Document

**Published in:** LIPIcs, Volume 279, 34th International Conference on Concurrency Theory (CONCUR 2023)

We introduce contextual behavioural metrics (CBMs) as a novel way of measuring the discrepancy in behaviour between processes, taking into account both quantitative aspects and contextual information. This way, process distances by construction take the environment into account: two (non-equivalent) processes may still exhibit very similar behaviour in some contexts, e.g., when certain actions are never performed. We first show how CBMs capture many well-known notions of equivalence and metric, including Larsen’s environmental parametrized bisimulation. We then study compositional properties of CBMs with respect to some common process algebraic operators, namely prefixing, restriction, non-deterministic sum, parallel composition and replication.

Ugo Dal Lago and Maurizio Murgia. Contextual Behavioural Metrics. In 34th International Conference on Concurrency Theory (CONCUR 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 279, pp. 38:1-38:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dallago_et_al:LIPIcs.CONCUR.2023.38, author = {Dal Lago, Ugo and Murgia, Maurizio}, title = {{Contextual Behavioural Metrics}}, booktitle = {34th International Conference on Concurrency Theory (CONCUR 2023)}, pages = {38:1--38:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-299-0}, ISSN = {1868-8969}, year = {2023}, volume = {279}, editor = {P\'{e}rez, Guillermo A. and Raskin, Jean-Fran\c{c}ois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2023.38}, URN = {urn:nbn:de:0030-drops-190320}, doi = {10.4230/LIPIcs.CONCUR.2023.38}, annote = {Keywords: Behavioural metrics, Labelled Transition Systems, Differential Semantics} }

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**Published in:** LIPIcs, Volume 269, 28th International Conference on Types for Proofs and Programs (TYPES 2022)

In the realm of quantum computing, circuit description languages represent a valid alternative to traditional QRAM-style languages. They indeed allow for finer control over the output circuit, without sacrificing flexibility nor modularity. We introduce a generalization of the paradigmatic lambda-calculus Proto-Quipper-M, which models the core features of the quantum circuit description language Quipper. The extension, called Proto-Quipper-K, is meant to capture a very general form of dynamic lifting. This is made possible by the introduction of a rich type and effect system in which not only computations, but also the very types are effectful. The main results we give for the introduced language are the classic type soundness results, namely subject reduction and progress.

Andrea Colledan and Ugo Dal Lago. On Dynamic Lifting and Effect Typing in Circuit Description Languages. In 28th International Conference on Types for Proofs and Programs (TYPES 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 269, pp. 3:1-3:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{colledan_et_al:LIPIcs.TYPES.2022.3, author = {Colledan, Andrea and Dal Lago, Ugo}, title = {{On Dynamic Lifting and Effect Typing in Circuit Description Languages}}, booktitle = {28th International Conference on Types for Proofs and Programs (TYPES 2022)}, pages = {3:1--3:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-285-3}, ISSN = {1868-8969}, year = {2023}, volume = {269}, editor = {Kesner, Delia and P\'{e}drot, Pierre-Marie}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2022.3}, URN = {urn:nbn:de:0030-drops-184468}, doi = {10.4230/LIPIcs.TYPES.2022.3}, annote = {Keywords: Circuit-Description Languages, \lambda-calculus, Dynamic lifting, Type and effect systems} }

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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 252, 31st EACSL Annual Conference on Computer Science Logic (CSL 2023)

We introduce a variation on Barthe et al.’s higher-order logic in which formulas are interpreted as predicates over open rather than closed objects. This way, concepts which have an intrinsically functional nature, like continuity, differentiability, or monotonicity, can be expressed and reasoned about in a very natural way, following the structure of the underlying program. We give open higher-order logic in distinct flavors, and in particular in its relational and local versions, the latter being tailored for situations in which properties hold only in part of the underlying function’s domain of definition.

Ugo Dal Lago, Francesco Gavazzo, and Alexis Ghyselen. Open Higher-Order Logic. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 17:1-17:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dallago_et_al:LIPIcs.CSL.2023.17, author = {Dal Lago, Ugo and Gavazzo, Francesco and Ghyselen, Alexis}, title = {{Open Higher-Order Logic}}, booktitle = {31st EACSL Annual Conference on Computer Science Logic (CSL 2023)}, pages = {17:1--17:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-264-8}, ISSN = {1868-8969}, year = {2023}, volume = {252}, editor = {Klin, Bartek and Pimentel, Elaine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.17}, URN = {urn:nbn:de:0030-drops-174785}, doi = {10.4230/LIPIcs.CSL.2023.17}, annote = {Keywords: Formal Verification, Relational Logic, First-Order Properties} }

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**Published in:** LIPIcs, Volume 243, 33rd International Conference on Concurrency Theory (CONCUR 2022)

A system of session types is introduced as induced by a Curry Howard correspondence applied to bounded linear logic, suitably extended with probabilistic choice operators and ground types. The resulting system satisfies some expected properties, like subject reduction and progress, but also unexpected ones, like a polynomial bound on the time needed to reduce processes. This makes the system suitable for modelling experiments and proofs from the so-called computational model of cryptography.

