Cited By
View all- Curzi GDas A(2023)Computational expressivity of (circular) proofs with fixed points2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS56636.2023.10175772(1-13)Online publication date: 26-Jun-2023
Canonical proof-objects for coinductive programming: infinets with infinitely many cuts
Article No.: 7, Pages 1 - 15
Abstract
Non-wellfounded and circular proofs have been recognised over the past decade as a valuable tool to study logics expressing (co)inductive properties, e.g. μ-calculi. Such proofs are non-wellfounded sequent derivations together with a global validity condition expressed in terms of progressing threads. While the cut-free fragment of circular proofs is satisfactory, cuts are poorly treated and the non-canonicity of sequent proofs becomes a major issue in the non-wellfounded setting. The present paper develops for (multiplicative linear logic with fixed points) the theory of infinets – proof-nets for non-wellfounded proofs. Our structures handles infinitely many cuts therefore solving a crucial shortcoming of the previous work [19]. We characterise correctness, define a more complete cut-reduction system and proving a cut-elimination theorem. To that end, we also provide an alternate cut reduction for non-wellfounded sequent calculus.
References
[1]
Martín Abadi, Luca Cardelli, Pierre-Louis Curien, and Jean-Jacques Lévy. 1991. Explicit Substitutions. Journal of Functional Programming 1, 4 (1991), 375–416.
[2]
Andreas Abel and Brigitte Pientka. 2013. Wellfounded recursion with copatterns: a unified approach to termination and productivity. In ACM SIGPLAN International Conference on Functional Programming, ICFP’13, Boston, MA, USA - September 25 - 27, 2013, Greg Morrisett and Tarmo Uustalu (Eds.). ACM, 185–196. https://doi.org/10.1145/2500365.2500591
[3]
Bahareh Afshari, Gerhard Jäger, and Graham E. Leigh. 2019. An Infinitary Treatment of Full Mu-Calculus. In Logic, Language, Information, and Computation, Rosalie Iemhoff, Michael Moortgat, and Ruy de Queiroz (Eds.). Springer Berlin Heidelberg, Berlin, Heidelberg, 17–34.
[4]
Jürgen Avenhaus, Ulrich Kühler, Tobias Schmidt-Samoa, and Claus-Peter Wirth. 2003. How to Prove Inductive Theorems? QUODLIBET!. In Automated Deduction - CADE-19, 19th International Conference on Automated Deduction Miami Beach, FL, USA, July 28 - August 2, 2003, Proceedings(Lecture Notes in Computer Science, Vol. 2741). Springer, 328–333. https://doi.org/10.1007/978-3-540-45085-6_29
[5]
David Baelde, Amina Doumane, Denis Kuperberg, and Alexis Saurin. 2020. Bouncing threads for circular and non-wellfounded proofs. (2020). https://arxiv.org/abs/2005.08257.
[6]
David Baelde, Amina Doumane, and Alexis Saurin. 2016. Infinitary Proof Theory: the Multiplicative Additive Case. In 25th EACSL Annual Conference on Computer Science Logic, CSL 2016, August 29 - September 1, 2016, Marseille, France(LIPIcs, Vol. 62). Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 42:1–42:17. http://www.dagstuhl.de/dagpub/978-3-95977-022-4
[7]
Marc Bagnol, Amina Doumane, and Alexis Saurin. 2015. On the Dependencies of Logical Rules. In Foundations of Software Science and Computation Structures - 18th International Conference, FoSSaCS 2015, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2015, London, UK, April 11-18, 2015. Proceedings. 436–450. https://doi.org/10.1007/978-3-662-46678-0_28
[8]
G. Bellin and J. van de Wiele. 1995. Subnets of Proof-Nets in MLL-. In Proceedings of the Workshop on Advances in Linear Logic. Cambridge University Press, USA, 249–270.
[9]
James Brotherston. 2005. Cyclic Proofs for First-Order Logic with Inductive Definitions. In Automated Reasoning with Analytic Tableaux and Related Methods, Bernhard Beckert (Ed.). Springer Berlin Heidelberg, Berlin, Heidelberg, 78–92.
