article

A Semantic Framework for Proof Evidence

Authors:

Zakaria Chihani,

Fabien RenaudAuthors Info & Claims

Journal of Automated Reasoning, Volume 59, Issue 3

Pages 287 - 330

https://doi.org/10.1007/s10817-016-9380-6

Published: 01 October 2017 Publication History

Abstract

Theorem provers produce evidence of proof in many different formats, such as proof scripts, natural deductions, resolution refutations, Herbrand expansions, and equational rewritings. In implemented provers, numerous variants of such formats are actually used: consider, for example, such variants of or restrictions to resolution refutations as binary resolution, hyper-resolution, ordered-resolution, paramodulation, etc. We propose the foundational proof certificates (FPC) framework for defining the semantics of a broad range of proof evidence. This framework allows both producers of proof certificates and the checkers of those certificates to have a clear formal definition of the semantics of a wide variety of proof evidence. Employing the FPC framework will allow one to separate a proof from its provenance and to allow anyone to construct their own proof checker for a given style of proof evidence. The foundation on which FPC relies is that of proof theory, particularly recent work into focused proof systems: such proof systems provide protocols by which a checker extracts information from the certificate (mediated by the so called clerks and experts) as well as performs various deterministic and non-deterministic computations. While we shall limit ourselves to first-order logic in this paper, we shall not limit ourselves in many other ways. The FPC framework is described for both classical and intuitionistic logics and for proof structures as diverse as resolution refutations, natural deduction, Frege proofs, and equality proofs.

References

[1]

Andreoli, J.-M.: Logic programming with focusing proofs in linear logic. J. Log. Comput. 2(3), 297---347 (1992)

[2]

Armand, M., Faure, G., Grégoire, B., Keller, C., Théry, L., Werner, B.: A modular integration of SAT/SMT solvers to Coq through proof witnesses. In: Jouannaud, J.-P., Shao, Z. (eds.) Certified Programs and Proofs (CPP 2011), LNCS 7086, pp. 135---150 (2011)

Digital Library

[3]

Baelde, D.: Least and greatest fixed points in linear logic. ACM Trans. Comput. Log. 13(1), 1---48 (2012)

Digital Library

[4]

Barendregt, H.: The impact of the lambda calculus in logic and computer science. Bull. Symb. Log. 3(2), 181---215 (1997)

[5]

Barendregt, H., Barendsen, E.: Autarkic computations in formal proofs. J. Autom. Reason. 28(3), 321---336 (2002)

Digital Library

[6]

Barendregt, H.P.: Introduction to generalized type systems. J. Funct. Program. 1(2), 125---154 (1991)

[7]

Bertot, Y., Castéran, P.: Interactive Theorem Proving and Program Development. Coq'Art: The Calculus of Inductive Constructions. Texts in Theoretical Computer Science. Springer, New York (2004)

Digital Library

[8]

Blanco, R., Miller, D.: Proof outlines as proof certificates: a system description. In: Cervesato, I., Schürmann, C. (eds.) Proceedings First International Workshop on Focusing, Suva, Fiji, 23rd November 2015, volume 197 of Electronic Proceedings in Theoretical Computer Science, pp. 7---14. Open Publishing Association (2015)

[9]

Boespflug, M., Carbonneaux, Q., Hermant, O.: The $$\lambda {\varPi }$$¿¿-calculus modulo as a universal proof language. In: Pichardie, D., Weber, T. (eds.) Proceedings of PxTP2012: Proof Exchange for Theorem Proving, pp. 28---43 (2012)

[10]

Böhme, S., Weber, T.: Designing proof formats: a user's perspective. In: Fontaine, P., Stump, A. (eds.) PxTP 2011: First International Workshop on Proof eXchange for Theorem Proving, pp 27---32 (2011)

[11]

Chaudhuri, K.: Classical and intuitionistic subexponential logics are equally expressive. In: Dawar, A., Veith, H. (eds.) CSL 2010: Computer Science Logic. LNCS 6247, Brno, Czech Republic, pp. 185---199. Springer (2010)

Digital Library

[12]

Chaudhuri, K., Hetzl, S., Miller, D.: A multi-focused proof system isomorphic to expansion proofs. J. Log. Comput. 26(2), 577---603 (2016)

[13]