Ugo Dal Lago and Giulia Giusti. On Session Typing, Probabilistic Polynomial Time, and Cryptographic Experiments. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 37:1-37:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dallago_et_al:LIPIcs.CONCUR.2022.37, author = {Dal Lago, Ugo and Giusti, Giulia}, title = {{On Session Typing, Probabilistic Polynomial Time, and Cryptographic Experiments}}, booktitle = {33rd International Conference on Concurrency Theory (CONCUR 2022)}, pages = {37:1--37:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-246-4}, ISSN = {1868-8969}, year = {2022}, volume = {243}, editor = {Klin, Bartek and Lasota, S{\l}awomir and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2022.37}, URN = {urn:nbn:de:0030-drops-171000}, doi = {10.4230/LIPIcs.CONCUR.2022.37}, annote = {Keywords: Session Types, Probabilistic Computation, Bounded Linear Logic, Cryptographic Experiments} }

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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 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Probabilistic complexity classes, despite capturing the notion of feasibility, have escaped any treatment by the tools of so-called implicit-complexity. Their inherently semantic nature is of course a barrier to the characterization of classes like BPP or ZPP, but not all classes are semantic. In this paper, we introduce a recursion-theoretic characterization of the probabilistic class PP, using recursion schemata with pointers.

Ugo Dal Lago, Reinhard Kahle, and Isabel Oitavem. A Recursion-Theoretic Characterization of the Probabilistic Class PP. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 35:1-35:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dallago_et_al:LIPIcs.MFCS.2021.35, author = {Dal Lago, Ugo and Kahle, Reinhard and Oitavem, Isabel}, title = {{A Recursion-Theoretic Characterization of the Probabilistic Class PP}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {35:1--35:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.35}, URN = {urn:nbn:de:0030-drops-144754}, doi = {10.4230/LIPIcs.MFCS.2021.35}, annote = {Keywords: Implicit complexity, tree-recursion, probabilistic classes, polynomial time, PP} }

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**Published in:** LIPIcs, Volume 195, 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)

We investigate program equivalence for linear higher-order (sequential) languages endowed with primitives for computational effects. More specifically, we study operationally-based notions of program equivalence for a linear λ-calculus with explicit copying and algebraic effects à la Plotkin and Power. Such a calculus makes explicit the interaction between copying and linearity, which are intensional aspects of computation, with effects, which are, instead, extensional. We review some of the notions of equivalences for linear calculi proposed in the literature and show their limitations when applied to effectful calculi where copying is a first-class citizen. We then introduce resource transition systems, namely transition systems whose states are built over tuples of programs representing the available resources, as an operational semantics accounting for both intensional and extensional interactive behaviours of programs. Our main result is a sound and complete characterization of contextual equivalence as trace equivalence defined on top of resource transition systems.

Ugo Dal Lago and Francesco Gavazzo. Resource Transition Systems and Full Abstraction for Linear Higher-Order Effectful Programs. In 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 195, pp. 23:1-23:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dallago_et_al:LIPIcs.FSCD.2021.23, author = {Dal Lago, Ugo and Gavazzo, Francesco}, title = {{Resource Transition Systems and Full Abstraction for Linear Higher-Order Effectful Programs}}, booktitle = {6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)}, pages = {23:1--23:19}, 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.2021.23}, URN = {urn:nbn:de:0030-drops-142618}, doi = {10.4230/LIPIcs.FSCD.2021.23}, annote = {Keywords: algebraic effects, linearity, program equivalence, full abstraction} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Type-two constructions abound in cryptography: adversaries for encryption and authentication schemes, if active, are modeled as algorithms having access to oracles, i.e. as second-order algorithms. But how about making cryptographic schemes themselves higher-order? This paper gives an answer to this question, by first describing why higher-order cryptography is interesting as an object of study, then showing how the concept of probabilistic polynomial time algorithm can be generalized so as to encompass algorithms of order strictly higher than two, and finally proving some positive and negative results about the existence of higher-order cryptographic primitives, namely authentication schemes and pseudorandom functions.

