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Document

DOI: 10.4230/LIPIcs.CSL.2026.29

Parametric Iteration in Resource Theories

Authors: Alessandro Di Giorgio, Pawel Sobocinski, and Niels Voorneveld

Published in: LIPIcs, Volume 363, 34th EACSL Annual Conference on Computer Science Logic (CSL 2026)

Abstract

Many algorithms are specified with respect to a fixed but unknown parameter. Examples of this are especially common in cryptography, where protocols often feature a security parameter such as the bit length of a secret key. Our aim is to capture this phenomenon in a more abstract setting. We focus on resource theories - general calculi of processes with a string diagrammatic syntax - introducing a general parametric iteration construction. By instantiating this construction within the Markov category of probabilistic Boolean circuits and equipping it with a suitable metric, we are able to capture the notion of negligibility via asymptotic equivalence, in a compositional way. This allows us to use diagrammatic reasoning to prove simple cryptographic theorems - for instance, proving that guessing a randomly generated key has negligible success.

Cite as

Alessandro Di Giorgio, Pawel Sobocinski, and Niels Voorneveld. Parametric Iteration in Resource Theories. In 34th EACSL Annual Conference on Computer Science Logic (CSL 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 363, pp. 29:1-29:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{digiorgio_et_al:LIPIcs.CSL.2026.29,
  author =	{Di Giorgio, Alessandro and Sobocinski, Pawel and Voorneveld, Niels},
  title =	{{Parametric Iteration in Resource Theories}},
  booktitle =	{34th EACSL Annual Conference on Computer Science Logic (CSL 2026)},
  pages =	{29:1--29:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-411-6},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{363},
  editor =	{Guerrini, Stefano and K\"{o}nig, Barbara},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2026.29},
  URN =		{urn:nbn:de:0030-drops-254541},
  doi =		{10.4230/LIPIcs.CSL.2026.29},
  annote =	{Keywords: Markov categories, Cryptography, String diagrams, Asymptotic equivalence}
}

Document

Invited Talk

DOI: 10.4230/LIPIcs.CALCO.2025.1

Effectful Mealy Machines: Coalgebraic and Causal Traces (Invited Talk)

Authors: Filippo Bonchi, Elena Di Lavore, and Mario Román

Published in: LIPIcs, Volume 342, 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025)

Abstract

We introduce effectful Mealy machines - a general notion of Mealy machine with global effects - and give them semantics in terms of both bisimilarity and traces. Bisimilarity of effectful Mealy machines is characterized syntactically, via free uniform feedback. Their traces are given a novel semantic coinductive universe in terms of effectful streams. We prove that this framework generalizes standard causal processes and captures existing flavours of Mealy machine, bisimilarity, and trace. This is an extended abstract for the manuscript Effectful Mealy Machines: Bisimulation and Trace that will appear in the proceedings of LiCS 2025 [Bonchi et al., 2025]; an extended version with proofs is also available (arxiv.org/abs/2410.10627) [Bonchi et al., 2025]. We additionally characterise causal processes as lax-natural transformations.

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Filippo Bonchi, Elena Di Lavore, and Mario Román. Effectful Mealy Machines: Coalgebraic and Causal Traces (Invited Talk). In 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 342, pp. 1:1-1:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bonchi_et_al:LIPIcs.CALCO.2025.1,
  author =	{Bonchi, Filippo and Di Lavore, Elena and Rom\'{a}n, Mario},
  title =	{{Effectful Mealy Machines: Coalgebraic and Causal Traces}},
  booktitle =	{11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025)},
  pages =	{1:1--1:7},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-383-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{342},
  editor =	{C\^{i}rstea, Corina and Knapp, Alexander},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2025.1},
  URN =		{urn:nbn:de:0030-drops-235596},
  doi =		{10.4230/LIPIcs.CALCO.2025.1},
  annote =	{Keywords: Mealy machines, coinduction, copy-discard categories, premonoidal categories}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2025.12

Ohana Trees and Taylor Expansion for the λI-Calculus: No variable gets left behind or forgotten!