Chaudhuri, K., Miller, D., Saurin, A.: Canonical sequent proofs via multi-focusing. In: Ausiello, G., Karhumäki, J., Mauri, G., Ong, L. (eds.) Fifth International Conference on Theoretical Computer Science, IFIP 273, pp. 383---396. Springer (2008)

[14]

Chaudhuri, K., Pfenning, F., Price, G.: A logical characterization of forward and backward chaining in the inverse method. J. Autom. Reason. 40(2---3), 133---177 (2008)

Digital Library

[15]

Chihani, Z.: Certification of first-order proofs in classical and intuitionistic logics. Ph.D. thesis, Ecole Polytechnique (2015)

[16]

Chihani, Z., Libal, T., Reis, G.: The proof certifier checkers. In: Nivelle, H.D. (ed.) Proceedings of the 24th Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX), LNCS 9323, pp. 201---210. Springer (2015)

Digital Library

[17]

Chihani, Z., Miller, D.: Proof certificates for equality reasoning. In: Benevides, M., Thiemann, R. (eds.) Post-proceedings of LSFA 2015: 10th Workshop on Logical and Semantic Frameworks, with Applications. Natal, Brazil, ENTCS 18612 (2016)

Digital Library

[18]

Chihani, Z., Miller, D., Renaud, F.: Checking foundational proof certificates for first-order logic (extended abstract). In: Blanchette, J.C., Urban, J. (eds.) Third International Workshop on Proof Exchange for Theorem Proving (PxTP 2013), volume 14 of EPiC Series, pp. 58---66. EasyChair (2013)

[19]

Chihani, Z., Miller, D., Renaud, F.: Foundational proof certificates in first-order logic. In: Bonacina, M.P. (ed.) CADE 24: Conference on Automated Deduction 2013, LNAI 7898, pp. 162---177 (2013)

Digital Library

[20]

Chihani, Z., Miller, D., Renaud, F.: Supporting $$\lambda $$¿Prolog code. http://www.lix.polytechnique.fr/Labo/Dale.Miller/papers/fpc-support.tar (2016)

[21]

Chomsky, N.: Three models for the description of language. IRE Trans. Inf. Theory 2(3), 113---124 (1956)

[22]

Church, A.: A formulation of the simple theory of types. J. Symb. Log. 5, 56---68 (1940)

[23]

Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. J. Symb. Log. 44(1), 36---50 (1979)

[24]

Cousineau, D., Dowek, G.: Embedding pure type systems in the lambda-Pi-calculus modulo. In: Rocca, S.R.D. (ed.) Proceedings of the Typed Lambda Calculi and Applications, 8th International Conference, TLCA 2007, Paris, France, 26---28 June 2007, LNCS 4583, pp. 102---117. Springer (2007)

Digital Library

[25]

Danos, V., Joinet, J.-B., Schellinx, H.: LKT and LKQ: sequent calculi for second order logic based upon dual linear decompositions of classical implication. In: Girard, J.-Y., Lafont, Y., Regnier, L. (eds.) Advances in Linear Logic. London Mathematical Society Lecture Note Series, vol. 222, pp. 211---224. Cambridge University Press, Cambridge (1995)

Digital Library

[26]

de Bruijn, N.G.: Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with an application to the Church---Rosser theorem. Indag. Math. 34(5), 381---392 (1972)

[27]

Delande, O., Miller, D., Saurin, A.: Proof and refutation in MALL as a game. Ann. Pure Appl. Log. 161(5), 654---672 (2010)

[28]

Dowek, G.: Skolemization in simple type theory: the logical and the theoretical points of view. In: Reasoning in Simple Type Theory: Festschrift in Honor of Peter B. Andrews on His 70th Birthday, number 17 in Studies in Logic, pp. 244---255. College Publications (2008)

[29]

Dowek, G., Hardin, T., Kirchner, C.: HOL-$$\lambda \sigma $$¿¿ an intentional first-order expression of higher-order logic. Math. Struct. Comput. Sci. 11(1), 1---25 (2001)

Digital Library

[30]

Dowek, G., Hardin, T., Kirchner, C.: Theorem proving modulo. J. Autom. Reason. 31(1), 31---72 (2003)

Digital Library

[31]

Dunchev, C., Guidi, F., Coen, C.S., Tassi, E.: ELPI: fast, embeddable, $$\lambda $$¿Prolog interpreter. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds.) Proceedings of the Logic for Programming, Artificial Intelligence, and Reasoning--20th International Conference, LPAR-20 2015, Suva, Fiji, 24---28 November 2015, pp. 460---468 (2015)