Boaz Barak, Raphaëlle Crubillé, and Ugo Dal Lago. On Higher-Order Cryptography. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 108:1-108:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{barak_et_al:LIPIcs.ICALP.2020.108, author = {Barak, Boaz and Crubill\'{e}, Rapha\"{e}lle and Dal Lago, Ugo}, title = {{On Higher-Order Cryptography}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {108:1--108:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.108}, URN = {urn:nbn:de:0030-drops-125153}, doi = {10.4230/LIPIcs.ICALP.2020.108}, annote = {Keywords: Higher-order computation, probabilistic computation, game semantics, cryptography} }

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Invited Talk

**Published in:** LIPIcs, Volume 167, 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)

The notion of solvability, crucial in the λ-calculus, is conservatively extended to a probabilistic setting, and a complete characterization of it is given. The employed technical tool is a type assignment system, based on non-idempotent intersection types, whose typable terms turn out to be precisely the terms which are solvable with nonnull probability. We also supply an operational characterization of solvable terms, through the notion of head normal form, and a denotational model of Λ_⊕, itself induced by the type system, which equates all the unsolvable terms.

Simona Ronchi Della Rocca, Ugo Dal Lago, and Claudia Faggian. Solvability in a Probabilistic Setting (Invited Talk). In 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 167, pp. 1:1-1:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ronchidellarocca_et_al:LIPIcs.FSCD.2020.1, author = {Ronchi Della Rocca, Simona and Dal Lago, Ugo and Faggian, Claudia}, title = {{Solvability in a Probabilistic Setting}}, booktitle = {5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)}, pages = {1:1--1:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-155-9}, ISSN = {1868-8969}, year = {2020}, volume = {167}, editor = {Ariola, Zena M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2020.1}, URN = {urn:nbn:de:0030-drops-123237}, doi = {10.4230/LIPIcs.FSCD.2020.1}, annote = {Keywords: Probabilistic Computation, Lambda Calculus, Solvability, Intersection Types} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We introduce a new form of logical relation which, in the spirit of metric relations, allows us to assign each pair of programs a quantity measuring their distance, rather than a boolean value standing for their being equivalent. The novelty of differential logical relations consists in measuring the distance between terms not (necessarily) by a numerical value, but by a mathematical object which somehow reflects the interactive complexity, i.e. the type, of the compared terms. We exemplify this concept in the simply-typed lambda-calculus, and show a form of soundness theorem. We also see how ordinary logical relations and metric relations can be seen as instances of differential logical relations. Finally, we show that differential logical relations can be organised in a cartesian closed category, contrarily to metric relations, which are well-known not to have such a structure, but only that of a monoidal closed category.

Ugo Dal Lago, Francesco Gavazzo, and Akira Yoshimizu. Differential Logical Relations, Part I: The Simply-Typed Case (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 111:1-111:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dallago_et_al:LIPIcs.ICALP.2019.111, author = {Dal Lago, Ugo and Gavazzo, Francesco and Yoshimizu, Akira}, title = {{Differential Logical Relations, Part I: The Simply-Typed Case}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {111:1--111:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.111}, URN = {urn:nbn:de:0030-drops-106879}, doi = {10.4230/LIPIcs.ICALP.2019.111}, annote = {Keywords: Logical Relations, lambda-Calculus, Program Equivalence, Semantics} }

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

We generalise Ehrhard and Regnier’s Taylor expansion from pure to probabilistic lambda-terms. We prove that the Taylor expansion is adequate when seen as a way to give semantics to probabilistic lambda-terms, and that there is a precise correspondence with probabilistic Böhm trees, as introduced by the second author. We prove this adequacy through notions of probabilistic resource terms and explicit Taylor expansion.

Ugo Dal Lago and Thomas Leventis. On the Taylor Expansion of Probabilistic lambda-terms. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 13:1-13:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dallago_et_al:LIPIcs.FSCD.2019.13, author = {Dal Lago, Ugo and Leventis, Thomas}, title = {{On the Taylor Expansion of Probabilistic lambda-terms}}, booktitle = {4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)}, pages = {13:1--13:16}, 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.2019.13}, URN = {urn:nbn:de:0030-drops-105206}, doi = {10.4230/LIPIcs.FSCD.2019.13}, annote = {Keywords: Probabilistic Lambda-Calculi, Taylor Expansion, Linear Logic} }

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**Published in:** LIPIcs, Volume 42, 26th International Conference on Concurrency Theory (CONCUR 2015)

In the ubiquitous presence of linear resources in quantum computation, program equivalence in linear contexts, where programs are used or executed once, is more important than in the classical setting. We introduce a linear contextual equivalence and two notions of bisimilarity, a state-based and a distribution-based, as proof techniques for reasoning about higher-order quantum programs. Both notions of bisimilarity are sound with respect to the linear contextual equivalence, but only the distribution-based one turns out to be complete. The completeness proof relies on a characterisation of the bisimilarity as a testing equivalence.

