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**Published in:** LIPIcs, Volume 239, 27th International Conference on Types for Proofs and Programs (TYPES 2021)

We show that there are connections between principal type schemata, cut-free λ-nets, and normal forms of the λ-calculus, and hence there are correspondences between the normalisation algorithms of the above structures, i.e. unification of principal types, cut-elimination of λ-nets, and normalisation of λ-terms. Once the above correspondences have been established, properties of the typing system, such as typability, subject reduction, and inhabitation, can be derived from properties of λ-nets, and vice-versa. We illustrate the above pattern on a specific type assignment system, we study principal types for this system, and we show that they correspond to λ-nets with a non-standard notion of cut-elimination. Properties of the type system are then derived from results on λ-nets.

Pietro Di Gianantonio and Marina Lenisa. Principal Types as Lambda Nets. In 27th International Conference on Types for Proofs and Programs (TYPES 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 239, pp. 5:1-5:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{digianantonio_et_al:LIPIcs.TYPES.2021.5, author = {Di Gianantonio, Pietro and Lenisa, Marina}, title = {{Principal Types as Lambda Nets}}, booktitle = {27th International Conference on Types for Proofs and Programs (TYPES 2021)}, pages = {5:1--5:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-254-9}, ISSN = {1868-8969}, year = {2022}, volume = {239}, editor = {Basold, Henning and Cockx, Jesper and Ghilezan, Silvia}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2021.5}, URN = {urn:nbn:de:0030-drops-167744}, doi = {10.4230/LIPIcs.TYPES.2021.5}, annote = {Keywords: Lambda calculus, Principal types, Linear logic, Lambda nets, Normalization, Cut elimination} }

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

We explore the possibility of extending Mardare et al.’s quantitative algebras to the structures which naturally emerge from Combinatory Logic and the λ-calculus. First of all, we show that the framework is indeed applicable to those structures, and give soundness and completeness results. Then, we prove some negative results clearly delineating to which extent categories of metric spaces can be models of such theories. We conclude by giving several examples of non-trivial higher-order quantitative algebras.

Ugo Dal Lago, Furio Honsell, Marina Lenisa, and Paolo Pistone. On Quantitative Algebraic Higher-Order Theories. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 4:1-4:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dallago_et_al:LIPIcs.FSCD.2022.4, author = {Dal Lago, Ugo and Honsell, Furio and Lenisa, Marina and Pistone, Paolo}, title = {{On Quantitative Algebraic Higher-Order Theories}}, booktitle = {7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)}, pages = {4:1--4:18}, 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.2022.4}, URN = {urn:nbn:de:0030-drops-162851}, doi = {10.4230/LIPIcs.FSCD.2022.4}, annote = {Keywords: Quantitative Algebras, Lambda Calculus, Combinatory Logic, Metric Spaces} }

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

We present the web portal Λ-symsym, available at http://158.110.146.197:31780/automata/, for experimenting with game semantics of λ^!-calculus, and its normalizing elementary sub-calculus, the λ^{EAL}-calculus. The λ^!-calculus is a generalization of the λ-calculus obtained by introducing a modal operator !, giving rise to a pattern β-reduction. Its sub-calculus corresponds to an applicatively closed class of terms normalizing in an elementary number of steps, in which all elementary functions can be encoded. The game model which we consider is the Geometry of Interaction model I introduced by Abramsky to study reversible computations, consisting of partial involutions over a very simple language of moves.
Given a λ^!- or a λ^{EAL}-term, M, Λ-symsym provides:
- an abstraction algorithm A^!, for compiling M into a term, A^!(M), of the linear combinatory logic CL^{!}, or the normalizing combinatory logic CL^{EAL};
- an interpretation algorithm [[ ]]^I yielding a specification of the partial involution [[A^!(M)]]^I in the model I;
- an algorithm, I2T, for synthesizing from [[A^!(M)]]^I a type, I2T([[A^!(M)]]^I), in a multimodal, intersection type assignment discipline, ⊢_!.
- an algorithm, T2I, for synthesizing a specification of a partial involution from a type in ⊢_!, which is an inverse to the former. We conjecture that ⊢_! M : I2T([[A^!(M)]]^I). Λ-symsym permits to investigate experimentally the fine structure of I, and hence the game semantics of the λ^!- and λ^{EAL}-calculi. For instance, we can easily verify that the model I is a λ^!-algebra in the case of strictly linear λ-terms, by checking all the necessary equations, and find counterexamples in the general case.
We make this tool available for readers interested to play with games (-semantics). The paper builds on earlier work by the authors, the type system being an improvement.

