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A focused linear logical framework and its application to metatheory of object logics

Published online by Cambridge University Press:  15 November 2021

Amy Felty
Affiliation:
University of Ottawa, Ottawa, Canada
Carlos Olarte*
Affiliation:
LIPN, CNRS UMR 7030, Université Sorbonne Paris Nord, Villetaneuse, France and ECT, Universidade Federal do Rio Grande do Norte, Natal, Brazil
Bruno Xavier
Affiliation:
DIMAp, Universidade Federal do Rio Grande do Norte, Natal, Brazil
*
*Corresponding author. Email: [email protected]

Abstract

Linear logic (LL) has been used as a foundation (and inspiration) for the development of programming languages, logical frameworks, and models for concurrency. LL’s cut-elimination and the completeness of focusing are two of its fundamental properties that have been exploited in such applications. This paper formalizes the proof of cut-elimination for focused LL. For that, we propose a set of five cut-rules that allows us to prove cut-elimination directly on the focused system. We also encode the inference rules of other logics as LL theories and formalize the necessary conditions for those logics to have cut-elimination. We then obtain, for free, cut-elimination for first-order classical, intuitionistic, and variants of LL. We also use the LL metatheory to formalize the relative completeness of natural deduction and sequent calculus in first-order minimal logic. Hence, we propose a framework that can be used to formalize fundamental properties of logical systems specified as LL theories.

