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A modal view of linear logic

Published online by Cambridge University Press:  12 March 2014

Simone Martini  and
Andrea Masini
Simone Martini
Affiliation:
Dipartimento di Informatica, Università di Pisa, Corso Italia, 40, 1-56125 Pisa, Italy, E-mail: [email protected]
Andrea Masini
Affiliation:
Dipartimento di Informatica, Università di Pisa, Corso Italia, 40, 1-56125 Pisa, Italy, E-mail: [email protected]
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Abstract

We present a sequent calculus for the modal logic S4, and building on some relevant features of this system (the absence of contraction rules and the confinement of weakenings into axioms and modal rules) we show how S4 can easily be translated into full prepositional linear logic, extending the Grishin-Ono translation of classical logic into linear logic. The translation introduces linear modalities (exponentials) only in correspondence with S4 modalities. We discuss the complexity of the decision problem for several classes of linear formulas naturally arising from the proposed translations.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 3 , September 1994 , pp. 888 - 899
Copyright
Copyright © Association for Symbolic Logic 1994

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References

REFERENCES

[1]Curry, H. B., Foundations of mathematical logic, Dover, New York, 1977.Google Scholar
[2]Dopp, J., Appendix Modal logics (Feys, , editor), Gauthier-Villars, Paris and Nauwelaerts, Louvain, 1965.Google Scholar
[3]Garey, M. R. and Johnson, D. S., Computers and intractability, W. H. Freeman and Co., San Francisco, California, 1979.Google Scholar
[4]Girard, J.-Y., Linear logic, Theoretical Computer Science, vol. 50, (1987), pp. 1–102.CrossRefGoogle Scholar
[5]Grishin, V. N., A nonstandard logic, and its application to set theory, Issledovanija po formalizovannym jazykam i néklassiceskim logikam (Studies in formalized languages and nonclassical logics), Nauka, Moscow, 1974, pp. 135–171. (Russian)Google Scholar
[6]Kanger, S. G., Provability in logic, Acta Universitatis Stockolmiensis, Stockolm Studies in Philosophy, vol. 1, Almquist & Wiksell, Stockolm, 1957.Google Scholar
[7]Ketonen, D., Untersucfumgen zum Pr&dikatenkalkül, Series A I. Mathematica Dissertationes, vol. 23, Annales Academiae Scientiarumicae Fenn, Helsinki, 1944.Google Scholar
[8]Lincoln, P., Mitchell, J. C., Scedrov, A., and Shankar, N., Decision problems for propositional linear logic, Annals of Pure and Applied Logic, vol. 56 (1992), pp. 239–311.CrossRefGoogle Scholar
[9]Lincoln, P., Scedrov, A., and Shankar, N., Linearizing intuitionistic implication, Annals of Pure and Applied Logic, vol. 60 (1993), pp. 151–177.CrossRefGoogle Scholar
[10]Ohnishi, M. and Matsumoto, K., Gentzen method in modal calculi I. Osaka Mathematical Journal, vol. 9 (1957), pp. 113–130.Google Scholar
[11]Ono, H., Structure rules and a logical hierarchy, Mathematical Logic (Petkov, P. P., editor), Plenum Press, New York, London, 1990, pp. 95–104.Google Scholar
[12]Prawitz, D. and Malmnäs, P.-E.. A survey on some connections between classical intuitionistic, and minimal logic, Contributions to mathematical logic, Logic Colloquium 1966 (Schmidt, H. A.. Schütte, K., and Thiele, H.-J., editors), North-Holland. Amsterdam, 1968, pp. 215–229.Google Scholar
[13]Troelstra, A. S., Lectures in linear logic, CSLI Lecture Notes, vol. 29, Chicago University Press. Chicago, Illinois. 1991.Google Scholar
[14]Zeman, J. J., Modal logic, the Lewis modal systems, Clarendon Press, Oxford. 1973.Google Scholar