Abstract
Recent developments in the theory of focused proof systems provide flexible means for structuring proofs within the sequent calculus. This structuring is organized around the construction of “macro” level inference rules based on the “micro” inference rules which introduce single logical connectives. After presenting focused proof systems for first-order classical logics (one with and one without fixed points and equality) we illustrate several examples of proof certificates formats that are derived naturally from the structure of such focused proof systems. In principle, a proof certificate contains two parts: the first part describes how macro rules are defined in terms of micro rules and the second part describes a particular proof object using the macro rules. The first part, which is based on the vocabulary of focused proof systems, describes a collection of macro rules that can be used to directly present the structure of proof evidence captured by a particular class of computational logic systems. While such proof certificates can capture a wide variety of proof structures, a proof checker can remain simple since it must only understand the micro-rules and the discipline of focusing. Since proofs and proof certificates are often likely to be large, there must be some flexibility in allowing proof certificates to elide subproofs: as a result, proof checkers will necessarily be required to perform (bounded) proof search in order to reconstruct missing subproofs. Thus, proof checkers will need to do unification and restricted backtracking search.
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References
Andreoli, J.-M.: Logic programming with focusing proofs in linear logic. J. of Logic and Computation 2(3), 297–347 (1992)
Andrews, P.B.: Theorem-proving via general matings. J. ACM 28, 193–214 (1981)
Appel, A.W., Felty, A.P.: Polymorphic lemmas and definitions in λProlog and Twelf. Theory and Practice of Logic Programming 4(1-2), 1–39 (2004)
Baelde, D.: A linear approach to the proof-theory of least and greatest fixed points. PhD thesis, Ecole Polytechnique (December 2008)
Baelde, D.: Least and greatest fixed points in linear logic. Accepted to the ACM Transactions on Computational Logic (September 2010)
Baelde, D., Miller, D., Snow, Z.: Focused Inductive Theorem Proving. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS, vol. 6173, pp. 278–292. Springer, Heidelberg (2010)
Barendregt, H.: Lambda calculus with types. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. 2, pp. 117–309. Oxford University Press (1992)
Barendregt, H., Barendsen, E.: Autarkic computations in formal proofs. J. of Automated Reasoning 28(3), 321–336 (2002)
Boespflug, M.: Conception d’un noyau de vérification de preuves pour le λΠ-calcul modulo. PhD thesis, Ecole Polytechnique (2011)
Dowek, G., Hardin, T., Kirchner, C.: Theorem proving modulo. J. of Automated Reasoning 31(1), 31–72 (2003)
Gentzen, G.: Investigations into logical deductions. In: Szabo, M.E. (ed.) The Collected Papers of Gerhard Gentzen, pp. 68–131. North-Holland, Amsterdam (1969); Translation of articles that appeared in 1934-1935
Liang, C., Miller, D.: Focusing and polarization in linear, intuitionistic, and classical logics. Theoretical Computer Science 410(46), 4747–4768 (2009)
Liang, C., Miller, D.: Kripke semantics and proof systems for combining intuitionistic logic and classical logic (September 2011) (submitted)
Martin-Löf, P.: Constructive mathematics and computer programming. In: Sixth International Congress for Logic, Methodology, and Philosophy of Science, Amsterdam, pp. 153–175. North-Holland (1982)
Miller, D.: Communicating and trusting proofs: The case for broad spectrum proof certificates (June 2011); Available from author’s website
Miller, D., Nadathur, G., Pfenning, F., Scedrov, A.: Uniform proofs as a foundation for logic programming. Annals of Pure and Applied Logic 51, 125–157 (1991)
Miller, D., Nigam, V.: Incorporating Tables into Proofs. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 466–480. Springer, Heidelberg (2007)
Nadathur, G., Mitchell, D.J.: System Description: Teyjus - A Compiler and Abstract Machine Based Implementation of λProlog. In: Ganzinger, H. (ed.) CADE 1999. LNCS (LNAI), vol. 1632, pp. 287–291. Springer, Heidelberg (1999)
Necula, G.C.: Proof-carrying code. In: Conference Record of the 24th Symposium on Principles of Programming Languages 1997, Paris, France, pp. 106–119. ACM Press (1997)
Pratt, V.R.: Every prime has a succinct certificate. SIAM Journal on Computing 4(3), 214–220 (1975)
Shankar, N.: Trust and Automation in Verification Tools. In: Cha, S(S.), Choi, J.-Y., Kim, M., Lee, I., Viswanathan, M. (eds.) ATVA 2008. LNCS, vol. 5311, pp. 4–17. Springer, Heidelberg (2008)
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Miller, D. (2011). A Proposal for Broad Spectrum Proof Certificates. In: Jouannaud, JP., Shao, Z. (eds) Certified Programs and Proofs. CPP 2011. Lecture Notes in Computer Science, vol 7086. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25379-9_6
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DOI: https://doi.org/10.1007/978-3-642-25379-9_6
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