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**Published in:** LIPIcs, Volume 260, 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)

This paper defines two decreasing measures for terms of the simply typed λ-calculus, called the 𝒲-measure and the 𝒯^{𝐦}-measure. A decreasing measure is a function that maps each typable λ-term to an element of a well-founded ordering, in such a way that contracting any β-redex decreases the value of the function, entailing strong normalization. Both measures are defined constructively, relying on an auxiliary calculus, a non-erasing variant of the λ-calculus. In this system, dubbed the λ^{𝐦}-calculus, each β-step creates a "wrapper" containing a copy of the argument that cannot be erased and cannot interact with the context in any other way. Both measures rely crucially on the observation, known to Turing and Prawitz, that contracting a redex cannot create redexes of higher degree, where the degree of a redex is defined as the height of the type of its λ-abstraction. The 𝒲-measure maps each λ-term to a natural number, and it is obtained by evaluating the term in the λ^{𝐦}-calculus and counting the number of remaining wrappers. The 𝒯^{𝐦}-measure maps each λ-term to a structure of nested multisets, where the nesting depth is proportional to the maximum redex degree.

Pablo Barenbaum and Cristian Sottile. Two Decreasing Measures for Simply Typed λ-Terms. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 11:1-11:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{barenbaum_et_al:LIPIcs.FSCD.2023.11, author = {Barenbaum, Pablo and Sottile, Cristian}, title = {{Two Decreasing Measures for Simply Typed \lambda-Terms}}, booktitle = {8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)}, pages = {11:1--11:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-277-8}, ISSN = {1868-8969}, year = {2023}, volume = {260}, editor = {Gaboardi, Marco and van Raamsdonk, Femke}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2023.11}, URN = {urn:nbn:de:0030-drops-179956}, doi = {10.4230/LIPIcs.FSCD.2023.11}, annote = {Keywords: Lambda Calculus, Rewriting, Termination, Strong Normalization, Simple Types} }

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

Proof terms are syntactic expressions that represent computations in term rewriting. They were introduced by Meseguer and exploited by van Oostrom and de Vrijer to study equivalence of reductions in (left-linear) first-order term rewriting systems. We study the problem of extending the notion of proof term to higher-order rewriting, which generalizes the first-order setting by allowing terms with binders and higher-order substitution. In previous works that devise proof terms for higher-order rewriting, such as Bruggink’s, it has been noted that the challenge lies in reconciling composition of proof terms and higher-order substitution (β-equivalence). This led Bruggink to reject "nested" composition, other than at the outermost level. In this paper, we propose a notion of higher-order proof term we dub rewrites that supports nested composition. We then define two notions of equivalence on rewrites, namely permutation equivalence and projection equivalence, and show that they coincide.

Pablo Barenbaum and Eduardo Bonelli. Reductions in Higher-Order Rewriting and Their Equivalence. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 8:1-8:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{barenbaum_et_al:LIPIcs.CSL.2023.8, author = {Barenbaum, Pablo and Bonelli, Eduardo}, title = {{Reductions in Higher-Order Rewriting and Their Equivalence}}, booktitle = {31st EACSL Annual Conference on Computer Science Logic (CSL 2023)}, pages = {8:1--8:18}, 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.2023.8}, URN = {urn:nbn:de:0030-drops-174694}, doi = {10.4230/LIPIcs.CSL.2023.8}, annote = {Keywords: Term Rewriting, Higher-Order Rewriting, Proof terms, Equivalence of Computations} }

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

The λ^{PRK}-calculus is a typed λ-calculus that exploits the duality between the notions of proof and refutation to provide a computational interpretation for classical propositional logic. In this work, we extend λ^{PRK} to encompass classical second-order logic, by incorporating parametric polymorphism and existential types. The system is shown to enjoy good computational properties, such as type preservation, confluence, and strong normalization, which is established by means of a reducibility argument. We identify a syntactic restriction on proofs that characterizes exactly the intuitionistic fragment of second-order λ^{PRK}, and we study canonicity results.

Pablo Barenbaum and Teodoro Freund. Proofs and Refutations for Intuitionistic and Second-Order Logic. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 9:1-9:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{barenbaum_et_al:LIPIcs.CSL.2023.9, author = {Barenbaum, Pablo and Freund, Teodoro}, title = {{Proofs and Refutations for Intuitionistic and Second-Order Logic}}, booktitle = {31st EACSL Annual Conference on Computer Science Logic (CSL 2023)}, pages = {9:1--9:18}, 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.2023.9}, URN = {urn:nbn:de:0030-drops-174707}, doi = {10.4230/LIPIcs.CSL.2023.9}, annote = {Keywords: lambda-calculus, propositions-as-types, classical logic, proof normalization} }

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**Published in:** LIPIcs, Volume 84, 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017)

We lift the theory of optimal reduction to a decomposition of the lambda calculus known as the Linear Substitution Calculus (LSC). LSC decomposes beta-reduction into finer steps that manipulate substitutions in two distinctive ways: it uses context rules that allow substitutions to act "at a distance" and rewrites modulo a set of equations that allow substitutions to "float" in a term. We propose a notion of redex family obtained by adapting Lévy labels to support these two distinctive features. This is followed by a proof of the finite family developments theorem (FFD). We then apply FFD to prove an optimal reduction theorem for LSC. We also apply FFD to deduce additional novel properties of LSC, namely an algorithm for standardisation by selection and normalisation of a linear call-by-need reduction strategy. All results are proved in the axiomatic setting of Glauert and Khashidashvili's Deterministic Residual Structures.

Pablo Barenbaum and Eduardo Bonelli. Optimality and the Linear Substitution Calculus. In 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 84, pp. 9:1-9:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{barenbaum_et_al:LIPIcs.FSCD.2017.9, author = {Barenbaum, Pablo and Bonelli, Eduardo}, title = {{Optimality and the Linear Substitution Calculus}}, booktitle = {2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017)}, pages = {9:1--9:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-047-7}, ISSN = {1868-8969}, year = {2017}, volume = {84}, editor = {Miller, Dale}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2017.9}, URN = {urn:nbn:de:0030-drops-77307}, doi = {10.4230/LIPIcs.FSCD.2017.9}, annote = {Keywords: Rewriting, Lambda Calculus, Explicit Substitutions, Optimal Reduction} }

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