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**Published in:** LIPIcs, Volume 263, 37th European Conference on Object-Oriented Programming (ECOOP 2023)

Resource-aware type systems statically approximate not only the expected result type of a program, but also the way external resources are used, e.g., how many times the value of a variable is needed. We extend the type system of Featherweight Java to be resource-aware, parametrically on an arbitrary grade algebra modeling a specific usage of resources. We prove that this type system is sound with respect to a resource-aware version of reduction, that is, a well-typed program has a reduction sequence which does not get stuck due to resource consumption. Moreover, we show that the available grades can be heterogeneous, that is, obtained by combining grades of different kinds, via a minimal collection of homomorphisms from one kind to another. Finally, we show how grade algebras and homomorphisms can be specified as Java classes, so that grade annotations in types can be written in the language itself.

Riccardo Bianchini, Francesco Dagnino, Paola Giannini, and Elena Zucca. Multi-Graded Featherweight Java. In 37th European Conference on Object-Oriented Programming (ECOOP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 263, pp. 3:1-3:27, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bianchini_et_al:LIPIcs.ECOOP.2023.3, author = {Bianchini, Riccardo and Dagnino, Francesco and Giannini, Paola and Zucca, Elena}, title = {{Multi-Graded Featherweight Java}}, booktitle = {37th European Conference on Object-Oriented Programming (ECOOP 2023)}, pages = {3:1--3:27}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-281-5}, ISSN = {1868-8969}, year = {2023}, volume = {263}, editor = {Ali, Karim and Salvaneschi, Guido}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ECOOP.2023.3}, URN = {urn:nbn:de:0030-drops-181960}, doi = {10.4230/LIPIcs.ECOOP.2023.3}, annote = {Keywords: Graded modal types, Java} }

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

Taking a quotient roughly means changing the notion of equality on a given object, set or type. In a quantitative setting, equality naturally generalises to a distance, measuring how much elements are similar instead of just stating their equivalence. Hence, quotients can be understood quantitatively as a change of distance. Quotients are crucial in many constructions both in mathematics and computer science and have been widely studied using categorical tools. Among them, Lawvere’s doctrines stand out, providing a fairly simple functorial framework capable to unify many notions of quotient and related constructions. However, abstracting usual predicate logics, they cannot easily deal with quantitative settings. In this paper, we show how, combining doctrines and the calculus of relations, one can unify quantitative and usual quotients in a common picture. More in detail, we introduce relational doctrines as a functorial description of (the core of) the calculus of relations. Then, we define quotients and a universal construction adding them to any relational doctrine, generalising the quotient completion of existential elementary doctrine and also recovering many quantitative examples. This construction deals with an intensional notion of quotient and breaks extensional equality of morphisms. Then, we describe another construction forcing extensionality, showing how it abstracts several notions of separation in metric and topological structures.

Francesco Dagnino and Fabio Pasquali. Quotients and Extensionality in Relational Doctrines. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 25:1-25:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dagnino_et_al:LIPIcs.FSCD.2023.25, author = {Dagnino, Francesco and Pasquali, Fabio}, title = {{Quotients and Extensionality in Relational Doctrines}}, booktitle = {8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)}, pages = {25:1--25:23}, 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.2023.25}, URN = {urn:nbn:de:0030-drops-180090}, doi = {10.4230/LIPIcs.FSCD.2023.25}, annote = {Keywords: Quantitative Reasoning, Calculus of Relations, Hyperdoctrines, Metric Spaces} }

Document

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

Logical relations built on top of an operational semantics are one of the most successful proof methods in programming language semantics. In recent years, more and more expressive notions of operationally-based logical relations have been designed and applied to specific families of languages. However, a unifying abstract framework for operationally-based logical relations is still missing. We show how fibrations can provide a uniform treatment of operational logical relations, using as reference example a λ-calculus with generic effects endowed with a novel, abstract operational semantics defined on a large class of categories. Moreover, this abstract perspective allows us to give a solid mathematical ground also to differential logical relations - a recently introduced notion of higher-order distance between programs - both pure and effectful, bringing them back to a common picture with traditional ones.