Authors: Rémy Cerda, Giulio Manzonetto, and Alexis Saurin

Published in: LIPIcs, Volume 337, 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)

Abstract

Although the λI-calculus is a natural fragment of the λ-calculus, obtained by forbidding the erasure, its equational theories did not receive much attention. The reason is that all proper denotational models studied in the literature equate all non-normalizable λI-terms, whence the associated theory is not very informative. The goal of this paper is to introduce a previously unknown theory of the λI-calculus, induced by a notion of evaluation trees that we call "Ohana trees". The Ohana tree of a λI-term is an annotated version of its Böhm tree, remembering all free variables that are hidden within its meaningless subtrees, or pushed into infinity along its infinite branches. We develop the associated theories of program approximation: the first approach - more classic - is based on finite trees and continuity, the second adapts Ehrhard and Regnier’s Taylor expansion. We then prove a Commutation Theorem stating that the normal form of the Taylor expansion of a λI-term coincides with the Taylor expansion of its Ohana tree. As a corollary, we obtain that the equality induced by Ohana trees is compatible with abstraction and application. We conclude by discussing the cases of Lévy-Longo and Berarducci trees, and generalizations to the full λ-calculus.

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Rémy Cerda, Giulio Manzonetto, and Alexis Saurin. Ohana Trees and Taylor Expansion for the λI-Calculus: No variable gets left behind or forgotten!. In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cerda_et_al:LIPIcs.FSCD.2025.12,
  author =	{Cerda, R\'{e}my and Manzonetto, Giulio and Saurin, Alexis},
  title =	{{Ohana Trees and Taylor Expansion for the \lambdaI-Calculus: No variable gets left behind or forgotten!}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{12:1--12:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-374-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{337},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2025.12},
  URN =		{urn:nbn:de:0030-drops-236277},
  doi =		{10.4230/LIPIcs.FSCD.2025.12},
  annote =	{Keywords: \lambda-calculus, program approximation, Taylor expansion, \lambdaI-calculus, persistent free variables, B\"{o}hm trees, Ohana trees}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2025.15

On the Metric Nature of (Differential) Logical Relations

Authors: Ugo Dal Lago, Naohiko Hoshino, and Paolo Pistone

Published in: LIPIcs, Volume 337, 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)

Abstract

Differential logical relations are a method to measure distances between higher-order programs. They differ from standard methods based on program metrics in that differences between functional programs are themselves functions, relating errors in input with errors in output, this way providing a more fine grained, contextual, information. The aim of this paper is to clarify the metric nature of differential logical relations. While previous work has shown that these do not give rise, in general, to (quasi-)metric spaces nor to partial metric spaces, we show that the distance functions arising from such relations, that we call quasi-quasi-metrics, can be related to both quasi-metrics and partial metrics, the latter being also captured by suitable relational definitions. Moreover, we exploit such connections to deduce some new compositional reasoning principles for program differences.

Cite as

Ugo Dal Lago, Naohiko Hoshino, and Paolo Pistone. On the Metric Nature of (Differential) Logical Relations. In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 15:1-15:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dallago_et_al:LIPIcs.FSCD.2025.15,
  author =	{Dal Lago, Ugo and Hoshino, Naohiko and Pistone, Paolo},
  title =	{{On the Metric Nature of (Differential) Logical Relations}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{15:1--15:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-374-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{337},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2025.15},
  URN =		{urn:nbn:de:0030-drops-236300},
  doi =		{10.4230/LIPIcs.FSCD.2025.15},
  annote =	{Keywords: Differential Logical Relations, Quantales, Quasi-Metrics, Partial Metrics}
}

Document

Invited Talk

DOI: 10.4230/LIPIcs.FSCD.2025.3

Unsolvable Terms in Filter Models (Invited Talk)

Authors: Mariangiola Dezani-Ciancaglini, Paola Giannini, and Furio Honsell

Published in: LIPIcs, Volume 337, 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)

Abstract

Intersection type theories (itt’s) and filter models, i.e. λ-calculus models generated by itt’s, are reviewed in full generality. In this framework, which subsumes most λ-calculus models in the literature based on Scott-continuous functions, we discuss the interpretation of unsolvable terms. We give a necessary, but not sufficient, condition on an itt for the interpretation of some unsolvable term to be non-trivial in the filter model it generates. This result is obtained building on a type theoretic characterisation of the fine structure of unsolvables.