Digital Library

[32]

Dyckhoff, R., Lengrand, S.: Call-by-value $$\lambda $$¿-calculus and LJQ. J. Log. Comput. 17(6), 1109---1134 (2007)

Digital Library

[33]

Felty, A.: Transforming specifications in a dependent-type lambda calculus to specifications in an intuitionistic logic. In: Huet, G., Plotkin, G.D. (eds.) Logical Frameworks. Cambridge University Press, Cambridge (1991)

[34]

Felty, A.: Encoding the calculus of constructions in a higher-order logic. In: Vardi, M. (ed.) 8th Symposium on Logic in Computer Science, pp. 233---244. IEEE (1993)

[35]

Felty, A.: Implementing tactics and tacticals in a higher-order logic programming language. J. Autom. Reason. 11(1), 43---81 (1993)

[36]

Fontaine, P., Marion, J.-Y., Merz, S., Nieto, L.P., Tiu, A.F.: Expressiveness + automation + soundness: towards combining SMT solvers and interactive proof assistants. In: Hermanns, H., Palsberg, J. (eds.) TACAS: Tools and Algorithms for the Construction and Analysis of Systems, 12th International Conference, LNCS 3920, pp. 167---181. Springer (2006)

Digital Library

[37]

Gallier, J.H.: Logic for Computer Science: Foundations of Automatic Theorem Proving. Harper & Row, New York (1986)

Digital Library

[38]

Gelder, A.V.: Producing and verifying extremely large propositional refutations: have your cake and eat it too. Ann. Math. Artif. Intell. 65(4), 329---372 (2012)

Digital Library

[39]

Gentzen, G.: Investigations into logical deduction. In: Szabo, M.E. (ed.) The Collected Papers of Gerhard Gentzen, pp. 68---131. North-Holland, Amsterdam (1935)

[40]

Gentzen, G.: Die widerspruchfreiheit der reinen zahlentheorie. Math. Ann. 112, 493---565 (1936). Reprinted in English translation as "The consistency of Elementary Number Theory" in The collected papers of Gerhard Gentzen, M. E. Szabo, ed

[41]

Girard, J.-Y.: Linear logic. Theoret. Comput. Sci. 50, 1---102 (1987)

Digital Library

[42]

Girard, J.-Y.: A new constructive logic: classical logic. Math. Struct. Comp. Sci. 1, 255---296 (1991)

[43]

Girard, J.-Y., Taylor, P., Lafont, Y.: Proofs and Types. Cambridge University Press, Cambridge (1989)

Digital Library

[44]

Gödel, K.: Zur intuitionistischen arithmetik und zahlentheorie. Ergeb. Eines Math. Kolloqu. 34---38 (1932). English translation in The Undecidable (M. Davis, ed.) 75---81 (1965)

[45]

Gödel, K.: Eine interpretation des intuitionistischen aussagenkalkuls. Ergeb. Eines Math. Kolloqu. 4, 39---40 (1933). Available in "Kurt Gödel: Collected Works. Volume 1" edited by S. Feferman and et al

[46]

Gordon, M.: From LCF to HOL: a short history. In: Plotkin, G.D., Stirling, C., Tofte, M. (eds.) Proof, Language, and Interaction: Essays in Honour of Robin Milner, pp. 169---186. MIT Press, Cambridge (2000)

Digital Library

[47]

Gordon, M.J., Milner, A.J., Wadsworth, C.P.: Edinburgh LCF: A Mechanised Logic of Computation, LNCS 78. Springer, New York (1979)

[48]

Harper, R., Honsell, F., Plotkin, G.: A framework for defining logics. J. ACM 40(1), 143---184 (1993)

Digital Library

[49]

Heath, Q., Miller, D.: A framework for proof certificates in finite state exploration. In: Kaliszyk, C., Paskevich, A. (eds.) Proceedings of the Fourth Workshop on Proof eXchange for Theorem Proving, number 186 in Electronic Proceedings in Theoretical Computer Science, pp. 11---26. Open Publishing Association (2015)

[50]

Herbelin, H.: Séquents qu'on calcule: de l'interprétation du calcul des séquents comme calcul de lambda-termes et comme calcul de stratégies gagnantes. Ph.D. thesis, Université Paris 7 (1995)

