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**Published in:** LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

When working in optimisation or privacy protection, one may need to estimate the sensitivity of computer programs, i.e., the maximum multiplicative increase in the distance between two inputs and the corresponding two outputs. In particular, differential privacy is a rigorous and widely used notion of privacy that is closely related to sensitivity. Several type systems for sensitivity and differential privacy based on linear logic have been proposed in the literature, starting with the functional language Fuzz. However, they are either limited to certain metrics (L¹ and L^∞), and thus to the associated privacy mechanisms, or they rely on a complex notion of type contexts that does not interact well with operational semantics. We therefore propose a graded linear type system - inspired by Bunched Fuzz [{w}under et al., 2023] - called Plurimetric Fuzz that handles L^p vector metrics (for 1 ≤ p ≤ +∞), uses standard type contexts, gives reasonable bounds on sensitivity, and has good metatheoretical properties. We also provide a denotational semantics in terms of metric complete partial orders, and translation mappings from and to Fuzz.

Victor Sannier and Patrick Baillot. A Linear Type System for L^p-Metric Sensitivity Analysis. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 12:1-12:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{sannier_et_al:LIPIcs.FSCD.2024.12, author = {Sannier, Victor and Baillot, Patrick}, title = {{A Linear Type System for L^p-Metric Sensitivity Analysis}}, booktitle = {9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)}, pages = {12:1--12:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-323-2}, ISSN = {1868-8969}, year = {2024}, volume = {299}, editor = {Rehof, Jakob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.12}, URN = {urn:nbn:de:0030-drops-203412}, doi = {10.4230/LIPIcs.FSCD.2024.12}, annote = {Keywords: type system, linear logic, sensitivity, vector metrics, differential privacy, lambda-calculus, functional programming, denotational semantics} }

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**Published in:** LIPIcs, Volume 203, 32nd International Conference on Concurrency Theory (CONCUR 2021)

We address the problem of analysing the complexity of concurrent programs written in Pi-calculus. We are interested in parallel complexity, or span, understood as the execution time in a model with maximal parallelism. A type system for parallel complexity has been recently proposed by the first two authors but it is too imprecise for non-linear channels and cannot analyse some concurrent processes. Aiming for a more precise analysis, we design a type system which builds on the concepts of sized types and usages. The sized types allow us to parametrize the complexity by the size of inputs, and the usages allow us to achieve a kind of rely-guarantee reasoning on the timing each process communicates with its environment. We prove that our new type system soundly estimates the parallel complexity, and show through examples that it is often more precise than the previous type system of the first two authors.

Patrick Baillot, Alexis Ghyselen, and Naoki Kobayashi. Sized Types with Usages for Parallel Complexity of Pi-Calculus Processes. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 34:1-34:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{baillot_et_al:LIPIcs.CONCUR.2021.34, author = {Baillot, Patrick and Ghyselen, Alexis and Kobayashi, Naoki}, title = {{Sized Types with Usages for Parallel Complexity of Pi-Calculus Processes}}, booktitle = {32nd International Conference on Concurrency Theory (CONCUR 2021)}, pages = {34:1--34:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-203-7}, ISSN = {1868-8969}, year = {2021}, volume = {203}, editor = {Haddad, Serge and Varacca, Daniele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2021.34}, URN = {urn:nbn:de:0030-drops-144111}, doi = {10.4230/LIPIcs.CONCUR.2021.34}, annote = {Keywords: Type Systems, Pi-calculus, Process Calculi, Complexity Analysis, Usages, Sized Types} }

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**Published in:** LIPIcs, Volume 119, 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)

Several type systems have been proposed to statically control the time complexity of lambda-calculus programs and characterize complexity classes such as FPTIME or FEXPTIME. A first line of research stems from linear logic and restricted versions of its !-modality controlling duplication. A second approach relies on the idea of tracking the size increase between input and output, and together with a restricted recursion scheme, to deduce time complexity bounds. However both approaches suffer from limitations : either a limited intensional expressivity, or linearity restrictions. In the present work we incorporate both approaches into a common type system, in order to overcome their respective constraints. Our system is based on elementary linear logic combined with linear size types, called sEAL, and leads to characterizations of the complexity classes FPTIME and 2k-FEXPTIME, for k >= 0.

Patrick Baillot and Alexis Ghyselen. Combining Linear Logic and Size Types for Implicit Complexity. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{baillot_et_al:LIPIcs.CSL.2018.9, author = {Baillot, Patrick and Ghyselen, Alexis}, title = {{Combining Linear Logic and Size Types for Implicit Complexity}}, booktitle = {27th EACSL Annual Conference on Computer Science Logic (CSL 2018)}, pages = {9:1--9:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-088-0}, ISSN = {1868-8969}, year = {2018}, volume = {119}, editor = {Ghica, Dan R. and Jung, Achim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2018.9}, URN = {urn:nbn:de:0030-drops-96763}, doi = {10.4230/LIPIcs.CSL.2018.9}, annote = {Keywords: Implicit computational complexity, lambda-calculus, linear logic, type systems, polynomial time complexity, size types} }

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**Published in:** LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)

We prove a general form of 'free-cut elimination' for first-order theories in linear logic, yielding normal forms of proofs where cuts are anchored to nonlogical steps. To demonstrate the usefulness of this result, we consider a version of arithmetic in linear logic, based on a previous axiomatisation by Bellantoni and Hofmann. We prove a witnessing theorem for a fragment of this arithmetic via the `witness function method', showing that the provably convergent functions are precisely the polynomial-time functions. The programs extracted are implemented in the framework of 'safe' recursive functions, due to Bellantoni and Cook, where the ! modality of linear logic corresponds to normal inputs of a safe recursive program.

Patrick Baillot and Anupam Das. Free-Cut Elimination in Linear Logic and an Application to a Feasible Arithmetic. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 40:1-40:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{baillot_et_al:LIPIcs.CSL.2016.40, author = {Baillot, Patrick and Das, Anupam}, title = {{Free-Cut Elimination in Linear Logic and an Application to a Feasible Arithmetic}}, booktitle = {25th EACSL Annual Conference on Computer Science Logic (CSL 2016)}, pages = {40:1--40:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-022-4}, ISSN = {1868-8969}, year = {2016}, volume = {62}, editor = {Talbot, Jean-Marc and Regnier, Laurent}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.40}, URN = {urn:nbn:de:0030-drops-65807}, doi = {10.4230/LIPIcs.CSL.2016.40}, annote = {Keywords: proof theory, linear logic, bounded arithmetic, polynomial time computation, implicit computational complexity} }

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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} }