Yuxin Deng, Yuan Feng, and Ugo Dal Lago. On Coinduction and Quantum Lambda Calculi. In 26th International Conference on Concurrency Theory (CONCUR 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 42, pp. 427-440, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{deng_et_al:LIPIcs.CONCUR.2015.427, author = {Deng, Yuxin and Feng, Yuan and Dal Lago, Ugo}, title = {{On Coinduction and Quantum Lambda Calculi}}, booktitle = {26th International Conference on Concurrency Theory (CONCUR 2015)}, pages = {427--440}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-91-0}, ISSN = {1868-8969}, year = {2015}, volume = {42}, editor = {Aceto, Luca and de Frutos Escrig, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2015.427}, URN = {urn:nbn:de:0030-drops-53883}, doi = {10.4230/LIPIcs.CONCUR.2015.427}, annote = {Keywords: Quantum lambda calculi, contextual equivalence, bisimulation} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

We study how the adoption of an evaluation mechanism with sharing and memoization impacts the class of functions which can be computed in polynomial time. We first show how a natural cost model in which lookup for an already computed result has no cost is indeed invariant. As a corollary, we then prove that the most general notion of ramified recurrence is sound for polynomial time, this way settling an open problem in implicit computational complexity.

Martin Avanzini and Ugo Dal Lago. On Sharing, Memoization, and Polynomial Time. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 62-75, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{avanzini_et_al:LIPIcs.STACS.2015.62, author = {Avanzini, Martin and Dal Lago, Ugo}, title = {{On Sharing, Memoization, and Polynomial Time}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {62--75}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.62}, URN = {urn:nbn:de:0030-drops-49042}, doi = {10.4230/LIPIcs.STACS.2015.62}, annote = {Keywords: implicit computational complexity, data-tiering, polynomial time} }

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**Published in:** LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012)

Polynomial interpretations and their generalizations like quasi-interpretations have been used in the setting of first-order functional languages to design criteria ensuring statically some
complexity bounds on programs. This fits in the area of implicit computational complexity, which aims at giving machine-free characterizations of complexity classes. In this paper, we extend this approach to the higher-order setting. For that we consider the notion of simply-typed term rewriting systems, we define higher-order polynomial interpretations for them and give a criterion ensuring that a program can be executed in polynomial time. In order to obtain a criterion flexible enough to validate interesting programs using higher-order primitives, we introduce a notion of polynomial quasi-interpretations, coupled with a simple termination criterion based on linear types and path-like orders.

Patrick Baillot and Ugo Dal Lago. Higher-Order Interpretations and Program Complexity. In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 16, pp. 62-76, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2012)

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@InProceedings{baillot_et_al:LIPIcs.CSL.2012.62, author = {Baillot, Patrick and Dal Lago, Ugo}, title = {{Higher-Order Interpretations and Program Complexity}}, booktitle = {Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL}, pages = {62--76}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-42-2}, ISSN = {1868-8969}, year = {2012}, volume = {16}, editor = {C\'{e}gielski, Patrick and Durand, Arnaud}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.62}, URN = {urn:nbn:de:0030-drops-36641}, doi = {10.4230/LIPIcs.CSL.2012.62}, annote = {Keywords: implicit complexity, higher-order rewriting, quasi-interpretations} }

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**Published in:** LIPIcs, Volume 15, 23rd International Conference on Rewriting Techniques and Applications (RTA'12) (2012)

The lambda-calculus is a widely accepted computational model of higher-order functional programs, yet there is not any direct and universally accepted cost model for it. As a consequence, the computational difficulty of reducing lambda-terms to their normal form is typically studied by reasoning on concrete implementation algorithms. In this paper, we show that when head reduction is the underlying dynamics, the unitary cost model is indeed invariant.
This improves on known results, which only deal with weak (call-by-value or call-by-name) reduction. Invariance is proved by way of a linear calculus of explicit substitutions, which allows to nicely decompose any head reduction step in the lambda-calculus into
more elementary substitution steps, thus making the combinatorics of
head-reduction easier to reason about. The technique is also a promising tool to attack what we see as the main open problem, namely understanding for which normalizing strategies the unitary cost model is invariant, if any.

Beniamino Accattoli and Ugo Dal Lago. On the Invariance of the Unitary Cost Model for Head Reduction. In 23rd International Conference on Rewriting Techniques and Applications (RTA'12). Leibniz International Proceedings in Informatics (LIPIcs), Volume 15, pp. 22-37, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2012)

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@InProceedings{accattoli_et_al:LIPIcs.RTA.2012.22, author = {Accattoli, Beniamino and Dal Lago, Ugo}, title = {{On the Invariance of the Unitary Cost Model for Head Reduction}}, booktitle = {23rd International Conference on Rewriting Techniques and Applications (RTA'12)}, pages = {22--37}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-38-5}, ISSN = {1868-8969}, year = {2012}, volume = {15}, editor = {Tiwari, Ashish}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2012.22}, URN = {urn:nbn:de:0030-drops-34820}, doi = {10.4230/LIPIcs.RTA.2012.22}, annote = {Keywords: lambda calculus, cost models, explicit substitutions, implicit computational complexity} }

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