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A general proof certification framework for modal logic

  • TOMER LIBAL (a1) and MARCO VOLPE (a2)

Abstract

One of the main issues in proof certification is that different theorem provers, even when designed for the same logic, tend to use different proof formalisms and produce outputs in different formats. The project ProofCert promotes the usage of a common specification language and of a small and trusted kernel in order to check proofs coming from different sources and for different logics. By relying on that idea and by using a classical focused sequent calculus as a kernel, we propose here a general framework for checking modal proofs. We present the implementation of the framework in a Prolog-like language and show how it is possible to specialize it in a simple and modular way in order to cover different proof formalisms, such as labelled systems, tableaux, sequent calculi and nested sequent calculi. We illustrate the method for the logic K by providing several examples and discuss how to further extend the approach.

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Andreoli, J. M. (1992). Logic programming with focusing proofs in linear logic. Journal of Logic and Computation 2 (3) 297347.
Avron, A. (1996). The method of hypersequents in the proof theory of propositional non-classical logics. In: Logic: From Foundations to Applications, European Logic Colloquium, Oxford University Press, 132.
Beckert, B. and Gor, R. (1997). Free-variable tableaux for propositional modal logics. Studia Logica 69 (1) 5996.
Benzmüller, C. and Woltzenlogel Paleo, B. (2015). Interacting with modal logics in the coq proof assistant. In: Proceedings of the 10th International Computer Science Symposium in Russia, Lecture Notes in Computer Science, vol. 9139, Springer, 398–411.
Blackburn, P. and Van Benthem, J. (2007). Modal logic: A semantic perspective. In: Wolter, F., Blackburn, P. and van Benthem, J. (eds.) Handbook of Modal Logic, Elsevier, 1–82.
Blanchette, J. C. and Paulson, L. C. (2018). ‘Hammering Away’. A Users Guide to Sledgehammer for Isabelle. Available at: https://isabelle.in.tum.de/dist/doc/sledgehammer.pdf
Brünnler, K. (2009). Deep sequent systems for modal logic. Archive for Mathematical Logic 48 (6) 551577.
Chaudhuri, K., Marin, S. and Straßburger, L. (2016). Focused and synthetic nested sequents. In: Jacobs, B. and Löding, C. (eds.) Foundations of Software Science and Computation Structures, Lecture Notes in Computer Science, vol. 9634, Springer, 390407.
Chihani, Z., Libal, T. and Reis, G. (2015). The proof certifier checkers. In: Automated Reasoning with Analytic Tableaux and Related Methods - 24th International Conference, Lecture Notes in Computer Science 9323, 201210.
Chihani, Z., Miller, D. and Renaud, F. (2017). A semantic framework for proof evidence. Journal of Automated Reasoning 59 (3) 287330.
de Bruijn, N.G. (1970). The mathematical language AUTOMATH, its usage, and some of its extensions. In: Proceedings of the Symposium on Automatic Demonstration. Lecture Notes in Mathematics 125, Springer, 29–61.
Doligez, D., Kriener, J., Lamport, L., Libal, T. and Merz, S. (2014). Coalescing: Syntactic abstraction for reasoning in first-order modal logics. In: Proceedings of “Automated Reasoning in Quantified Non-Classical Logics”, EPiC Series in Computing, vol. 33, 1–16.
Fitting, M. (1972). Tableau methods of proof for modal logics. Notre Dame Journal of Formal Logic 13 (2) 237247.
Fitting, M. (2007). Modal proof theory. In: Blackburn, P., van Benthem, J. and Wolter, F. (eds.) Handbook of Modal Logic, Elsevier, 85138.
Fitting, M. (2012). Prefixed tableaus and nested sequents. Annals of Pure and Applied Logic 163 (3) 291313.
Gabbay, D. (1996). Labelled Deductive Systems, Clarendon Press.
Goré, R. and Ramanayake, R. (2012). Labelled tree sequents, tree hypersequents and nested (deep) sequents. In: Proceedings of the 9th conference on “Advances in Modal Logic”, College Publications, 279–299.
Indrzejczak, A. (2010). Natural Deduction, Hybrid Systems and Modal Logics, Springer.
Kashima, R. (1994). Cut-free sequent calculi for some tense logics. Studia Logica 53 (1) 119136.
Lellmann, B. (2015). Linear nested sequents, 2-sequents and hypersequents. In: Proceedings of the 24th International Conference on Automated Reasoning with Analytic Tableaux and Related Methods, Lecture Notes in Computer Science, vol. 9323, Springer, 135–150.
Lellmann, B. and Pimentel, E. (2015). Proof search in nested sequent calculi. In: Proceedings of the 20th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning, Lecture Notes in Computer Science, vol. 9450, Springer, 558–574.
Liang, C. and Miller, D. (2009). Focusing and polarization in linear, intuitionistic, and classical logics. Theoretical Computer Science 410 (46) 47474768.
Libal, T. and Volpe, M. (2016). Certification of prefixed tableau proofs for modal logic. In: Proceedings of the 7th International Symposium on Games, Automata, Logics and Formal Verification, EPTCS Series, vol. 226, 257–271.
Marin, S., Miller, D. and Volpe, M. (2016). A focused framework for emulating modal proof systems. In: Proceedings of the 11th Conference on ‘Advances in Modal Logic’, College Publications, 469–488.
Miller, D. (2011). Proofcert: Broad spectrum proof certificates. An ERC Advanced Grant funded for the five years 2012–2016. Technical description, available online at: http://www.lix.polytechnique.fr/Labo/Dale.Miller/ProofCert/ProofCert.pdf
Miller, D. and Nadathur, G. (2012). Programming With Higher-Order Logic, Cambridge University Press.
Miller, D. and Volpe, M. (2015). Focused labeled proof systems for modal logic. In: Proceedings of the 20th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning, Lecture Notes in Computer Science, vol. 9450, Springer, 266–280.
Negri, S. (2005). Proof analysis in modal logic. Journal of Philosophical Logic 34 (5–6) 507544.
Poggiolesi, F. (2011). Gentzen Calculi for Modal Propositional Logic, Springer.
Stewart, C. and Stouppa, P. (2004). A systematic proof theory for several modal logics. In: Proceedings of the 5th Conference on ‘Advances in Modal logic’, King's College Publications, 309–333.
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Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
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