# Classical Natural Deduction for S4 Modal Logic

Article

First Online:

Received:

Revised:

- 109 Downloads

## Abstract

This paper proposes a natural deduction system CNDS4 for classical S4 modal logic with necessity and possibility modalities. This new system is an extension of Parigot’s Classical Natural Deduction with dualcontext to formulate S4 modal logic. The modal λ*μ*-calculus is also introduced as a computational extraction of CNDS4. It is an extension of both the λ*μ*-calculus and the modal λ-calculus. Subject reduction, confluency, and strong normalization of the modal λ*μ*-calculus are shown. Finally, the computational interpretation of the modal λ*μ*-calculus, especially the computational meaning of the modal possibility operator, is discussed.

## Keywords:

Functional Programming Classical S4 Modal Logic Curry-Howard Isomorphism Lambda-Calculus Staged Computation## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Bierman, G. M. and De Paiva, V., “On an intuitionistic modal logic,”
*Studia Logica, 65, 3*, pp. 383-416, 2000.CrossRefMATHMathSciNetGoogle Scholar - 2.Curien, P. L. and Herbelin, H., “The duality of computation,” in
*Proc. of the 5th ACM SIGPLAN International Conference on Functional Programming, ICFP*, pp. 233-243, 2000.Google Scholar - 3.Davies, R., “A temporal-logic approach to binding-time analysis,” in
*Proc. of 11 th Annual IEEE Symposium on Logic in Computer Science*, IEEE Computer Society Press, pp. 184-195, 1996.Google Scholar - 4.Davies, R. and Pfenning, F., “A modal analysis of staged computation,”
*Journal of the ACM, 48, 3*, pp. 555-604, 2001.CrossRefMathSciNetGoogle Scholar - 5.De Groote, P., “On the relation between the lambda-mu-calculus and the syntactic theory of sequential control,” in
*Proc. of the 5th International Conference on Logic Programming and Automated Reasoning, LPAR ’94, LNCS, 822*, pp. 31-43, 1994.Google Scholar - 6.Ghani, N., De Paiva, V. and Ritter, E., “Explicit Substitutions for Constructive Necessity,” in
*The 25th International Colloquium, Automata, Languages and Programming, Denmark*, pp. 743-754, 1998.Google Scholar - 7.Griffin, T. G., “A formulae-as-types notion of control,” in
*Proc. of the 1990 Principles of Programming Languages Conference*, IEEE Computer Society Press, pp. 47-58, 1990.Google Scholar - 8.Groote, P. D., “An environment machine for the λ
*μ*-calculus,”*Mathematical Structures in Computer Science, 8, 6*, pp. 637-669, 1998.CrossRefMATHMathSciNetGoogle Scholar - 9.Kakutani, Y., “Duality between Call-by-Name Recursion and Call-by-Value Iteration,” in
*Proc. of the 16th International Workshop on Computer Science Logic, CSL, LNCS, 2471*, pp. 506-521, 2002.Google Scholar - 10.Kakutani, Y., “Call-by-Name and Call-by-Value in Normal Modal Logic,” in
*Proc. of Programming Languages and Systems, 5th Asian Symposium, APLAS, LNCS, 4807*, pp. 399-414, 2007.Google Scholar - 11.Kimura, D., “Duality between Call-by-value Reductions and Call-by-name Reductions,”
*IPSJ Journal, 48, 4*, pp. 1721-1757, 2007.MathSciNetGoogle Scholar - 12.De Paz, M., Medeiros, N., “A new S4 classical modal logic in natural deduction,”
*Journal of Symbolic Logic, 71, 3*, pp. 799-809, 2006CrossRefMathSciNetGoogle Scholar - 13.Nanevski, A., “A Modal Calculus for Exception Handling,” in
*the 3rd intuitionistic modal logics and applications workshop*, 2005.Google Scholar - 14.Newman, M. H. A., “On theories with a combinatorial definition of “equivalence”,”
*Annals of Mathematics, 43, 2*, pp. 223-243, 1942.CrossRefMathSciNetGoogle Scholar - 15.Ong, C.-H. L. and Stewart, C. A., “A Curry-Howard foundation for functional computation with control,” in
*Proc. of the Symposium on Principles of Programming Languages*, pp. 215-227, 1997.Google Scholar - 16.Parigot, M., “λ
*μ*-calculus: an algorithmic interpretation of classical natural deduction,” in*Proc. of International Conference on Logic Programming and Automated Deduction, LNCS, 624*, pp. 190-201, 1992.Google Scholar - 17.Parigot, M., “Strong normalization for second order classical natural deduction,” in
*Proc. of Eighth Annual IEEE Symposium on Logic in Computer Science*, pp. 39-46, 1993.Google Scholar - 18.Pfenning, F. and Davies, R., “A judgmental reconstruction of modal logic,”
*Mathematical Structures in Computer Science, 11*, pp. 511-540, 2001.CrossRefMATHMathSciNetGoogle Scholar - 19.Prawitz, D.,
*Natural Deduction: A Proof-Theoretical Study*, Almqvist and Wiksell, Stockholm, 1965.Google Scholar - 20.Selinger, P., “Control Categories and Duality: on the Categorical Semantics of the Lambda-Mu Calculus,”
*Mathematical Structures in Computer Science*, pp. 207-260, 2001.Google Scholar - 21.Shan, C.-C., “A Computational Interpretation of Classical S4 Modal Logic,” in
*The 3rd intuitionistic modal logics and applications workshop*, 2005.Google Scholar - 22.Taha, W. and Sheard, T., “MetaML and multi-stage programming with explicit annotations,”
*Theoretical Computer Science, 248, 1-2*, pp. 211-242, 2000.CrossRefMATHGoogle Scholar - 23.Wadler, P., “Call-by-Value is Dual to Call-by-Name - Reloaded,” in
*Rewriting Techniques and Applications, Japan, LNCS, 3467*, pp. 185-203, 2005.Google Scholar - 24.Yuse, Y. and Igarashi, A., “A modal type system for multi-level generating extensions with persistent code,” in
*Proc. of the 8th ACM SIGPLAN Symposium on Principles and Practice of Declarative Programming*, pp. 201-212, 2006.Google Scholar

## Copyright information

© Ohmsha and Springer Japan jointly hold copyright of the journal. 2011