Type
Paper
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Ambler, S., Crole, R. L. and Momigliano, A. (2002). Combining higher order abstract syntax with tactical theorem proving and (co)induction. In: Carreño, V., Muñoz, C. A. and Tahar, S. (eds.), 15th International Conference, TPHOLs 2002, vol. 2410. Lecture Notes in Computer Science. Springer, 13–30.CrossRefGoogle Scholar
Andreoli, J.-M. (1992). Logic programming with focusing proofs in linear logic. Journal of Logic and Computation 2 (3).CrossRefGoogle Scholar
Bertot, Y. and Castéran, P. (2004). Interactive Theorem Proving and Program Development - Coq’Art: The Calculus of Inductive Constructions. Texts in Theoretical Computer Science. An EATCS Series, Springer.Google Scholar
Caires, L., Pfenning, F. and Toninho, B. (2016). Linear logic propositions as session types. Mathematical Structures in Computer Science 26 (3) 367423.CrossRefGoogle Scholar
Cervesato, I. and Pfenning, F. (2002). A linear logical framework. Information & Computation 179 (1) 1975.CrossRefGoogle Scholar
Chaudhuri, K., Despeyroux, J., Olarte, C. and Pimentel, E. (2019a). Hybrid linear logic, revisited. Mathematical Structures in Computer Science 29 (8) 11511176.CrossRefGoogle Scholar
Chaudhuri, K., Lima, L. and Reis, G. (2019b). Formalized meta-theory of sequent calculi for linear logics. Theoretical Computer Science 781 2438.Google Scholar
Chlipala, A. (2008). Parametric higher-order abstract syntax for mechanized semantics. In: Hook, J. and Thiemann, P., (eds.), Proc. of ICFP, pp. 143–156. ACM.CrossRefGoogle Scholar
Crole, R. L. 2011. The representational adequacy of Hybrid. Mathematical Structures in Computer Science, 21(3):585646.CrossRefGoogle Scholar
Dawson, J. E. and Goré, R. (2010). Generic methods for formalising sequent calculi applied to provability logic. In: Logic for Programming, Artificial Intelligence, and Reasoning - 17th International Conference, LPAR-17.Google Scholar
Despeyroux, J. and Chaudhuri, K. (2014). A hybrid linear logic for constrained transition systems. In: Post-Proc. of TYPES 2013, vol. 26. Leibniz Intl. Procs. in Informatics. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 150–168.Google Scholar
Despeyroux, J., Felty, A. P. and Hirschowitz, A. (1995). Higher-order abstract syntax in Coq. In: Typed Lambda Calculi and Applications, Second International Conference on Typed Lambda Calculi and Applications, TLCA.Google Scholar
Despeyroux, J., Felty, A. P., Liò, P. and Olarte, C. (2018). A logical framework for modelling breast cancer progression. In: Chaves, M. and Martins, M. A. (eds.), MLCSB 2018, vol. 11415. LNCS. Springer, 121–141.Google Scholar
Felty, A. and Momigliano, A. (2009). Reasoning with hypothetical judgments and open terms in Hybrid. In: Principles and Practives of Declarative Programming (PPDP). ACM, 83–92.CrossRefGoogle Scholar
Felty, A. P. and Momigliano, A. (2012). Hybrid - A definitional two-level approach to reasoning with higher-order abstract syntax. Journal of Automated Reasoning 48 (1) 43105.CrossRefGoogle Scholar
Gentzen, G. (1969). Investigations into logical deductions. In: The Collected Papers of Gerhard Gentzen. North-Holland, 68–131.Google Scholar
Girard, J.-Y. (1987). Linear logic. Theoretical Computer Science 50 1102.CrossRefGoogle Scholar
Kalvala, S. and Paiva, V. D. (1995). Mechanizing linear logic in Isabelle. In: In 10th International Congress of Logic, Philosophy and Methodology of Science.Google Scholar
Lellmann, B., Olarte, C. and Pimentel, E. (2017). A uniform framework for substructural logics with modalities. In Eiter, T. and Sands, D. (eds.), LPAR-21, vol. 46. EPiC Series in Computing. EasyChair, 435–455.Google Scholar
Liang, C. and Miller, D. (2009). Focusing and polarization in linear, intuitionistic, and classical logics. Theoretical Computer Science 410 (46) 47474768.CrossRefGoogle Scholar
Mahmoud, M. Y. and Felty, A. P. (2019). Formalization of metatheory of the Quipper quantum programming language in a linear logic. Journal of Automated Reasoning 63 (4) 9671002.CrossRefGoogle Scholar
McDowell, R. and Miller, D. (2002). Reasoning with higher-order abstract syntax in a logical framework. ACM Transactions on Computational Logic 3 (1) 80136.CrossRefGoogle Scholar
Miller, D. and Palamidessi, C. (1999). Foundational aspects of syntax. ACM Computing Surveys 31.CrossRefGoogle Scholar
Miller, D. and Pimentel, E. (2013). A formal framework for specifying sequent calculus proof systems. Theoretical Computer Science 474 98116.CrossRefGoogle Scholar
Miller, D. and Saurin, A. (2007). From proofs to focused proofs: A modular proof of focalization in linear logic. In: Duparc, J. and Henzinger, T. A. (eds.), CSL, vol. 4646. LNCS. Springer, 405–419.CrossRefGoogle Scholar
Nigam, V. and Miller, D. (2010). A framework for proof systems. Journal of Automated Reasoning 45 (2) 157188.CrossRefGoogle Scholar
Nigam, V., Olarte, C. and Pimentel, E. (2017). On subexponentials, focusing and modalities in concurrent systems. Theoretical Computer Science 693 3558.CrossRefGoogle Scholar
Nigam, V., Pimentel, E. and Reis, G. (2016). An extended framework for specifying and reasoning about proof systems. J. Log. Comput., 26(2):539576.CrossRefGoogle Scholar
Olarte, C., Pimentel, E. and Rueda, C. (2018). A concurrent constraint programming interpretation of access permissions. Theory and Practice of Logic Programming 18 (2) 252295.CrossRefGoogle Scholar
Olarte, C., Pimentel, E. and Xavier, B. (2020). A fresh view of linear logic as a logical framework. Electronic Notes in Theoretical Computer Science 351 143165.CrossRefGoogle Scholar
Pfenning, F. (2000). Structural cut elimination. Information and Computation 157(1–2) 84141.CrossRefGoogle Scholar
Pfenning, F. and Elliott, C. (1988). Higher-order Abstract Syntax. In: ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI).Google Scholar
Pimentel, E., Nigam, V. and Neto, J. (2015). Multi-focused proofs with different polarity assignments. In: Benevides, M. R. F. and Thiemann, R. (eds.) Proceedings of LSFA 2015, vol. 323. ENTCS. Elsevier, 163–179.Google Scholar
Power, J. and Webster, C. (1999). Working with linear logic in COQ. In 12th International Conference on Theorem Proving in Higher Order Logics, 1–16.Google Scholar
Reed, J. (2009). A Hybrid Logical Framework. PhD thesis, Carnegie Mellon University.Google Scholar
Simmons, R. J. (2014). Structural focalization. ACM Transactions on Computational Logic 15 (3), 21:121:33.CrossRefGoogle Scholar
Xavier, B., Olarte, C., Reis, G. and Nigam, V. (2017). Mechanizing focused linear logic in coq. In Alves, S. and Wasserman, R. (eds.), LSFA 2017, vol. 338. ENTCS. Elsevier, 219–236.Google Scholar