structures in Computer Science
• ISSN: 0960-1295
• EISSN: 1469-8072
• URL: /core/journals/mathematical-structures-in-computer-science
Who would you like to send this to? *

×

## Full text viewsFull text views reflects the number of PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 13 *