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**Published in:** LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

We consider a minimal extension of the language of arithmetic, such that the bounded formulas provably total in a suitably-defined theory à la Buss (expressed in this new language) precisely capture polytime random functions. Then, we provide two new characterizations of the semantic class BPP obtained by internalizing the error-bound check within a logical system: the first relies on measure-sensitive quantifiers, while the second is based on standard first-order quantification. This leads us to introduce a family of effectively enumerable subclasses of BPP, called BPP_T and consisting of languages captured by those probabilistic Turing machines whose underlying error can be proved bounded in T. As a paradigmatic example of this approach, we establish that polynomial identity testing is in BPP_T, where T = IΔ₀+Exp is a well-studied theory based on bounded induction.

Melissa Antonelli, Ugo Dal Lago, Davide Davoli, Isabel Oitavem, and Paolo Pistone. Enumerating Error Bounded Polytime Algorithms Through Arithmetical Theories. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 10:1-10:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{antonelli_et_al:LIPIcs.CSL.2024.10, author = {Antonelli, Melissa and Dal Lago, Ugo and Davoli, Davide and Oitavem, Isabel and Pistone, Paolo}, title = {{Enumerating Error Bounded Polytime Algorithms Through Arithmetical Theories}}, booktitle = {32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)}, pages = {10:1--10:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-310-2}, ISSN = {1868-8969}, year = {2024}, volume = {288}, editor = {Murano, Aniello and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.10}, URN = {urn:nbn:de:0030-drops-196538}, doi = {10.4230/LIPIcs.CSL.2024.10}, annote = {Keywords: Bounded Arithmetic, Randomized Computation, Implicit Computational Complexity} }

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**Published in:** LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

We study the interpretation of the lambda-calculus in a framework based on tropical mathematics, and we show that it provides a unifying framework for two well-developed quantitative approaches to program semantics: on the one hand program metrics, based on the analysis of program sensitivity via Lipschitz conditions, on the other hand resource analysis, based on linear logic and higher-order program differentiation. To do that, we focus on the semantics arising from the relational model weighted over the tropical semiring, and we discuss its application to the study of "best case" program behavior for languages with probabilistic and non-deterministic effects. Finally, we show that a general foundation for this approach is provided by an abstract correspondence between tropical algebra and Lawvere’s theory of generalized metric spaces.

Davide Barbarossa and Paolo Pistone. Tropical Mathematics and the Lambda-Calculus I: Metric and Differential Analysis of Effectful Programs. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 14:1-14:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{barbarossa_et_al:LIPIcs.CSL.2024.14, author = {Barbarossa, Davide and Pistone, Paolo}, title = {{Tropical Mathematics and the Lambda-Calculus I: Metric and Differential Analysis of Effectful Programs}}, booktitle = {32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)}, pages = {14:1--14:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-310-2}, ISSN = {1868-8969}, year = {2024}, volume = {288}, editor = {Murano, Aniello and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.14}, URN = {urn:nbn:de:0030-drops-196577}, doi = {10.4230/LIPIcs.CSL.2024.14}, annote = {Keywords: Relational semantics, Differential lambda-calculus, Tropical semiring, Program metrics, Lawvere quantale} }

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

In this paper we are concerned with understanding the nature of program metrics for calculi with higher-order types, seen as natural generalizations of program equivalences. Some of the metrics we are interested in are well-known, such as those based on the interpretation of terms in metric spaces and those obtained by generalizing observational equivalence. We also introduce a new one, called the interactive metric, built by applying the well-known Int-Construction to the category of metric complete partial orders. Our aim is then to understand how these metrics relate to each other, i.e., whether and in which cases one such metric refines another, in analogy with corresponding well-studied problems about program equivalences. The results we obtain are twofold. We first show that the metrics of semantic origin, i.e., the denotational and interactive ones, lie in between the observational and equational metrics and that in some cases, these inclusions are strict. Then, we give a result about the relationship between the denotational and interactive metrics, revealing that the former is less discriminating than the latter. All our results are given for a linear lambda-calculus, and some of them can be generalized to calculi with graded comonads, in the style of Fuzz.

Ugo Dal Lago, Naohiko Hoshino, and Paolo Pistone. On the Lattice of Program Metrics. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 20:1-20:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dallago_et_al:LIPIcs.FSCD.2023.20, author = {Dal Lago, Ugo and Hoshino, Naohiko and Pistone, Paolo}, title = {{On the Lattice of Program Metrics}}, booktitle = {8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)}, pages = {20:1--20:19}, 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.2023.20}, URN = {urn:nbn:de:0030-drops-180049}, doi = {10.4230/LIPIcs.FSCD.2023.20}, annote = {Keywords: Metrics, Lambda Calculus, Linear Types} }

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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 195, 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)

Due to the undecidability of most type-related properties of System F like type inhabitation or type checking, restricted polymorphic systems have been widely investigated (the most well-known being ML-polymorphism). In this paper we investigate System Fat, or atomic System F, a very weak predicative fragment of System F whose typable terms coincide with the simply typable ones. We show that the type-checking problem for Fat is decidable and we propose an algorithm which sheds some new light on the source of undecidability in full System F. Moreover, we investigate free theorems and contextual equivalence in this fragment, and we show that the latter, unlike in the simply typed lambda-calculus, is undecidable.