Francesco Dagnino and Francesco Gavazzo. A Fibrational Tale of Operational Logical Relations. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 3:1-3:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dagnino_et_al:LIPIcs.FSCD.2022.3, author = {Dagnino, Francesco and Gavazzo, Francesco}, title = {{A Fibrational Tale of Operational Logical Relations}}, booktitle = {7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)}, pages = {3:1--3:21}, 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.2022.3}, URN = {urn:nbn:de:0030-drops-162840}, doi = {10.4230/LIPIcs.FSCD.2022.3}, annote = {Keywords: logical relations, operational semantics, fibrations, generic effects, program distance} }

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**Published in:** LIPIcs, Volume 222, 36th European Conference on Object-Oriented Programming (ECOOP 2022)

There exists a broad family of multiparty sessions in which the progress of one session participant is not unconditional, but depends on the choices performed by other participants. These sessions fall outside the scope of currently available session type systems that guarantee progress. In this work we propose the first type system ensuring that well-typed multiparty sessions, including those exhibiting the aforementioned dependencies, fairly terminate. Fair termination is termination under a fairness assumption that disregards those interactions deemed unfair and therefore unrealistic. Fair termination, combined with the usual safety properties ensured within sessions, not only is desirable per se, but it entails progress and enables a compositional form of static analysis such that the well-typed composition of fairly terminating sessions results in a fairly terminating program.

Luca Ciccone, Francesco Dagnino, and Luca Padovani. Fair Termination of Multiparty Sessions. In 36th European Conference on Object-Oriented Programming (ECOOP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 222, pp. 26:1-26:26, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ciccone_et_al:LIPIcs.ECOOP.2022.26, author = {Ciccone, Luca and Dagnino, Francesco and Padovani, Luca}, title = {{Fair Termination of Multiparty Sessions}}, booktitle = {36th European Conference on Object-Oriented Programming (ECOOP 2022)}, pages = {26:1--26:26}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-225-9}, ISSN = {1868-8969}, year = {2022}, volume = {222}, editor = {Ali, Karim and Vitek, Jan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ECOOP.2022.26}, URN = {urn:nbn:de:0030-drops-162544}, doi = {10.4230/LIPIcs.ECOOP.2022.26}, annote = {Keywords: Multiparty sessions, fair termination, fair subtyping, deadlock freedom} }

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

We provide an Agda library for inference systems, also supporting their recent generalization allowing flexible coinduction, that is, interpretations which are neither inductive, nor purely coinductive. A specific inference system can be obtained as an instance by writing a set of meta-rules, in an Agda format which closely resembles the usual one. In this way, the user gets for free the related properties, notably the inductive and coinductive intepretation and the corresponding proof principles. Moreover, a significant modularity is achieved. Indeed, rather than being defined from scratch and with a built-in interpretation, an inference system can also be obtained by composition operators, such as union and restriction to a smaller universe, and its semantics can be modularly chosen as well. In particular, flexible coinduction is obtained by composing in a certain way the interpretations of two inference systems. We illustrate the use of the library by several examples. The most significant one is a big-step semantics for the λ-calculus, where flexible coinduction allows to obtain a special result (∞) for all and only the diverging computations, and the proof of equivalence with small-step semantics is carried out by relying on the proof principles offered by the library.

