research-article

Fixed Points In Quantitative Semantics

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J. LairdAuthors Info & Claims

LICS '16: Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science

Pages 347 - 356

https://doi.org/10.1145/2933575.2934569

Published: 05 July 2016 Publication History

Abstract

We describe an interpretation of recursive computation in a symmetric monoidal category with infinite biproducts and cofree commutative comonoids (for instance, the category of free modules over a complete semiring). Such categories play a significant role in "quantitative" models of computation: they bear a canonical complete monoid enrichment, but may not be cpo-enriched, making standard techniques for reasoning about fixed points unavailable. By constructing a bifree algebra for the cofree exponential, we obtain fixed points for morphisms in its co-Kleisli category without requiring any order-theoretic structure. These fixed points corresponding to infinite sums of finitary approximants indexed over the nested finite multisets, each representing a unique call-pattern for computation of the fixed point. We illustrate this construction by using it to give a denotational semantics for PCF with non-deterministic choice and scalar weights from a complete semiring, proving that this is computationally adequate with respect to an operational semantics which evaluates a term by taking a weighted sum of the residues of its terminating reduction paths.

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Cited By

Galal ZPacaud Lemay JSobocinski PLago UEsparza J(2024)Combining fixpoint and differentiation theoryProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3661814.3662108(1-14)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3661814.3662108
Tsukada TAsada K(2022)Linear-Algebraic Models of Linear Logic as Categories of Modules over Σ-Semirings✱Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3531130.3533373(1-13)Online publication date: 2-Aug-2022
https://dl.acm.org/doi/10.1145/3531130.3533373
Laird J(2020)Weighted models for higher-order computationInformation and Computation10.1016/j.ic.2020.104645(104645)Online publication date: Nov-2020
https://doi.org/10.1016/j.ic.2020.104645
Show More Cited By

Index Terms

Fixed Points In Quantitative Semantics
1. Theory of computation
  1. Semantics and reasoning
    1. Program semantics

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cover image ACM Conferences

LICS '16: Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science

July 2016

901 pages

ISBN:9781450343916

DOI:10.1145/2933575

Conference Chair:
Eric Koskinen,
General Chair:
Martin Grohe,
Program Chair:
Natarajan Shankar

Copyright © 2016 ACM.

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SIGLOG: ACM Special Interest Group on Logic and Computation
EACSL: European Association for Computer Science Logic
IEEE-CS\DATC: IEEE Computer Society

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New York, NY, United States

Publication History

Published: 05 July 2016

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Engineering and Physical Sciences Research Council

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LICS '16

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SIGLOG
EACSL
IEEE-CS\DATC

LICS '16: 31st Annual ACM/IEEE Symposium on Logic in Computer Science

July 5 - 8, 2016

NY, New York, USA

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Cited By

Galal ZPacaud Lemay JSobocinski PLago UEsparza J(2024)Combining fixpoint and differentiation theoryProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3661814.3662108(1-14)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3661814.3662108
Tsukada TAsada K(2022)Linear-Algebraic Models of Linear Logic as Categories of Modules over Σ-Semirings✱Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3531130.3533373(1-13)Online publication date: 2-Aug-2022
https://dl.acm.org/doi/10.1145/3531130.3533373
Laird J(2020)Weighted models for higher-order computationInformation and Computation10.1016/j.ic.2020.104645(104645)Online publication date: Nov-2020
https://doi.org/10.1016/j.ic.2020.104645
Tsukada TAsada KOng C(2018)Species, Profunctors and Taylor Expansion Weighted by SMCCProceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3209108.3209157(889-898)Online publication date: 9-Jul-2018
https://dl.acm.org/doi/10.1145/3209108.3209157
Ong CAceto LIngólfsdóttir A(2017)Quantitative semantics of the lambda calculusProceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science10.5555/3329995.3329999(1-12)Online publication date: 20-Jun-2017
https://dl.acm.org/doi/10.5555/3329995.3329999
Ong C(2017)Quantitative semantics of the lambda calculus: Some generalisations of the relational model2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS.2017.8005064(1-12)Online publication date: Jun-2017
https://doi.org/10.1109/LICS.2017.8005064
Laird J(2017)From Qualitative to Quantitative SemanticsProceedings of the 20th International Conference on Foundations of Software Science and Computation Structures - Volume 1020310.1007/978-3-662-54458-7_3(36-52)Online publication date: 22-Apr-2017
https://dl.acm.org/doi/10.1007/978-3-662-54458-7_3

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