Paolo Pistone and Luca Tranchini. What’s Decidable About (Atomic) Polymorphism?. In 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 195, pp. 27:1-27:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{pistone_et_al:LIPIcs.FSCD.2021.27, author = {Pistone, Paolo and Tranchini, Luca}, title = {{What’s Decidable About (Atomic) Polymorphism?}}, booktitle = {6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)}, pages = {27:1--27:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-191-7}, ISSN = {1868-8969}, year = {2021}, volume = {195}, editor = {Kobayashi, Naoki}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2021.27}, URN = {urn:nbn:de:0030-drops-142652}, doi = {10.4230/LIPIcs.FSCD.2021.27}, annote = {Keywords: Atomic System F, Predicative Polymorphism, ML-Polymorphism, Type-Checking, Contextual Equivalence, Free Theorems} }

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**Published in:** LIPIcs, Volume 183, 29th EACSL Annual Conference on Computer Science Logic (CSL 2021)

Program semantics is traditionally concerned with program equivalence. However, in fields like approximate, incremental and probabilistic computation, it is often useful to describe to which extent two programs behave in a similar, although non equivalent way. This has motivated the study of program (pseudo)metrics, which have found widespread applications, e.g. in differential privacy. In this paper we show that the standard metric on real numbers can be lifted to higher-order types in a novel way, yielding a metric semantics of the simply typed lambda-calculus in which types are interpreted as quantale-valued partial metric spaces. Using such metrics we define a class of higher-order denotational models, called diameter space models, that provide a quantitative semantics of approximate program transformations. Noticeably, the distances between objects of higher-types are elements of functional, thus non-numerical, quantales. This allows us to model contextual reasoning about arbitrary functions, thus deviating from classic metric semantics.

Guillaume Geoffroy and Paolo Pistone. A Partial Metric Semantics of Higher-Order Types and Approximate Program Transformations. In 29th EACSL Annual Conference on Computer Science Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 183, pp. 23:1-23:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{geoffroy_et_al:LIPIcs.CSL.2021.23, author = {Geoffroy, Guillaume and Pistone, Paolo}, title = {{A Partial Metric Semantics of Higher-Order Types and Approximate Program Transformations}}, booktitle = {29th EACSL Annual Conference on Computer Science Logic (CSL 2021)}, pages = {23:1--23:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-175-7}, ISSN = {1868-8969}, year = {2021}, volume = {183}, editor = {Baier, Christel and Goubault-Larrecq, Jean}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2021.23}, URN = {urn:nbn:de:0030-drops-134578}, doi = {10.4230/LIPIcs.CSL.2021.23}, annote = {Keywords: Simply typed \lambda-calculus, program metrics, approximate program transformations, partial metric spaces} }

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**Published in:** LIPIcs, Volume 183, 29th EACSL Annual Conference on Computer Science Logic (CSL 2021)

In this paper we explore a family of type isomorphisms in System F whose validity corresponds, semantically, to some form of the Yoneda isomorphism from category theory. These isomorphisms hold under theories of equivalence stronger than βη-equivalence, like those induced by parametricity and dinaturality. We show that the Yoneda type isomorphisms yield a rewriting over types, that we call Yoneda reduction, which can be used to eliminate quantifiers from a polymorphic type, replacing them with a combination of monomorphic type constructors. We establish some sufficient conditions under which quantifiers can be fully eliminated from a polymorphic type, and we show some application of these conditions to count the inhabitants of a type and to compute program equivalence in some fragments of System F.

Paolo Pistone and Luca Tranchini. The Yoneda Reduction of Polymorphic Types. In 29th EACSL Annual Conference on Computer Science Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 183, pp. 35:1-35:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{pistone_et_al:LIPIcs.CSL.2021.35, author = {Pistone, Paolo and Tranchini, Luca}, title = {{The Yoneda Reduction of Polymorphic Types}}, booktitle = {29th EACSL Annual Conference on Computer Science Logic (CSL 2021)}, pages = {35:1--35:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-175-7}, ISSN = {1868-8969}, year = {2021}, volume = {183}, editor = {Baier, Christel and Goubault-Larrecq, Jean}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2021.35}, URN = {urn:nbn:de:0030-drops-134696}, doi = {10.4230/LIPIcs.CSL.2021.35}, annote = {Keywords: System F, Type isomorphisms, Yoneda isomorphism, Program equivalence} }

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

We present two results which relate dinaturality with a syntactic property (typability) and a semantic one (interpretability by beta-eta-stable sets). First, we prove that closed dinatural lambda-terms are simply typable, that is, the converse of the well-known fact that simply typable closed terms are dinatural. The argument exposes a syntactical aspect of dinaturality, as lambda-terms are type-checked by computing their associated dinaturality equation. Second, we prove that a closed lambda-term belonging to all beta-eta-stable interpretations of a simple type must be dinatural, that is, we prove dinaturality by semantical means. To do this, we show that such terms satisfy a suitable version of binary parametricity and we derive dinaturality from it.
By combining the two results we obtain a new proof of the completeness of beta-eta-stable semantics with respect to simple types. While the completeness of this semantics is well-known in the literature, the technique here developed suggests that dinaturality might be exploited to prove completeness also for other, less manageable, semantics (like saturated families or reducibility candidates) for which a direct argument is not yet known.

Paolo Pistone. On Dinaturality, Typability and beta-eta-Stable Models. In 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 84, pp. 29:1-29:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{pistone:LIPIcs.FSCD.2017.29, author = {Pistone, Paolo}, title = {{On Dinaturality, Typability and beta-eta-Stable Models}}, booktitle = {2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017)}, pages = {29:1--29:17}, 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.2017.29}, URN = {urn:nbn:de:0030-drops-77291}, doi = {10.4230/LIPIcs.FSCD.2017.29}, annote = {Keywords: Dinaturality, simply-typed lambda-calculus, beta-eta-stable semantics, completeness} }

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