Furio Honsell, Marina Lenisa, and Ivan Scagnetto. Λ-Symsym: An Interactive Tool for Playing with Involutions and Types. In 26th International Conference on Types for Proofs and Programs (TYPES 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 188, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{honsell_et_al:LIPIcs.TYPES.2020.7, author = {Honsell, Furio and Lenisa, Marina and Scagnetto, Ivan}, title = {{\Lambda-Symsym: An Interactive Tool for Playing with Involutions and Types}}, booktitle = {26th International Conference on Types for Proofs and Programs (TYPES 2020)}, pages = {7:1--7:18}, 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.2020.7}, URN = {urn:nbn:de:0030-drops-138867}, doi = {10.4230/LIPIcs.TYPES.2020.7}, annote = {Keywords: game semantics, lambda calculus, involutions, linear logic, implicit computational complexity} }

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

Abramsky’s affine combinatory algebras are models of affine combinatory logic, which refines standard combinatory logic in the direction of Linear Logic. Abramsky introduced various universal models of computation based on affine combinatory algebras, consisting of partial involutions over a suitable formal language {of moves}, in order to discuss reversible computation in a Geometry of Interaction setting. We investigate partial involutions from the point of view of the model theory of lambda!-calculus. The latter is a refinement of the standard lambda-calculus, corresponding to affine combinatory logic. We introduce intersection type systems for the lambda!-calculus, by extending standard intersection types with a !_u-operator. These induce affine combinatory algebras, and, via suitable quotients, models of the lambda!-calculus. In particular, we introduce an intersection type system for assigning principal types to lambda!-terms, and we state a correspondence between the partial involution interpreting a combinator and the principal type of the corresponding lambda!-term. This analogy allows for explaining as unification between principal types the somewhat awkward linear application of involutions arising from Geometry of Interaction.

Alberto Ciaffaglione, Pietro Di Gianantonio, Furio Honsell, Marina Lenisa, and Ivan Scagnetto. lambda!-calculus, Intersection Types, and Involutions. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ciaffaglione_et_al:LIPIcs.FSCD.2019.15, author = {Ciaffaglione, Alberto and Di Gianantonio, Pietro and Honsell, Furio and Lenisa, Marina and Scagnetto, Ivan}, title = {{lambda!-calculus, Intersection Types, and Involutions}}, booktitle = {4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)}, pages = {15:1--15:16}, 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.2019.15}, URN = {urn:nbn:de:0030-drops-105228}, doi = {10.4230/LIPIcs.FSCD.2019.15}, annote = {Keywords: Affine Combinatory Algebra, Affine Lambda-calculus, Intersection Types, Geometry of Interaction} }

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**Published in:** LIPIcs, Volume 23, Computer Science Logic 2013 (CSL 2013)

The aim of this work is to correlate two different approaches to the semantics of programming languages: game semantics and intersection type assignment systems (ITAS). Namely, we present an ITAS that provides the description of the semantic interpretation of a typed lambda calculus in a game model based on innocent strategies. Compared to the traditional ITAS used to describe the semantic interpretation in domain theoretic models, the ITAS presented in this paper has two main differences: the introduction of a notion of labelling on moves, and the omission of several rules, i.e. the subtyping rules and some structural rules.

Pietro Di Gianantonio and Marina Lenisa. Innocent Game Semantics via Intersection Type Assignment Systems. In Computer Science Logic 2013 (CSL 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 23, pp. 231-247, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{digianantonio_et_al:LIPIcs.CSL.2013.231, author = {Di Gianantonio, Pietro and Lenisa, Marina}, title = {{Innocent Game Semantics via Intersection Type Assignment Systems}}, booktitle = {Computer Science Logic 2013 (CSL 2013)}, pages = {231--247}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-60-6}, ISSN = {1868-8969}, year = {2013}, volume = {23}, editor = {Ronchi Della Rocca, Simona}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2013.231}, URN = {urn:nbn:de:0030-drops-42005}, doi = {10.4230/LIPIcs.CSL.2013.231}, annote = {Keywords: Game Semantics, Intersection Type Assignment System, Lambda Calculus} }

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