Reflecting BDDs in Coq
We describe an implementation and a proof of correctness of binary decision diagrams (BDDs), completely formalized in Coq. This allows us to run BDD-based algorithms inside Coq and paves the way for a smooth integration of symbolic model checking in the Coq proof assistant by using reflection. It also gives us, by Coq’s extraction mechanism, certified BDD algorithms implemented in Caml. We also implement and prove correct a garbage collector for our implementation of BDDs inside Coq. Our experiments show that this approach works in practice, and is able to solve both relatively hard propositional problems and actual industrial hardware verification tasks.
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- 1.S. F. Allen, R. L. Constable, D. J. Howe, and W. E. Aitken. The semantics of reflected proof. In LICS’90. IEEE Computer Society Press, June 1990.Google Scholar
- 2.B. Barras. Auto-validation d’un système de preuves avec familles inductives. PhD thesis, University Paris VII, Nov. 1999. Code and Coq proofs available at http://www.cl.cam.ac.uk/~bb236/home/coq-in-coq.tar.gz.
- 3.B. Barras, S. Boutin, C. Cornes, J. Courant, Y. Coscoy, D. Delahaye, D. de Rauglaudre, J.-C. Filliâtre, E. Giménez, H. Herbelin, G. Huet, H. Laulhère, C. Muñoz, C. Murthy, C. Parent-Vigouroux, P. Loiseleur, C. Paulin-Mohring, A. Saïbi, and B. Werner. The Coq proof assistant reference manual. Version 6.2.41, available at http://coq.inria.fr/doc/main.html, Jan. 1999.
- 4.D. A. Basin and R. Constable. Metalogical frameworks. In G. Huet and G. Plotkin, editors, Logical Environments, pages 1–29. Cambridge University Press, 1993. Also available as Technical Report MPI-I-92-205.Google Scholar
- 5.J.-P. Billon. Perfect normal forms for discrete functions. Technical Report 87019, Bull S. A. Research Center, June 1987.Google Scholar
- 6.S. Boutin. Using reflection to build efficient and certified decision procedures. In M. Abadi and T. Ito, editors, TACS’97. Springer-Verlag LNCS 1281, 1997.Google Scholar
- 7.R. Boyer and J. S. Moore. Metafunctions: Proving them correct and using them efficiently as new proof procedures. In The Correctness Problem in Computer Science, London, 1981. Academic Press.Google Scholar
- 8.K. S. Brace, R. L. Rudell, and R. E. Bryant. Efficient implementation of a BDD package. In DAC’90. ACM/IEEE, 1990.Google Scholar
- 9.R. E. Bryant. Graph-based algorithms for Boolean function manipulation. IEEE Transactions on Computers, C35(8), Aug. 1986.Google Scholar
- 10.O. Coudert. SIAM: Une Boîte à Outils Pour la Preuve Formelle de Systèmes Séquentiels. PhD thesis, Ecole Nationale Supérieure des Télécommunications, Paris, Oct. 1991.Google Scholar
- 11.H. Goguen, R. Brooksby, and R. Burstall. Memory management: An abstract formulation of incremental tracing. In Types’99. Springer Verlag LNCS, 1999. Submitted.Google Scholar
- 12.G. Gonthier. Verifying the safety of a practical concurrent garbage collector. In CAV’96. Springer Verlag LNCS 1102, July 1996.Google Scholar
- 13.M. Gordon. Programming combinations of deduction and BDD-based symbolic calculation. Technical Report 480, University of Cambridge Computer Laboratory, Dec. 1999.Google Scholar
- 14.M. Gordon and K. F. Larsen. Combining the Hol98 proof assistant with the BuDDy BDD package. Technical Report 481, University of Cambridge Computer Laboratory, Dec. 1999.Google Scholar
- 15.M. J. C. Gordon and T. F. Melham. Introduction to HOL: A theorem proving environment for higher order logic. Cambridge University Press, 1993.Google Scholar
- 16.M. J. C. Gordon, R. Milner, and C. Wadsworth. Edinburgh LCF, a mechanical logic of computation. Report CSR-11-77 (in 2 parts), Dept. of Computer Science, U. Edinburgh, 1977.Google Scholar
- 17.J. Goubault. Standard ML with fast sets and maps. In ML’94. ACM Press, June 1994.Google Scholar
- 18.J. Goubault-Larrecq. Satisfiability of inequality constraints and detection of cycles with negative weight in graphs. Part of the Coq contribs, available at http://pauillac.inria.fr/coq/contribs/graphs.html, 1998.
- 19.J. Harrison. Binary decision diagrams as a HOL derived rule. The Computer Journal, 38, 1995.Google Scholar
- 20.G. Huet, G. Kahn, and C. Paulin-Mohring. The Coq Proof Assistant, A Tutorial. Coq Project, Inria, 1998. Draft, version 6.2.4. Available at http://coq.inria.fr/doc/tutorial.html.
- 21.P. Jackson. Verifying a garbage collection algorithm. In TPHOL’98. Springer Verlag LNCS 1479, 1998.Google Scholar
- 22.R. Kumar, K. Schneider, and T. Kropf. Structuring and automating hardware proofs in a higher-order theorem-proving environment. Journal of Formal System Design, 1993.Google Scholar
- 23.Z. Luo and R. Pollack. The LEGO proof development system: A user’s manual. Technical Report ECS-LFCS-92-211, U. of Edinburgh, May 1992.Google Scholar
- 24.K. L. McMillan. Symbolic Model Checking: An Approach to the State Explosion Problem. Kluwer Academic Publishers, 1993.Google Scholar
- 25.G. Morrisett, M. Felleisen, and R. Harper. Abstract models of memory management. In Functional Programming and Computer Architecture, June 1995.Google Scholar
- 26.NIST. Common criteria for information technology security evaluation. ISO International Standard (IS) 15408, Jan. 2000. Version 2.1.Google Scholar
- 27.S. Owre, J. M. Rushby, and N. Shankar. PVS: A prototype verification system. In D. Kapur, editor, CADE’92. Springer Verlag LNAI 607, June 1992.Google Scholar
- 29.D. M. Russino.. A mechanically verified garbage collector. Formal Aspects of Computing, 6, 1994.Google Scholar
- 30.A. Urquhart. Hard examples for resolution. Journal of the ACM, 34(1), 1987.Google Scholar
- 31.D. Verkest and L. Claesen. Special benchmark session on tautology checking. In L. Claesen, editor, IMEC-IFIP Workshop on Applied Formal Methods for Correct VLSI Design, 1990.Google Scholar
- 32.R. W. Weyhrauch. Prolegomena to a theory of mechanized formal reasoning. Artifical Intelligence, 13(1, 2), 1980.Google Scholar
- 33.S. Yu and Z. Luo. Implementing a model checker for LEGO. In J. Fitzgerald, C. B. Jones, and P. Lucas, editors, FME’97. Springer-Verlag LNCS 1313, Sept. 1997.Google Scholar