[51]

Hodges, W.: Logic and games. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Stanford University, Stanford (2013)

[52]

Honsell, F., Lenisa, M., Liquori, L., Maksimovic, P., Scagnetto, I.: LFP: a logical framework with external predicates. In: Chlipala, A., Schürmann, C. (eds.) LFMTP 2012: Proceedings of the Seventh International Workshop on Logical Frameworks and Meta-Languages, Theory and Practice, pp. 13---22. ACM, New York (2012)

Digital Library

[53]

Howe, J.M.: Proof search issues in some non-classical logics. Ph.D. thesis, University of St Andrews (1998). Available as University of St Andrews Research Report CS/99/1

[54]

Hughes, D.J.D.: Proofs without syntax. Ann. Math. 143(3), 1065---1076 (2006)

[55]

Hurd, J.: The OpenTheory standard theory library. In: Bobaru, M.G., Havelund, K., Holzmann, G.J., Joshi, R. (eds.) The Third International Symposium on NASA Formal Methods, LNCS 6617, pp. 177---191 (2011)

Digital Library

[56]

Johnson, S.C.: Yacc: Yet Another Compiler---Compiler, vol. 32. Bell Laboratories, Murray Hill (1975)

[57]

Kahn, G.: Natural semantics. In: Brandenburg, F.-J., Vidal-Naquet, G., Wirsing, M. (eds.) Proceedings of the Symposium on Theoretical Aspects of Computer Science, LNCS 247, pp. 22---39. Springer (1987)

Digital Library

[58]

Kolmogorov, A .N.: On the principle of the excluded middle. Mat. Sb. 32, 646---667 (1925). English translation by Jean van Heijenoort in From Frege to Gödel

[59]

Laurent, O.: Etude de la polarisation en logique. Ph.D. thesis, Université Aix-Marseille II (2002)

[60]

Liang, C., Miller, D.: Focusing and polarization in linear, intuitionistic, and classical logics. Theoret. Comput. Sci. 410(46), 4747---4768 (2009)

Digital Library

[61]

Liang, C., Miller, D.: A focused approach to combining logics. Ann. Pure Appl. Log. 162(9), 679---697 (2011)

[62]

Lorenzen, P.: Ein dialogisches konstruktivitätskriterium. In: Infinitistic Methods: Proceedings of the Symposium on the Foundations of Mathematics, pp. 193---200. PWN (1961)

[63]

Miller, D.: A compact representation of proofs. Stud. Log. 46(4), 347---370 (1987)

[64]

Miller, D.: Communicating and trusting proofs: the case for broad spectrum proof certificates. In: Schroeder-Heister, P., Hodges, W., Heinzmann, G., Bour, P.E. (eds.) Logic, Methodology, and Philosophy of Science. Proceedings of the Fourteenth International Congress, pp. 323---342. College Publications (2014)

[65]

Miller, D.: Proof checking and logic programming. In: Falaschi, M. (ed.) Logic-Based Program Synthesis and Transformation (LOPSTR), LNCS 9527, pp. 3---17. Springer, New York (2015)

Digital Library

[66]

Miller, D., Nadathur, G.: Programming with Higher-Order Logic. Cambridge University Press, Cambridge (2012)

Digital Library

[67]

Miller, D., Nadathur, G., Pfenning, F., Scedrov, A.: Uniform proofs as a foundation for logic programming. Ann. Pure Appl. Log. 51, 125---157 (1991)

[68]

Miller, D., Saurin, A.: From proofs to focused proofs: a modular proof of focalization in linear logic. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007: Computer Science Logic, LNCS 4646, pp. 405---419. Springer, New York (2007)

Digital Library

[69]

Miller, D., Volpe, M.: Focused labeled proof systems for modal logic. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds.) Logic for Programming, Artificial Intelligence, and Reasoning (LPAR), LNCS 9450, pp. 266---280. Springer (2015)

Digital Library

[70]

Milner, R., Tofte, M., Harper, R., MacQueen, D.: The Definition of Standard ML (Revised). The MIT Press, Cambridge (1997)

Digital Library

[71]

Nadathur, G., Mitchell, D.J.: System description: Teyjus--a compiler and abstract machine based implementation of $$\lambda $$¿Prolog. In: Ganzinger, H. (ed.) 16th Conference on Automated Deduction (CADE), LNAI 1632, pp. 287---291. Springer, Trento (1999)