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Mariangiola Dezani-Ciancaglini, Paola Giannini, and Furio Honsell. Unsolvable Terms in Filter Models (Invited Talk). In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 3:1-3:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dezaniciancaglini_et_al:LIPIcs.FSCD.2025.3,
  author =	{Dezani-Ciancaglini, Mariangiola and Giannini, Paola and Honsell, Furio},
  title =	{{Unsolvable Terms in Filter Models}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{3:1--3:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-374-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{337},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2025.3},
  URN =		{urn:nbn:de:0030-drops-236181},
  doi =		{10.4230/LIPIcs.FSCD.2025.3},
  annote =	{Keywords: \lambda-calculus, Intersection Types, Unsolvable Terms, Filter Models}
}

Document

DOI: 10.4230/LIPIcs.STACS.2025.40

Identity-Preserving Lax Extensions and Where to Find Them

Authors: Sergey Goncharov, Dirk Hofmann, Pedro Nora, Lutz Schröder, and Paul Wild

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract

Generic notions of bisimulation for various types of systems (nondeterministic, probabilistic, weighted etc.) rely on identity-preserving (normal) lax extensions of the functor encapsulating the system type, in the paradigm of universal coalgebra. It is known that preservation of weak pullbacks is a sufficient condition for a functor to admit a normal lax extension (the Barr extension, which in fact is then even strict); in the converse direction, nothing is currently known about necessary (weak) pullback preservation conditions for the existence of normal lax extensions. In the present work, we narrow this gap by showing on the one hand that functors admitting a normal lax extension preserve 1/4-iso pullbacks, i.e. pullbacks in which at least one of the projections is an isomorphism. On the other hand, we give sufficient conditions, showing that a functor admits a normal lax extension if it weakly preserves either 1/4-iso pullbacks and 4/4-epi pullbacks (i.e. pullbacks in which all morphisms are epic) or inverse images. We apply these criteria to concrete examples, in particular to functors modelling neighbourhood systems and weighted systems.

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Sergey Goncharov, Dirk Hofmann, Pedro Nora, Lutz Schröder, and Paul Wild. Identity-Preserving Lax Extensions and Where to Find Them. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 40:1-40:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{goncharov_et_al:LIPIcs.STACS.2025.40,
  author =	{Goncharov, Sergey and Hofmann, Dirk and Nora, Pedro and Schr\"{o}der, Lutz and Wild, Paul},
  title =	{{Identity-Preserving Lax Extensions and Where to Find Them}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{40:1--40:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.40},
  URN =		{urn:nbn:de:0030-drops-228665},
  doi =		{10.4230/LIPIcs.STACS.2025.40},
  annote =	{Keywords: (Bi-)simulations, lax extensions, modal logics, coalgebra}
}

Document

DOI: 10.4230/LIPIcs.CSL.2025.28

Strong Induction Is an Up-To Technique

Authors: Filippo Bonchi, Elena Di Lavore, and Anna Ricci

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract

Up-to techniques are enhancements of the coinduction proof principle which, in lattice theoretic terms, is the dual of induction. What is the dual of coinduction up-to? By means of duality, we illustrate a theory of induction up-to and we observe that an elementary proof technique, commonly known as strong induction, is an instance of induction up-to. We also show that, when generalising our theory from lattices to categories, one obtains an enhancement of the induction definition principle known in the literature as comonadic recursion.

Cite as

Filippo Bonchi, Elena Di Lavore, and Anna Ricci. Strong Induction Is an Up-To Technique. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 28:1-28:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bonchi_et_al:LIPIcs.CSL.2025.28,
  author =	{Bonchi, Filippo and Di Lavore, Elena and Ricci, Anna},
  title =	{{Strong Induction Is an Up-To Technique}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{28:1--28:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.28},
  URN =		{urn:nbn:de:0030-drops-227856},
  doi =		{10.4230/LIPIcs.CSL.2025.28},
  annote =	{Keywords: Induction, Coinduction, Up-to Techniques, Induction up-to, Lattices, Algebras}
}

Document

DOI: 10.4230/LIPIcs.CSL.2025.34

The Lambda Calculus Is Quantifiable

Authors: Valentin Maestracci and Paolo Pistone

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract

In this paper we introduce several quantitative methods for the lambda-calculus based on partial metrics, a well-studied variant of standard metric spaces that have been used to metrize non-Hausdorff topologies, like those arising from Scott domains. First, we study quantitative variants, based on program distances, of sensible equational theories for the λ-calculus, like those arising from Böhm trees and from the contextual preorder. Then, we introduce applicative distances capturing higher-order Scott topologies, including reflexive objects like the D_∞ model. Finally, we provide a quantitative insight on the well-known connection between the Böhm tree of a λ-term and its Taylor expansion, by showing that the latter can be presented as an isometric transformation.

