Abstract
We discuss a scheme for defining and reasoning about partial recursive functions within a classical two-valued logic in which all terms denote. We show how a total extension of the partial function introduced by a recursive declaration may be axiomatized within a classical logic, and illustrate by an example the kind of reasoning that our scheme supports. By presenting a naive set-theoretic semantics, we show that the system we propose is logically consistent. Our work is motivated largely by the pragmatic issues arising from mechanical theorem proving – we discuss some of the practical benefits and limitations of our scheme for mechanical verification of software and hardware systems.
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References
Barringer, H., Cheng, J. H., and Jones, C. B.: A logic covering undefinedness in program proofs, Acta Informatica 21(1984), 251–269.
Boyer, R. S. and Moore, J S.: A Computational Logic, ACM Monograph Series, Academic Press, 1979.
Cheng, J. H. and Jones, C. B.: On the usability of logics which handle partial functions, in C. Morgan and J. Woodcock (eds), Proceedings of the Third Refinement Workshop, Springer-Verlag, 1990.
Finn, S. and Fourman, M. P.: Logic Manual for the Lambda System, Abstract Hardware Limited, version 3.2, 1990.
Finn, S. and Fourman, M. P.: Logic Manual for the Lambda System, Abstract Hardware Limited, version 4.2.1, 1992.
Farmer, W. M.: A partial functions version of Church’s simple theory of types, J. Symbolic Logic 55(3) (1990), 1269–1291.
Gordon, M. J. C.: HOL: A proof generating system for higher-order logic, in G. Birtwistle and P. A. Subrahmanyam (eds), VLSI Specification, Verification and Synthesis,Kluwer Academic Publishers, 1987, pp. 73–128.
Gordon, M., Milner, R., and Wadsworth, C.: Edinburgh LCF, Lecture Notes in Computer Science 78, Springer-Verlag, 1979.
Jones, C. B.: Systematic Software Development Using VDM, Prentice Hall International, 1986.
Jones, C. B. et al. (eds), Mural: A Formal Development Support System, Springer-Verlag, 1991.
Luo, Z. and Pollack, R.: LEGO Proof Development System: User’s Manual, LFCS Report Series ECS-LFCS-92-211, Department of Computer Science, University of Edinburgh, 1992.
Owre, S. et al.: Formal verification for fault-tolerant architectures: Prolegomena to the design of PVS, IEEE Transactions on Software Engineering 21(2) (1995), 107–125.
Rushby, J., von Henke, F., and Owre, S.: An Introduction to Formal Specification and Verification using EHDM, Technical Report SRI-CSL-91-02, SRI International, 1991.
Scott, D. S.: Identity and existence in intuitionistic logic, in Lecture Notes in Mathematics, Vol. 735, Springer-Verlag, 1979, pp. 661–696.
Spivey, J. M.: Understanding Z–A Specification Language and Its Formal Semantics,Cam-bridge Tracts in Computer Science 3, CUP, 1988.
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Finn, S., Fourman, M.P. & Longley, J. Partial Functions in a Total Setting. Journal of Automated Reasoning 18, 85–104 (1997). https://doi.org/10.1023/A:1005702928286
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DOI: https://doi.org/10.1023/A:1005702928286