[10]
James Brotherston, Nikos Gorogiannis, and Rasmus Lerchedahl Petersen. 2012. A Generic Cyclic Theorem Prover. In Programming Languages and Systems - 10th Asian Symposium, APLAS 2012, Kyoto, Japan, December 11-13, 2012. Proceedings(Lecture Notes in Computer Science, Vol. 7705). Springer, 350–367. https://doi.org/10.1007/978-3-642-35182-2_25
[11]
James Brotherston and Alex Simpson. 2010. Sequent calculi for induction and infinite descent. Journal of Logic and Computation 21, 6 (10 2010), 1177–1216. https://doi.org/10.1093/logcom/exq052 arXiv:https://academic.oup.com/logcom/article-pdf/21/6/1177/2787531/exq052.pdf
[12]
Alan Bundy. 2001. The Automation of Proof by Mathematical Induction. In Handbook of Automated Reasoning (in 2 volumes). Elsevier and MIT Press, 845–911.
[13]
Alonzo Church and J. B. Rosser. 1936. Some Properties of Conversion. Trans. Amer. Math. Soc. 39, 3 (1936), 472–482. http://www.jstor.org/stable/1989762
[14]
Pierre-Louis Curien. 2006. Introduction to linear logic and ludics, part II., 44 pages.
[15]
Anupam Das. 2019. Structure vs. Invariants in Proofs: project announcement. (2019). Talk at CiSS-19, http://www.cse.chalmers.se/~bahafs/CiSS2019/programme.html.
[16]
Anupam Das, Amina Doumane, and Damien Pous. 2018. Left-Handed Completeness for Kleene algebra, via Cyclic Proofs. In LPAR(EPiC Series in Computing, Vol. 57). EasyChair, 271–289.
[17]
Anupam Das and Damien Pous. 2017. A Cut-Free Cyclic Proof System for Kleene Algebra. In Automated Reasoning with Analytic Tableaux and Related Methods, Renate A. Schmidtand Cláudia Nalon (Eds.). Springer International Publishing, Cham, 261–277.
[18]
Abhishek De, Luc Pellissier, and Alexis Saurin. [n.d.]. Eliminating infinitely many cuts in non-wellfounded MLL proof-nets. ([n. d.]). https://hal.archives-ouvertes.fr/hal-03235591 Available at: https://hal.archives-ouvertes.fr/hal-03235591.
[19]
Abhishek De and Alexis Saurin. 2019. Infinets: The Parallel Syntax for Non-wellfounded Proof-Theory. In TABLEAUX 2019, Serenella Cerrito and Andrei Popescu (Eds.). Springer International Publishing, 297–316. https://doi.org/10.1007/978-3-030-29026-9_17
[20]
Farzaneh Derakhshan and Frank Pfenning. 2019. Circular Proofs as Session-Typed Processes: A Local Validity Condition. arXiv e-prints, Article arXiv:1908.01909 (Aug. 2019), arXiv:1908.01909 pages. arxiv:1908.01909 [cs.LO]
[21]
Simon Docherty and Reuben N. S. Rowe. 2019. A Non-wellfounded, Labelled Proof System for Propositional Dynamic Logic. In Automated Reasoning with Analytic Tableaux and Related Methods, Serenella Cerrito and Andrei Popescu (Eds.). Springer International Publishing, Cham, 335–352.
[22]
Amina Doumane. 2017. Constructive completeness for the linear-time μ-calculus. In LICS. IEEE Computer Society, 1–12.
[23]
Amina Doumane. 2017. On the infinitary proof theory of logics with fixed points. (Théorie de la démonstration infinitaire pour les logiques à points fixes). Ph.D. Dissertation. Paris Diderot University, France. https://tel.archives-ouvertes.fr/tel-01676953
[24]
Thomas Ehrhard and Farzad Jafar-Rahmani. 2021. Categorical models of Linear Logic with fixed points of formulas. In 36th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2021, Rome, Italy, June 29 - July 2, 2021. IEEE, 1–13. https://doi.org/10.1109/LICS52264.2021.9470664
[25]
Jérôme Fortier and Luigi Santocanale. 2013. Cuts for circular proofs: semantics and cut-elimination. In CSL.
[26]
Eduardo Giménez. 1998. Structural Recursive Definitions in Type Theory. In Proceedings 25th Int. Coll. on Automata, Languages and Programming, ICALP’98, Aalborg, Denmark, 13–17 July 1998, K. G. Larsen, S. Skyum, and G. Winskel (Eds.). LNCS, Vol. 1443. Springer-Verlag, Berlin, 397–408.
[27]
Jean-Yves Girard. 1987. Linear Logic. Theor. Comput. Sci. 50, 1 (Jan. 1987), 1–102. https://doi.org/10.1016/0304-3975(87)90045-4
[28]
Jean-Yves Girard. 1996. Proof-nets: The parallel syntax for proof-theory. In Logic and Algebra. Marcel Dekker, 97–124.
