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The undecidability of second order linear logic without exponentials

Published online by Cambridge University Press:  12 March 2014

Yves Lafont
Yves Lafont*
Affiliation:
Laboratoire de Mathématiques Discrètes, UPR 9016 du CNRS, 163 Avenue de Luminy, Case 930, 13288 Marseille Cedex 9, France, E-mail: lafont@lmd.univ-mrs.fr
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Abstract

Recently, Lincoln, Scedrov and Shankar showed that the multiplicative fragment of second order intuitionistic linear logic is undecidable, using an encoding of second order intuitionistic logic. Their argument applies to the multiplicative-additive fragment, but it does not work in the classical case, because second order classical logic is decidable. Here we show that the multiplicative-additive fragment of second order classical linear logic is also undecidable, using an encoding of two-counter machines originally due to Kanovich. The faithfulness of this encoding is proved by means of the phase semantics.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 2 , June 1996 , pp. 541 - 548
Copyright
Copyright © Association for Symbolic Logic 1996

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References

REFERENCES

[1]Girard, J.-Y., Linear logic, Theoretical Computer Science, vol. 50 (1987), pp. 1–102.Google Scholar
[2]Girard, J.-Y., Linear logic: its syntax and semantics, Advances in Linear Logic (Girard, J.-Y., Lafont, Y., and Regnier, L., editors), London Mathematical Society Lecture Note Series, 222, Cambridge University Press, 1995, pp. 1–42.Google Scholar
[3]Kanovich, M., The direct simulation of Minsky machines in linear logic, Advances in Linear Logic (Girard, J.-Y., Lafont, Y., and Regnier, L., editors), London Mathematical Society Lecture Note Series, no. 222, Cambridge University Press, 1995, pp. 123–145.Google Scholar
[4]Lafont, Y. and Scedrov, A., The undecidability of second order multiplicative linear logic, to appear in Information and Computation, 1996.Google Scholar
[5]Lambek, J., How to program an infinite abacus, Canadian Mathematical Bulletin, vol. 4 (1961), pp. 295–302.Google Scholar
[6]Lincoln, P., Mitchell, J., Scedrov, A., and Shankar, N., Decision problems for propositional linear logic, Annals of Pure and Applied Logic, vol. 56 (1992), pp. 239–311.Google Scholar
[7]Minsky, M., Recursive unsolvability of Post's problem of ‘tag’ and other topics in the theory of Turing machines, Annals of Mathematics, vol. 74 (1961), pp. 437–455.Google Scholar
[8]Scedrov, A.Lincoln, P. and Shankar, N., Decision problems for second order linear logic, 10-th annual IEEE Symposium on logic in computer science (Girard, J.-Y., editor), IEEE Computer Society Press, San Diego, California, 1995, pp. 239–311.Google Scholar
[9]Scedrov, A., A brief guide to linear logic, Current trends in theoretical computer science (Rozenberg, G. and Salomaa, A., editors), World Scientific Publishing Co., 1993, pp. 377–394.Google Scholar
[10]Schellinx, H., Some syntactical observations on linear logic, Journal of Logic and Computation, vol. 1 (1991), no. 4, pp. 537–559.Google Scholar