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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Continuity is a key principle of intuitionistic logic that is generally accepted by constructivists but is inconsistent with classical logic. Most commonly, continuity states that a function from the Baire space to numbers, only needs approximations of the points in the Baire space to compute. More recently, another formulation of the continuity principle was put forward. It states that for any function F from the Baire space to numbers, there exists a (dialogue) tree that contains the values of F at its leaves and such that the modulus of F at each point of the Baire space is given by the length of the corresponding branch in the tree. In this paper we provide the first internalization of this "inductive" continuity principle within a computational setting. Concretely, we present a class of intuitionistic theories that validate this formulation of continuity thanks to computations that construct such dialogue trees internally to the theories using effectful computations. We further demonstrate that this inductive continuity principle implies other forms of continuity principles.

Liron Cohen, Bruno da Rocha Paiva, Vincent Rahli, and Ayberk Tosun. Inductive Continuity via Brouwer Trees. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 37:1-37:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cohen_et_al:LIPIcs.MFCS.2023.37, author = {Cohen, Liron and da Rocha Paiva, Bruno and Rahli, Vincent and Tosun, Ayberk}, title = {{Inductive Continuity via Brouwer Trees}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {37:1--37:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.37}, URN = {urn:nbn:de:0030-drops-185718}, doi = {10.4230/LIPIcs.MFCS.2023.37}, annote = {Keywords: Continuity, Dialogue trees, Stateful computations, Intuitionistic Logic, Extensional Type Theory, Constructive Type Theory, Realizability, Theorem proving, Agda} }

Document

**Published in:** LIPIcs, Volume 252, 31st EACSL Annual Conference on Computer Science Logic (CSL 2023)

The principle of continuity is a seminal property that holds for a number of intuitionistic theories such as System T. Roughly speaking, it states that functions on real numbers only need approximations of these numbers to compute. Generally, continuity principles have been justified using semantical arguments, but it is known that the modulus of continuity of functions can be computed using effectful computations such as exceptions or reference cells. This paper presents a class of intuitionistic theories that features stateful computations, such as reference cells, and shows that these theories can be extended with continuity axioms. The modulus of continuity of the functionals on the Baire space is directly computed using the stateful computations enabled in the theory.

Liron Cohen and Vincent Rahli. Realizing Continuity Using Stateful Computations. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 15:1-15:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cohen_et_al:LIPIcs.CSL.2023.15, author = {Cohen, Liron and Rahli, Vincent}, title = {{Realizing Continuity Using Stateful Computations}}, booktitle = {31st EACSL Annual Conference on Computer Science Logic (CSL 2023)}, pages = {15:1--15:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-264-8}, ISSN = {1868-8969}, year = {2023}, volume = {252}, editor = {Klin, Bartek and Pimentel, Elaine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.15}, URN = {urn:nbn:de:0030-drops-174761}, doi = {10.4230/LIPIcs.CSL.2023.15}, annote = {Keywords: Continuity, Stateful computations, Intuitionism, Extensional Type Theory, Constructive Type Theory, Realizability, Theorem proving, Agda} }

Document

**Published in:** LIPIcs, Volume 228, 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)

Time-progressing expressions, i.e., expressions that compute to different values over time such as Brouwerian choice sequences or reference cells, are a common feature in many frameworks. For type theories to support such elements, they usually employ sheaf models. In this paper, we provide a general framework in the form of an extensional type theory incorporating various time-progressing elements along with a general possible-worlds forcing interpretation parameterized by modalities. The modalities can, in turn, be instantiated with topological spaces of bars, leading to a general sheaf model. This parameterized construction allows us to capture a distinction between theories that are "agnostic", i.e., compatible with classical reasoning in the sense that classical axioms can be validated, and those that are "intuitionistic", i.e., incompatible with classical reasoning in the sense that classical axioms can be proven false. This distinction is made via properties of the modalities selected to model the theory and consequently via the space of bars instantiating the modalities. We further identify a class of time-progressing elements that allows deriving "intuitionistic" theories that include not only choice sequences but also simpler operators, namely reference cells.

Liron Cohen and Vincent Rahli. Constructing Unprejudiced Extensional Type Theories with Choices via Modalities. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 10:1-10:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cohen_et_al:LIPIcs.FSCD.2022.10, author = {Cohen, Liron and Rahli, Vincent}, title = {{Constructing Unprejudiced Extensional Type Theories with Choices via Modalities}}, booktitle = {7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)}, pages = {10:1--10:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-233-4}, ISSN = {1868-8969}, year = {2022}, volume = {228}, editor = {Felty, Amy P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2022.10}, URN = {urn:nbn:de:0030-drops-162917}, doi = {10.4230/LIPIcs.FSCD.2022.10}, annote = {Keywords: Intuitionism, Extensional Type Theory, Constructive Type Theory, Realizability, Choice sequences, References, Classical Logic, Theorem proving, Agda} }