Digital Library

[72]

Necula, G.C., Rahul, S.P.: Oracle-based checking of untrusted software. In: Hankin, C., Schmidt, D. (eds.) 28th ACM Symposium on Principles of Programming Languages, pp. 142---154 (2001)

Digital Library

[73]

Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL--A Proof Assistant for Higher-Order Logic. LNCS 2283. Springer, New York (2002)

Digital Library

[74]

Paulson, L.C.: The foundation of a generic theorem prover. J. Autom. Reason. 5, 363---397 (1989)

Digital Library

[75]

Pereira, F.C.N., Shieber, S.M.: Prolog and Natural-Language Analysis, vol. 10. CLSI, Stanford (1987)

Digital Library

[76]

Pfenning, F., Schürmann, C.: System description: Twelf--a meta-logical framework for deductive systems. In: Ganzinger, H. (ed.) 16th Conference on Automated Deduction (CADE), LNAI 1632, pp. 202---206. Springer, Trento (1999)

Digital Library

[77]

Plotkin, G.D.: A Structural Approach to Operational Semantics. DAIMI FN-19. Aarhus University, Aarhus (1981)

[78]

Plotkin, G.D.: The origins of structural operational semantics. J. Log. Algebraic Program. 60, 3---15 (2004)

[79]

Prawitz, D.: Natural Deduction. Almqvist & Wiksell, Uppsala (1965)

[80]

Qi, X., Gacek, A., Holte, S., Nadathur, G., Snow, Z.: The Teyjus system--version 2. http://teyjus.cs.umn.edu/ (2015)

[81]

Rabe, F.: The future of logic: foundation-independence. Log. Universalis. 10(1), 1---20 (2016)

[82]

Saillard, R.: Towards explicit rewrite rules in the $$\lambda {\varPi }$$¿¿-calculus modulo. In: Schulz, S., Sutcliffe, G., Konev, B. (eds.) IWIL-10th International Workshop on the Implementation of Logics, (2013)

[83]

Schwichtenberg, H.: Minlog. In: Wiedijk, F. (ed.) The Seventeen Provers of the World, LNCS 3600, pp. 151---157. Springer, New York (2006)

Digital Library

[84]

Shieber, S.M., Schabes, Y., Pereira, F.C.N.: Principles and implementation of deductive parsing. J. Log. Program. 24(1---2), 3---36 (1995)

[85]

Slaney, J.: Solution to a problem of Ono and Komori. J. Philos. Log. 18, 103---111 (1989)

[86]

Snow, Z., Baelde, D., Nadathur, G.: A meta-programming approach to realizing dependently typed logic programming. In: Kutsia, T., Schreiner, W., Fernández, M. (eds.) ACM SIGPLAN Conference on Principles and Practice of Declarative Programming (PPDP), pp. 187---198 (2010)

Digital Library

[87]

Stoy, J.E.: Denotational Semantics: The Scott-Strachey Approach to Programming Language Theory. MIT Press, Cambridge (1977)

Digital Library

[88]

Stump, A.: Proof checking technology for satisfiability modulo theories. Electron. Notes Theor. Comput. Sci. 228, 121---133 (2009)

Digital Library

[89]

Stump, A., Oe, D., Reynolds, A., Hadarean, L., Tinelli, C.: SMT proof checking using a logical framework. Form. Methods Syst. Des. 42(1), 91---118 (2013)

Digital Library

[90]

Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory, 2nd edn. Cambridge University Press, Cambridge (2000)

Digital Library

[91]

Wetzler, N., Heule, M.J.H., Hunt, J.W.A.: DRAT-trim: efficient checking and trimming using expressive clausal proofs. In: Sinz, C., Egly, U. (eds.) Theory and Applications of Satisfiability Testing SAT 2014, LNCS 8561, pp. 422---429. Springer, New York (2014)

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cover image Journal of Automated Reasoning

Journal of Automated Reasoning Volume 59, Issue 3

October 2017

101 pages

ISSN:0168-7433

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Mantovani MMomigliano A(2021)Towards Substructural Property-Based TestingLogic-Based Program Synthesis and Transformation10.1007/978-3-030-98869-2_6(92-112)Online publication date: 7-Sep-2021
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