Luca Ciccone, Francesco Dagnino, and Elena Zucca. Flexible Coinduction in Agda. In 12th International Conference on Interactive Theorem Proving (ITP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 193, pp. 13:1-13:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ciccone_et_al:LIPIcs.ITP.2021.13, author = {Ciccone, Luca and Dagnino, Francesco and Zucca, Elena}, title = {{Flexible Coinduction in Agda}}, booktitle = {12th International Conference on Interactive Theorem Proving (ITP 2021)}, pages = {13:1--13:19}, 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.2021.13}, URN = {urn:nbn:de:0030-drops-139083}, doi = {10.4230/LIPIcs.ITP.2021.13}, annote = {Keywords: inference systems, induction, coinduction} }

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**Published in:** LIPIcs, Volume 166, 34th European Conference on Object-Oriented Programming (ECOOP 2020)

The aim of the paper is to provide solid foundations for a programming paradigm natively supporting the creation and manipulation of cyclic data structures. To this end, we describe coFJ, a Java-like calculus where objects can be infinite and methods are equipped with a codefinition (an alternative body). We provide an abstract semantics of the calculus based on the framework of inference systems with corules. In coFJ with this semantics, FJ recursive methods on finite objects can be extended to infinite objects as well, and behave as desired by the programmer, by specifying a codefinition. We also describe an operational semantics which can be directly implemented in a programming language, and prove the soundness of such semantics with respect to the abstract one.

Davide Ancona, Pietro Barbieri, Francesco Dagnino, and Elena Zucca. Sound Regular Corecursion in coFJ. In 34th European Conference on Object-Oriented Programming (ECOOP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 166, pp. 1:1-1:28, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ancona_et_al:LIPIcs.ECOOP.2020.1, author = {Ancona, Davide and Barbieri, Pietro and Dagnino, Francesco and Zucca, Elena}, title = {{Sound Regular Corecursion in coFJ}}, booktitle = {34th European Conference on Object-Oriented Programming (ECOOP 2020)}, pages = {1:1--1:28}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-154-2}, ISSN = {1868-8969}, year = {2020}, volume = {166}, editor = {Hirschfeld, Robert and Pape, Tobias}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ECOOP.2020.1}, URN = {urn:nbn:de:0030-drops-131582}, doi = {10.4230/LIPIcs.ECOOP.2020.1}, annote = {Keywords: Operational semantics, coinduction, programming paradigms, regular terms} }

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SCICO Journal-first

**Published in:** LIPIcs, Volume 166, 34th European Conference on Object-Oriented Programming (ECOOP 2020)

The known is finite, the unknown infinite - Thomas Henry Huxley
The behaviour of programs can be described by the final results of computations, and/or their interactions with the context, also seen as observations. For instance, a function call can terminate and return a value, as well as have output effects during its execution.
Here, we deal with semantic definitions covering both results and observations. Often, such definitions are provided for finite computations only. Notably, in big-step style, infinite computations are simply not modelled, hence diverging and stuck terms are not distinguished. This becomes even more unsatisfactory if we have observations, since a non-terminating program may have significant infinite behaviour.
Recently, examples of big-step semantics modeling divergence have been provided [Davide Ancona et al., 2017; Davide Ancona et al., 2018] by means of generalized inference systems [Davide Ancona et al., 2017; Francesco Dagnino, 2019], which allow corules to control coinduction. Indeed, modeling infinite behaviour by a purely coinductive interpretation of big-step rules would lead to spurious results [Xavier Leroy and Hervé Grall, 2009] and undetermined observation, whereas, by adding appropriate corules, we can correctly get divergence (∞) as the only result, and a uniquely determined observation. This approach has been adopted in [Davide Ancona et al., 2017; Davide Ancona et al., 2018] to design big-step definitions including infinite behaviour for lambda-calculus and a simple imperative Java-like language. However, in such works the designer of the semantics is in charge of finding the appropriate corules, and this is a non-trivial task.
In this paper, we show a general construction that extends a given big-step semantics, modeling finite computations, to include infinite behaviour as well, notably by generating appropriate corules. The construction consists of two steps:
1) Starting from a monoid O modeling finite observations (e.g., finite traces), we construct an ω-monoid ⟨O, O_∞⟩ also modeling infinite observations (e.g., infinite traces). The latter structure is a variation of the notion of ω-semigroup [Dominique Perrin and Jean-Eric Pin, 2004], including a mixed product composing a finite with a possibly infinite observation, and an infinite product mapping an infinite sequence of finite observations into a single one (possibly infinite).
2) Starting from an inference system defining a big-step judgment c⇒⟨r, o⟩, with c denoting a configuration, r ∈ R a result, and o ∈ O a finite observation, we construct an inference system with corules defining an extended big-step judgment c⇒c ⇒ ⟨r_∞, o_∞⟩ with r_∞ ∈ R_∞ = R+{∞}, and o_∞ ∈ O_∞ a "possibly infinite" observation. The construction generates additional rules for propagating divergence, and corules for introducing divergence in a controlled way.
The exact corules added in the construction depend on the type of observations that one starts with. To show the effectiveness of our approach, we provide several instances of the framework, with different kinds of (finite) observations.
Finally, we prove a correctness result for the construction. To this end, we assume the original big-step semantics to be equivalent to (finite sequences of steps in) a reference small-step semantics, and we show that, by applying the construction, we obtain an extended big-step semantics which is still equivalent to the small-step semantics, where we consider possibly infinite sequences of steps.} As hypotheses, rather than {just} equivalence in the finite case {(which would be not enough)}, we assume a set of equivalence conditions between individual big-step rules and the small-step relation. This proof of equivalence holds for deterministic semantics; issues arising in the non-deterministic case and a possible solution are sketched in the conclusion of the full paper.

