A timing refinement of intuitionistic proofs and its application to the timing analysis of combinational circuits
Up until now classical logic has been the logic of choice in formal hardware verification. This paper advances the application of intuitionistic logic to the timing analysis of digital circuits. The intuitionistic setting serves two purposes at the same time. The model-theoretic properties are exploited to handle the second-order nature of bounded delays in a purely propositional setting without need to introduce explicit time and temporal operators. The proof theoretic properties are exploited to extract quantitative timing information and to reintroduce explicit time in a convenient and systematic way.
We present a natural Kripke-style semantics for intuitionistic propositional logic, as a special case of a Kripke constraint model for Propositional Lax Logic , in which validity is validity up to stabilization. We show that this semantics is equivalently characterized in terms of stabilization bounds so that implication ⊃ comes out as “boundedly gives rise to.” An intensional semantics for proofs is presented which allows us effectively to compute quantitative stabilization bounds.
We discuss the application of the theory to the timing analysis of combinational circuits. To test our ideas we have implemented an experimental prototype tool and run several simple examples. Proofs are omitted as they appear in an extended technical report .
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- 1.ACM. International Workshop on Timing Issues in the Specification and Synthesis of Digital Systems, Intermar Hotel Malente, Germany, September 1993.Google Scholar
- 2.A. G. Dragalin. Mathematical Intuitionism. Introduction to Proof Theory. American Mathematical Society, 1988.Google Scholar
- 3.R. Dyckhoff. Contraction-free sequent calculi for intuitionistic logic. The Journal of Symbolic Logic, 57(3):795–807, September 1992.Google Scholar
- 4.M. Fairtlough and M. Mendler. An intuitionistic modal logic with applications to the formal verification of hardware. In Proceedings of the 1994 Annual Conference of the European Association for Computer Science Logic, pages 354–368. University of Warsawa, Springer, LNCS 933, 1995.Google Scholar
- 5.Torkel Franzén. Algorithmic aspects of intuitionistic propositional logic. Research Report SICS R87010, Swedish Institute of Computer Science, 1987.Google Scholar
- 6.G. Gentzen. Untersuchungen über das Logische Schließen. Math. Z., 39:176–210, 405–431, 1934–1935.Google Scholar
- 7.J. Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge University Press, 1989.Google Scholar
- 8.C. T. Gray, W. Liu, and R. K. Cavin III. Exact timing analysis considering data dependent delays. .Google Scholar
- 9.D. Gurr. Semantic Frameworks for Complexity. PhD thesis, Edinburgh University, Department of Computer Science, January 1991.Google Scholar
- 10.Lego. The lego proof assistant. http://www.dcs.ed.ac.uk/packages/lego on the World Wide Web, 1995.Google Scholar
- 11.M. Mendler. Constrained proofs: a logic for dealing with behavioural constraints in formal hardware verification. In G. Jones and M. Sheeran, editors, Workshop on Designing Correct Circuits. Springer, 1991.Google Scholar
- 12.M. Mendler. A Modal Logic for Handling Behavioural Constraints in Formal Hardware Verification. PhD thesis, Department of Computer Science, University of Edinburgh, ECS-LFCS-93-255, March 1993.Google Scholar
- 13.M. Mendler. A timing refinement of intuitionistic proofs and its application to the timing analysis of combinational circuits. Technical Report MIP-9518, University of Passau, November 1995.Google Scholar
- 14.M. Mendler and M. Fairtlough. Ternary simulation: A refinement of binary functions or an abstraction of real-time behaviour? Technical Report MIP-9605, University of Passau, March 1996.Google Scholar
- 15.E. Moggi. Computational lambda-calculus and monads. In Proceedings LIGS'89, pages 14–23, June 1989.Google Scholar
- 16.P. Tharau. BinProlog 3.45 User Guide. Departement d'Informatique, Université de Moncton, Canada, June 1995.Google Scholar