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Complete Volume

**Published in:** LIPIcs, Volume 193, 12th International Conference on Interactive Theorem Proving (ITP 2021)

LIPIcs, Volume 193, ITP 2021, Complete Volume

Liron Cohen and Cezary Kaliszyk. LIPIcs, Volume 193, ITP 2021, Complete Volume. In 12th International Conference on Interactive Theorem Proving (ITP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 193, pp. 1-560, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@Proceedings{cohen_et_al:LIPIcs.ITP.2021, title = {{LIPIcs, Volume 193, ITP 2021, Complete Volume}}, booktitle = {12th International Conference on Interactive Theorem Proving (ITP 2021)}, pages = {1--560}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-188-7}, ISSN = {1868-8969}, year = {2021}, volume = {193}, editor = {Cohen, Liron and Kaliszyk, Cezary}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2021}, URN = {urn:nbn:de:0030-drops-138943}, doi = {10.4230/LIPIcs.ITP.2021}, annote = {Keywords: LIPIcs, Volume 193, ITP 2021, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 193, 12th International Conference on Interactive Theorem Proving (ITP 2021)

Front Matter, Table of Contents, Preface, Conference Organization

Liron Cohen and Cezary Kaliszyk. Front Matter, Table of Contents, Preface, Conference Organization. In 12th International Conference on Interactive Theorem Proving (ITP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 193, pp. 0:i-0:viii, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cohen_et_al:LIPIcs.ITP.2021.0, author = {Cohen, Liron and Kaliszyk, Cezary}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {12th International Conference on Interactive Theorem Proving (ITP 2021)}, pages = {0:i--0:viii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-188-7}, ISSN = {1868-8969}, year = {2021}, volume = {193}, editor = {Cohen, Liron and Kaliszyk, Cezary}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2021.0}, URN = {urn:nbn:de:0030-drops-138955}, doi = {10.4230/LIPIcs.ITP.2021.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

**Published in:** LIPIcs, Volume 183, 29th EACSL Annual Conference on Computer Science Logic (CSL 2021)

One of the differences between Brouwerian intuitionistic logic and classical logic is their treatment of time. In classical logic truth is atemporal, whereas in intuitionistic logic it is time-relative. Thus, in intuitionistic logic it is possible to acquire new knowledge as time progresses, whereas the classical Law of Excluded Middle (LEM) is essentially flattening the notion of time stating that it is possible to decide whether or not some knowledge will ever be acquired. This paper demonstrates that, nonetheless, the two approaches are not necessarily incompatible by introducing an intuitionistic type theory along with a Beth-like model for it that provide some middle ground. On one hand they incorporate a notion of progressing time and include evolving mathematical entities in the form of choice sequences, and on the other hand they are consistent with a variant of the classical LEM. Accordingly, this new type theory provides the basis for a more classically inclined Brouwerian intuitionistic type theory.

Mark Bickford, Liron Cohen, Robert L. Constable, and Vincent Rahli. Open Bar - a Brouwerian Intuitionistic Logic with a Pinch of Excluded Middle. In 29th EACSL Annual Conference on Computer Science Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 183, pp. 11:1-11:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bickford_et_al:LIPIcs.CSL.2021.11, author = {Bickford, Mark and Cohen, Liron and Constable, Robert L. and Rahli, Vincent}, title = {{Open Bar - a Brouwerian Intuitionistic Logic with a Pinch of Excluded Middle}}, booktitle = {29th EACSL Annual Conference on Computer Science Logic (CSL 2021)}, pages = {11:1--11:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-175-7}, ISSN = {1868-8969}, year = {2021}, volume = {183}, editor = {Baier, Christel and Goubault-Larrecq, Jean}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2021.11}, URN = {urn:nbn:de:0030-drops-134455}, doi = {10.4230/LIPIcs.CSL.2021.11}, annote = {Keywords: Intuitionism, Extensional type theory, Constructive Type Theory, Realizability, Choice sequences, Classical Logic, Law of Excluded Middle, Theorem proving, Coq} }

Document

**Published in:** LIPIcs, Volume 119, 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)

Transitive closure logic is a known extension of first-order logic obtained by introducing a transitive closure operator. While other extensions of first-order logic with inductive definitions are a priori parametrized by a set of inductive definitions, the addition of the transitive closure operator uniformly captures all finitary inductive definitions. In this paper we present an infinitary proof system for transitive closure logic which is an infinite descent-style counterpart to the existing (explicit induction) proof system for the logic. We show that, as for similar systems for first-order logic with inductive definitions, our infinitary system is complete for the standard semantics and subsumes the explicit system. Moreover, the uniformity of the transitive closure operator allows semantically meaningful complete restrictions to be defined using simple syntactic criteria. Consequently, the restriction to regular infinitary (i.e. cyclic) proofs provides the basis for an effective system for automating inductive reasoning.

Liron Cohen and Reuben N. S. Rowe. Uniform Inductive Reasoning in Transitive Closure Logic via Infinite Descent. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 17:1-17:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{cohen_et_al:LIPIcs.CSL.2018.17, author = {Cohen, Liron and Rowe, Reuben N. S.}, title = {{Uniform Inductive Reasoning in Transitive Closure Logic via Infinite Descent}}, booktitle = {27th EACSL Annual Conference on Computer Science Logic (CSL 2018)}, pages = {17:1--17:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-088-0}, ISSN = {1868-8969}, year = {2018}, volume = {119}, editor = {Ghica, Dan R. and Jung, Achim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2018.17}, URN = {urn:nbn:de:0030-drops-96841}, doi = {10.4230/LIPIcs.CSL.2018.17}, annote = {Keywords: Induction, Transitive Closure, Infinitary Proof Systems, Cyclic Proof Systems, Soundness, Completeness, Standard Semantics, Henkin Semantics} }

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