[29]
Farzad Jafarrahmani. 2018. Denotational semantics of linear logic with least and greatest fixpoint. Master’s thesis. Université Paris Diderot.
[30]
Ralph Matthes. 1999. Monotone Fixed-Point Types and Strong Normalization. In Computer Science Logic, Georg Gottlob, Etienne Grandjean, and Katrin Seyr (Eds.). Springer Berlin Heidelberg, Berlin, Heidelberg, 298–312.
[31]
Grigori E Mints. 1978. Finite investigations of transfinite derivations. Journal of Soviet Mathematics 10, 4 (1978), 548–596.
[32]
Rémi Nollet, Alexis Saurin, and Christine Tasson. 2018. Local Validity for Circular Proofs in Linear Logic with Fixed Points. In CSL(LIPIcs, Vol. 119). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 35:1–35:23.
[33]
Dag Prawitz. 1965. Natural Deduction: A Proof-Theoretical Study. Dover Publications.
[34]
Martin Protzen. 1994. Lazy generation of induction hypotheses. In Automated Deduction — CADE-12, Alan Bundy (Ed.). Springer Berlin Heidelberg, Berlin, Heidelberg, 42–56.
[35]
Luigi Santocanale. 2002. A Calculus of Circular Proofs and Its Categorical Semantics. In Foundations of Software Science and Computation Structures(Lecture Notes in Computer Science, Vol. 2303), Mogens Nielsen and Uffe Engberg (Eds.). Springer, 357–371.
[36]
Ulrich Schöpp and Alex K. Simpson. 2002. Verifying Temporal Properties Using Explicit Approximants: Completeness for Context-free Processes. In FoSSaCS(Lecture Notes in Computer Science, Vol. 2303). Springer, 372–386.
[37]
Christoph Sprenger and Mads Dam. 2003. On the Structure of Inductive Reasoning: Circular and Tree-Shaped Proofs in the μCalculus. In Foundations of Software Science and Computation Structures, Andrew D. Gordon (Ed.). Springer Berlin Heidelberg, Berlin, Heidelberg, 425–440.
[38]
Sorin Stratulat. 2017. Cyclic Proofs with Ordering Constraints. In Automated Reasoning with Analytic Tableaux and Related Methods - 26th International Conference, TABLEAUX 2017, Brasília, Brazil, September 25-28, 2017, Proceedings. 311–327.
Recommendations
Bouncing Threads for Circular and Non-Wellfounded Proofs: Towards Compositionality with Circular Proofs
LICS '22: Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer ScienceGiven that (co)inductive types are naturally modelled as fixed points, it is unsurprising that fixed-point logics are of interest in the study of programming languages, via the Curry-Howard (or proofs-as-programs) correspondence. This motivates ...
Least and Greatest Fixed Points in Linear Logic
The first-order theory of MALL (multiplicative, additive linear logic) over only equalities is a well-structured but weak logic since it cannot capture unbounded (infinite) behavior. Instead of accounting for unbounded behavior via the addition of the ...
Infinets: The Parallel Syntax for Non-wellfounded Proof-Theory
Automated Reasoning with Analytic Tableaux and Related MethodsAbstractLogics based on the -calculus are used to model inductive and coinductive reasoning and to verify reactive systems. A well-structured proof-theory is needed in order to apply such logics to the study of programming languages with (co)inductive ...
Comments
Information & Contributors
Information
Published In
September 2021
277 pages
Copyright © 2021 ACM.
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 07 October 2021
Check for updates
Author Tags
Qualifiers
- Research-article
- Research
- Refereed limited
Funding Sources
- Marie Sk?odowska-Curie grant
Conference
PPDP 2021
PPDP 2021: 23rd International Symposium on Principles and Practice of Declarative Programming
September 6 - 8, 2021
Tallinn, Estonia
Acceptance Rates
Overall Acceptance Rate 230 of 486 submissions, 47%
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- View Citations1Total Citations
- 46Total Downloads
- Downloads (Last 12 months)2
- Downloads (Last 6 weeks)0
Reflects downloads up to 04 Feb 2025
Other Metrics
Citations
Cited By
View all- Curzi GDas A(2023)Computational expressivity of (circular) proofs with fixed points2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS56636.2023.10175772(1-13)Online publication date: 26-Jun-2023
View Options
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign inFull Access
View options
View or Download as a PDF file.
PDFeReader
View online with eReader.
eReaderHTML Format
View this article in HTML Format.
HTML Format