Abstract
As the systems we have to specify and verify become larger and more complex, there is a mounting need to combine different tools and decision procedures to accomplish large proof tasks. The problem, then, is how to be sure that we can trust heterogeneous proofs produced by different tools based on different formalisms. In this work we focus on certification and synthesis of equational proofs, that are pervasive in most proof tasks and for which many tools are poorly equipped. Fortunately, equational proof engines like ELAN and Maude can perform millions of equational proof steps per second which, if properly certified, can be trusted by other tools. We present a general method to certify and synthesize proofs in membership equational logic, where the synthesis may involve generating full proofs from proof traces modulo combinations of associativity, commutativity, and identity axioms. We propose a simple representation for proof objects and give algorithms that can synthesize space-efficient, machine-checkable proof objects from proof traces.
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References
Barendregt, H., Barendsen, E.: Autarkik computations and formal proofs. Journal of Automated Reasoning 28(3), 321–336 (2002)
Borovanský, P., Kirchner, C., Kirchner, H., Moreau, P.-E.: ELAN from a rewriting logic point of view. Theoretical Computer Science 285, 155–185 (2002)
Bouhoula, A., Jouannaud, J.-P., Meseguer, J.: Specification and proof in membership equational logic. Theoretical Computer Science 236, 35–132 (2000)
Bruni, R., Meseguer, J.: Generalized rewrite theories. Manuscript (January 2003), http://maude.cs.uiuc.edu
Clavel, M., Durán, F., Eker, S., Lincoln, P., Martí-Oliet, N., Meseguer, J., Quesada, J.: Maude: specification and programming in rewriting logic. Theoretical Computer Science 285, 187–243 (2002)
Constable, R.: Implementing Mathematics with the Nuprl Proof Development System. Prentice-Hall, Englewood Cliffs (1987)
Goguen, J., Roşu, G.: Institution morphisms. Formal Aspects of Computing 13(3-5), 274–307 (2002)
Gordon, M., Melham, T. (eds.): Introduction to HOL: A theorem proving environment for higher order logic. Cambridge University Press, Cambridge (1993)
Kaufmann, M., Manolios, P., Moore, J.: Computer-Aided Reasoning: An Approach. Kluwer, Dordrecht (2000)
Martí-Oliet, N., Meseguer, J.: Rewriting logic as a logical and semantic framework. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, 2nd edn., pp. 1–87. Kluwer Academic Publishers, Dordrecht (2002); First published as SRI Tech. Report SRI-CSL-93-05 (August 1993)
Meseguer, J.: General logics. In: Ebbinghaus, H.-D., et al. (eds.) Logic Colloquium 1987, pp. 275–329. North-Holland, Amsterdam (1989)
Meseguer, J.: A logical theory of concurrent objects and its realization in the Maude language. In: Agha, G., Wegner, P., Yonezawa, A. (eds.) Research Directions in Concurrent Object-Oriented Programming, pp. 314–390. MIT Press, Cambridge (1993)
Meseguer, J.: Membership algebra as a logical framework for equational specification. In: Parisi-Presicce, F. (ed.) WADT 1997. LNCS, vol. 1376, pp. 18–61. Springer, Heidelberg (1998)
Meseguer, J., Martí-Oliet, N.: From abstract data types to logical frameworks. In: Reggio, G., Astesiano, E., Tarlecki, A. (eds.) Abstract Data Types 1994 and COMPASS 1994. LNCS, vol. 906, pp. 48–80. Springer, Heidelberg (1995)
Mossakowski, T.: Heterogeneous development graphs and heterogeneous borrowing. In: Nielsen, M., Engberg, U. (eds.) FOSSACS 2002. LNCS, vol. 2303, pp. 326–341. Springer, Heidelberg (2002)
Nguyen, Q., Kirchner, C., Kirchner, H.: External rewriting for skeptical proof assistants. Journal of Automated Reasoning 29(3-4), 309–336 (2002)
Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)
Owre, S., Rajan, S., Rushby, J., Shankar, N., Srivas, M.: PVS: Combining specification, proof checking, and model checking. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 411–414. Springer, Heidelberg (1996)
Roşu, G.: Complete categorical equational deduction. In: Fribourg, L. (ed.) CSL 2001 and EACSL 2001. LNCS, vol. 2142, pp. 528–538. Springer, Heidelberg (2001)
Stehr, M.-O.: Programming, Specification, and Interactive Theorem Proving — Towards a Unified Language based on Equational Logic, Rewriting Logic, and Type Theory. Doctoral Thesis, Universität Hamburg, Fachbereich Informatik, Germany (2002), http://www.sub.uni-hamburg.de/disse/810/
Stehr, M.-O., Naumov, P., Meseguer, J.: The HOL/NuPRl proof translator—A practical approach to formal interoperability. In: Boulton, R.J., Jackson, P.B. (eds.) TPHOLs 2001. LNCS, vol. 2152, pp. 329–345. Springer, Heidelberg (2001)
Tarlecki, A.: Towards heterogeneous specifications. In: Proc. Workshop on Frontiers of Combining Systems FroCoS 1998, Amsterdam, October 1998. Applied Logic Series. Kluwer Academic Publishers, Dordrecht (1998)
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Roşu, G., Eker, S., Lincoln, P., Meseguer, J. (2003). Certifying and Synthesizing Membership Equational Proofs. In: Araki, K., Gnesi, S., Mandrioli, D. (eds) FME 2003: Formal Methods. FME 2003. Lecture Notes in Computer Science, vol 2805. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45236-2_21
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DOI: https://doi.org/10.1007/978-3-540-45236-2_21
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