Davide Ancona, Francesco Dagnino, Jurriaan Rot, and Elena Zucca. A Big Step from Finite to Infinite Computations (SCICO Journal-first). In 34th European Conference on Object-Oriented Programming (ECOOP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 166, pp. 32:1-32:2, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ancona_et_al:LIPIcs.ECOOP.2020.32, author = {Ancona, Davide and Dagnino, Francesco and Rot, Jurriaan and Zucca, Elena}, title = {{A Big Step from Finite to Infinite Computations}}, booktitle = {34th European Conference on Object-Oriented Programming (ECOOP 2020)}, pages = {32:1--32:2}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-154-2}, ISSN = {1868-8969}, year = {2020}, volume = {166}, editor = {Hirschfeld, Robert and Pape, Tobias}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ECOOP.2020.32}, URN = {urn:nbn:de:0030-drops-131895}, doi = {10.4230/LIPIcs.ECOOP.2020.32}, annote = {Keywords: Operational semantics, coinduction, infinite behaviour} }

Document

**Published in:** LIPIcs, Volume 109, 32nd European Conference on Object-Oriented Programming (ECOOP 2018)

Generalized inference systems have been recently introduced, and used, among other applications, to define semantic judgments which uniformly model terminating computations and divergence. We show that the approach can be successfully extended to more sophisticated notions of infinite behaviour, that is, to express that a diverging computation produces some possibly infinite result. This also provides a motivation to smoothly extend the theory of generalized inference systems to include, besides coaxioms, also corules, a more general notion for which significant examples were missing until now. We first illustrate the approach on a lambda-calculus with output effects, for which we also provide an alternative semantics based on standard notions, and a complete proof of the equivalence of the two semantics. Then, we consider a more involved example, that is, an imperative Java-like language with I/O primitives.

Davide Ancona, Francesco Dagnino, and Elena Zucca. Modeling Infinite Behaviour by Corules. In 32nd European Conference on Object-Oriented Programming (ECOOP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 109, pp. 21:1-21:31, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ancona_et_al:LIPIcs.ECOOP.2018.21, author = {Ancona, Davide and Dagnino, Francesco and Zucca, Elena}, title = {{Modeling Infinite Behaviour by Corules}}, booktitle = {32nd European Conference on Object-Oriented Programming (ECOOP 2018)}, pages = {21:1--21:31}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-079-8}, ISSN = {1868-8969}, year = {2018}, volume = {109}, editor = {Millstein, Todd}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ECOOP.2018.21}, URN = {urn:nbn:de:0030-drops-92267}, doi = {10.4230/LIPIcs.ECOOP.2018.21}, annote = {Keywords: Operational semantics, coinduction, trace semantics} }

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