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**Published in:** LIPIcs, Volume 188, 26th International Conference on Types for Proofs and Programs (TYPES 2020)

The approach to proof search dubbed "coinductive proof search", and previously developed by the authors for implicational intuitionistic logic, is in this paper extended to LJP, a focused sequent-calculus presentation of polarized intuitionistic logic, including an array of positive and negative connectives. As before, this includes developing a coinductive description of the search space generated by a sequent, an equivalent inductive syntax describing the same space, and decision procedures for inhabitation problems in the form of predicates defined by recursion on the inductive syntax. We prove the decidability of existence of focused inhabitants, and of finiteness of the number of focused inhabitants for polarized intuitionistic logic, by means of such recursive procedures. Moreover, the polarized logic can be used as a platform from which proof search for other logics is understood. We illustrate the technique with LJT, a focused sequent calculus for full intuitionistic propositional logic (including disjunction). For that, we have to work out the "negative translation" of LJT into LJP (that sees all intuitionistic types as negative types), and verify that the translation gives a faithful representation of proof search in LJT as proof search in the polarized logic. We therefore inherit decidability of both problems studied for LJP and thus get new proofs of these results for LJT.

José Espírito Santo, Ralph Matthes, and Luís Pinto. Coinductive Proof Search for Polarized Logic with Applications to Full Intuitionistic Propositional Logic. In 26th International Conference on Types for Proofs and Programs (TYPES 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 188, pp. 4:1-4:24, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{espiritosanto_et_al:LIPIcs.TYPES.2020.4, author = {Esp{\'\i}rito Santo, Jos\'{e} and Matthes, Ralph and Pinto, Lu{\'\i}s}, title = {{Coinductive Proof Search for Polarized Logic with Applications to Full Intuitionistic Propositional Logic}}, booktitle = {26th International Conference on Types for Proofs and Programs (TYPES 2020)}, pages = {4:1--4:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-182-5}, ISSN = {1868-8969}, year = {2021}, volume = {188}, editor = {de'Liguoro, Ugo and Berardi, Stefano and Altenkirch, Thorsten}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2020.4}, URN = {urn:nbn:de:0030-drops-138837}, doi = {10.4230/LIPIcs.TYPES.2020.4}, annote = {Keywords: Inhabitation problems, Coinduction, Lambda-calculus, Polarized logic} }

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Complete Volume

**Published in:** LIPIcs, Volume 130, 24th International Conference on Types for Proofs and Programs (TYPES 2018)

LIPIcs, Volume 130, TYPES'18, Complete Volume

Peter Dybjer, José Espírito Santo, and Luís Pinto. LIPIcs, Volume 130, TYPES'18, Complete Volume. In 24th International Conference on Types for Proofs and Programs (TYPES 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 130, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@Proceedings{dybjer_et_al:LIPIcs.TYPES.2018, title = {{LIPIcs, Volume 130, TYPES'18, Complete Volume}}, booktitle = {24th International Conference on Types for Proofs and Programs (TYPES 2018)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-106-1}, ISSN = {1868-8969}, year = {2019}, volume = {130}, editor = {Dybjer, Peter and Esp{\'\i}rito Santo, Jos\'{e} and Pinto, Lu{\'\i}s}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2018}, URN = {urn:nbn:de:0030-drops-114507}, doi = {10.4230/LIPIcs.TYPES.2018}, annote = {Keywords: Theory of computation,Type theory; Constructive mathematics; Logic and verification; Program verification, Software and its engineering} }

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Front Matter

**Published in:** LIPIcs, Volume 130, 24th International Conference on Types for Proofs and Programs (TYPES 2018)

Front Matter, Table of Contents, Preface, Conference Organization

Peter Dybjer, José Espírito Santo, and Luís Pinto. Front Matter, Table of Contents, Preface, Conference Organization. In 24th International Conference on Types for Proofs and Programs (TYPES 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 130, pp. 0:i-0:x, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dybjer_et_al:LIPIcs.TYPES.2018.0, author = {Dybjer, Peter and Esp{\'\i}rito Santo, Jos\'{e} and Pinto, Lu{\'\i}s}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {24th International Conference on Types for Proofs and Programs (TYPES 2018)}, pages = {0:i--0:x}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-106-1}, ISSN = {1868-8969}, year = {2019}, volume = {130}, editor = {Dybjer, Peter and Esp{\'\i}rito Santo, Jos\'{e} and Pinto, Lu{\'\i}s}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2018.0}, URN = {urn:nbn:de:0030-drops-114045}, doi = {10.4230/LIPIcs.TYPES.2018.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)

