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# A coding theoretic study of MLL proof nets

Abstract

In this paper we propose a novel approach for analysing proof nets of Multiplicative Linear Logic (MLL) using coding theory. We define families of proof structures called PS-families and introduce a metric space for each family. In each family: (1)

an MLL proof net is a true code element; and

(2)

a proof structure that is not an MLL proof net is a false (or corrupted) code element.

The definition of our metrics elegantly reflects the duality of the multiplicative connectives. We show that in our framework one-error-detection is always possible but one-error-correction is always impossible. We also demonstrate the importance of our main result by presenting two proof-net enumeration algorithms for a given PS-family: the first searches proof nets naively and exhaustively without help from our main result, while the second uses our main result to carry out an intelligent search. In some cases, the first algorithm visits proof structures exponentially, while the second does so only polynomially.

References
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Baylis, J. (1998) Error-Correcting Codes: A Mathematical Introduction, Chapman and Hall.
Bellin, G. and van de Wiele, J. (1995) Subnets of Proof-nets in MLL. In: Girard, J.-Y., Lafont, Y. and Regnier, L. (eds.) Advances in Linear Logic, Cambridge University Press 249270.
Danos, V. and Regnier, R. (1989) The structure of multiplicatives. Archive for Mathematical Logic 28 181203.
Girard, J.-Y. (1987) Linear Logic. Theoretical Computer Science 50 1102.
Girard, J.-Y. (1996) Proof-nets: the parallel syntax for proof-theory. In: Ursini, A. and Agliano, P. (eds.) Logic and Algebra, Marcel Dekker 97124.
Girard, J.-Y. (2006) Le Point Aveugle: Tome I, vers la perfection, Hermann.
Guerrini, S. (1999) Correctness of Multiplicative Proof Nets Is Linear. In: 14th Annual IEEE Symposium on Logic in Computer Science (LICS'99), IEEE Computer Society Press 454463.
Matsuoka, S. (2007) Weak Typed Böhm Theorem on IMLL. Annals of Pure and Applied Logic 145 3790.
MacWilliams, F. J. and Sloane, N. J. A. (1993) The Theory of Error-correcting Codes, North-Holland.
Recommend this journal

Mathematical Structures in Computer Science
• ISSN: 0960-1295
• EISSN: 1469-8072
• URL: /core/journals/mathematical-structures-in-computer-science
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