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Valentin Maestracci and Paolo Pistone. The Lambda Calculus Is Quantifiable. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 34:1-34:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{maestracci_et_al:LIPIcs.CSL.2025.34,
  author =	{Maestracci, Valentin and Pistone, Paolo},
  title =	{{The Lambda Calculus Is Quantifiable}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{34:1--34:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.34},
  URN =		{urn:nbn:de:0030-drops-227911},
  doi =		{10.4230/LIPIcs.CSL.2025.34},
  annote =	{Keywords: Lambda-calculus, Scott semantics, Partial metric spaces, B\"{o}hm trees, Taylor expansion}
}

Document

DOI: 10.4230/LIPIcs.FSTTCS.2024.26

Two Views on Unification: Terms as Strategies

Authors: Furio Honsell, Marina Lenisa, and Ivan Scagnetto

Published in: LIPIcs, Volume 323, 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)

Abstract

In [Furio Honsell et al., 2024], the authors have shown that linear application in Geometry of Interaction (GoI) models of λ-calculus amounts to resolution between principal types of linear λ-terms. This analogy also works in the reverse direction. Indeed, an alternative definition of unification between algebraic terms can be given by viewing the terms to be unified as strategies, i.e. sets of pairs of occurrences of the same variable, and verifying the termination of the GoI interaction obtained by playing the two strategies. In this paper we prove that such a criterion of unification is equivalent to the standard one. It can be viewed as a local, bottom-up, definition of unification. Dually, it can be understood as the GoI interpretation of unification. The proof requires generalizing earlier work to arbitrary algebraic constructors and allowing for multiple occurrences of the same variable in terms. In particular, we show that two terms σ and τ unify if and only if R(σ) ⊆̂(τ) ;̂ ({R}(σ) ;̂ {R}(τ))^* and (τ) ⊆̂(σ) ;̂ ({R}(τ) ;̂ {R}(σ))^*, where {R}(σ) denotes the set of pairs of paths leading to the same variable in the term σ, ⊆̂ denotes "inclusion up to substitution" and ;̂ denotes "composition up to substitution".

Cite as

Furio Honsell, Marina Lenisa, and Ivan Scagnetto. Two Views on Unification: Terms as Strategies. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{honsell_et_al:LIPIcs.FSTTCS.2024.26,
  author =	{Honsell, Furio and Lenisa, Marina and Scagnetto, Ivan},
  title =	{{Two Views on Unification: Terms as Strategies}},
  booktitle =	{44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-355-3},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{323},
  editor =	{Barman, Siddharth and Lasota, S{\l}awomir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2024.26},
  URN =		{urn:nbn:de:0030-drops-222158},
  doi =		{10.4230/LIPIcs.FSTTCS.2024.26},
  annote =	{Keywords: unification, geometry of interaction, games}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2021.5

Principal Types as Lambda Nets

Authors: Pietro Di Gianantonio and Marina Lenisa

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

Abstract

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.

Cite as

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}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2022.4

On Quantitative Algebraic Higher-Order Theories

Authors: Ugo Dal Lago, Furio Honsell, Marina Lenisa, and Paolo Pistone

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

Abstract

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.

Cite as

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}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2020.7

Λ-Symsym: An Interactive Tool for Playing with Involutions and Types

Authors: Furio Honsell, Marina Lenisa, and Ivan Scagnetto

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

Abstract

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.

Cite as

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}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2019.15

lambda!-calculus, Intersection Types, and Involutions

Authors: Alberto Ciaffaglione, Pietro Di Gianantonio, Furio Honsell, Marina Lenisa, and Ivan Scagnetto

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

Abstract

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.

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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}
}

Document

DOI: 10.4230/LIPIcs.CSL.2013.231

Innocent Game Semantics via Intersection Type Assignment Systems

Authors: Pietro Di Gianantonio and Marina Lenisa

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

Abstract

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.

Cite as

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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