We study the computational interpretation of the two standard modal embeddings, usually named after Girard and Gödel, of intuitionistic logic into IS4. As source system we take either the call-by-name (cbn) or the call-by-value (cbv) lambda-calculus with simple types. The target system can be taken to be the, arguably, simplest fragment of IS4, here recast as a very simple lambda-calculus equipped with an indeterminate lax monoidal comonad. A slight refinement of the target and of the embeddings shows that: the target is a calculus indifferent to the calling paradigms cbn/cbv, obeying a new paradigm that we baptize call-by-box (cbb), and enjoying standardization; and that Girard’s (resp. Gödel’s) embbedding is a translation of cbn (resp. cbv) lambda-calculus into this calculus, using a compilation technique we call protecting-by-a-box, enjoying the preservation and reflection properties known for cps translations - but in a stronger form that allows the extraction of standardization for cbn or cbv as consequence of standardization for cbb. The modal target and embeddings achieve thus an unification of call-by-name and call-by-value as call-by-box.

José Espírito Santo, Luís Pinto, and Tarmo Uustalu. Modal Embeddings and Calling Paradigms. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 18:1-18:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{espiritosanto_et_al:LIPIcs.FSCD.2019.18, author = {Esp{\'\i}rito Santo, Jos\'{e} and Pinto, Lu{\'\i}s and Uustalu, Tarmo}, title = {{Modal Embeddings and Calling Paradigms}}, booktitle = {4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)}, pages = {18:1--18:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-107-8}, ISSN = {1868-8969}, year = {2019}, volume = {131}, editor = {Geuvers, Herman}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.18}, URN = {urn:nbn:de:0030-drops-105256}, doi = {10.4230/LIPIcs.FSCD.2019.18}, annote = {Keywords: intuitionistic S4, call-by-name, call-by-value, comonadic lambda-calculus, standardization, indifference property} }

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**Published in:** LIPIcs, Volume 97, 22nd International Conference on Types for Proofs and Programs (TYPES 2016)

This paper gives a comprehensive and coherent view on permutability in the intuitionistic sequent calculus with cuts. Specifically we show that, once permutability is packaged into appropriate global reduction procedures, it organizes the internal structure of the system and determines fragments with computational interest, both for the computation-as-proof-normalization and the computation-as-proof-search paradigms. The vehicle of the study is a lambda-calculus of multiary proof terms with generalized application, previously developed by the authors (the paper argues this system represents the simplest fragment of ordinary sequent calculus that does not fall into mere natural deduction). We start by adapting to our setting the concept of normal proof, developed by Mints, Dyckhoff, and Pinto, and by defining natural proofs, so that a proof is normal iff it is natural and cut-free. Natural proofs form a subsystem with a transparent Curry-Howard interpretation (a kind of formal vector notation for lambda-terms with vectors consisting of lists of lists of arguments), while searching for normal proofs corresponds to a slight relaxation of focusing (in the sense of LJT). Next, we define a process of permutative conversion to natural form, and show that its combination with cut elimination gives a concept of normalization for the sequent calculus. We derive a systematic picture of the full system comprehending a rich set of reduction procedures (cut elimination, flattening, permutative conversion, normalization, focalization), organizing the relevant subsystems and the important subclasses of cut-free, normal, and focused proofs.

José Espírito Santo, Maria João Frade, and Luís Pinto. Permutability in Proof Terms for Intuitionistic Sequent Calculus with Cuts. In 22nd International Conference on Types for Proofs and Programs (TYPES 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 97, pp. 10:1-10:27, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{espiritosanto_et_al:LIPIcs.TYPES.2016.10, author = {Esp{\'\i}rito Santo, Jos\'{e} and Frade, Maria Jo\~{a}o and Pinto, Lu{\'\i}s}, title = {{Permutability in Proof Terms for Intuitionistic Sequent Calculus with Cuts}}, booktitle = {22nd International Conference on Types for Proofs and Programs (TYPES 2016)}, pages = {10:1--10:27}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-065-1}, ISSN = {1868-8969}, year = {2018}, volume = {97}, editor = {Ghilezan, Silvia and Geuvers, Herman and Ivetic, Jelena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2016.10}, URN = {urn:nbn:de:0030-drops-98523}, doi = {10.4230/LIPIcs.TYPES.2016.10}, annote = {Keywords: sequent calculus, permutative conversion, Curry-Howard isomorphism, vector of arguments, generalized application, normal proof, natural proof, cut eli} }

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