Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-12T08:58:23.890Z Has data issue: false hasContentIssue false
  • English
  • Français

A general proof certification framework for modal logic

Published online by Cambridge University Press:  26 March 2019

TOMER LIBAL  and
MARCO VOLPE
TOMER LIBAL
Affiliation:
The American University of Paris, Paris, France Email: tlibal@aup.edu
MARCO VOLPE
Affiliation:
Fortiss GmbH, Munich, Germany Email: volpe@fortiss.org
Get access Rights & Permissions [Opens in a new window]

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.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 29 , Special Issue 8: A special issue on structural proof theory, automated reasoning and computation in celebration of Dale Miller’s 60th birthday , September 2019 , pp. 1344 - 1378
Copyright
© Cambridge University Press 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andreoli, J. M. (1992). Logic programming with focusing proofs in linear logic. Journal of Logic and Computation 2 (3) 297–347.CrossRefGoogle Scholar
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, 1–32.Google Scholar
Beckert, B. and Gor, R. (1997). Free-variable tableaux for propositional modal logics. Studia Logica 69 (1) 59–96.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.pdfGoogle Scholar
Brünnler, K. (2009). Deep sequent systems for modal logic. Archive for Mathematical Logic 48 (6) 551–577.CrossRefGoogle Scholar
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, 390–407.CrossRefGoogle Scholar
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, 201–210.CrossRefGoogle Scholar
Chihani, Z., Miller, D. and Renaud, F. (2017). A semantic framework for proof evidence. Journal of Automated Reasoning 59 (3) 287–330.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.Google Scholar
Fitting, M. (1972). Tableau methods of proof for modal logics. Notre Dame Journal of Formal Logic 13 (2) 237–247.CrossRefGoogle Scholar
Fitting, M. (2007). Modal proof theory. In: Blackburn, P., van Benthem, J. and Wolter, F. (eds.) Handbook of Modal Logic, Elsevier, 85–138.CrossRefGoogle Scholar
Fitting, M. (2012). Prefixed tableaus and nested sequents. Annals of Pure and Applied Logic 163 (3) 291–313.CrossRefGoogle Scholar
Gabbay, D. (1996). Labelled Deductive Systems, Clarendon Press.Google Scholar
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.Google Scholar
Indrzejczak, A. (2010). Natural Deduction, Hybrid Systems and Modal Logics, Springer.CrossRefGoogle Scholar
Kashima, R. (1994). Cut-free sequent calculi for some tense logics. Studia Logica 53 (1) 119–136.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Liang, C. and Miller, D. (2009). Focusing and polarization in linear, intuitionistic, and classical logics. Theoretical Computer Science 410 (46) 4747–4768.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.Google Scholar
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.pdfGoogle Scholar
Miller, D. and Nadathur, G. (2012). Programming With Higher-Order Logic, Cambridge University Press.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Negri, S. (2005). Proof analysis in modal logic. Journal of Philosophical Logic 34 (5–6) 507–544.CrossRefGoogle Scholar
Poggiolesi, F. (2011). Gentzen Calculi for Modal Propositional Logic, Springer.CrossRefGoogle Scholar
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.Google Scholar