research-article

Equations, Contractions, and Unique Solutions

Author:

Davide SangiorgiAuthors Info & Claims

ACM Transactions on Computational Logic (TOCL), Volume 18, Issue 1

Article No.: 4, Pages 1 - 30

https://doi.org/10.1145/2971339

Published: 16 March 2017 Publication History

Abstract

One of the most studied behavioural equivalences is bisimilarity. Its success is much due to the associated bisimulation proof method, which can be further enhanced by means of “bisimulation up-to” techniques such as “up-to context.”

A different proof method is discussed, based on a unique solution of special forms of inequations called contractions and inspired by Milner’s theorem on unique solution of equations. The method is as powerful as the bisimulation proof method and its “up-to context” enhancements. The definition of contraction can be transferred onto other behavioural equivalences, possibly contextual and non-coinductive. This enables a coinductive reasoning style on such equivalences, either by applying the method based on unique solution of contractions or by injecting appropriate contraction preorders into the bisimulation game.

The techniques are illustrated in CCS-like languages; an example dealing with higher-order languages is also shown.

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Cited By

Sangiorgi D(2024)An Abstract Account of Up-to Techniques for Inductive Behavioural RelationsLeveraging Applications of Formal Methods, Verification and Validation. REoCAS Colloquium in Honor of Rocce De Nicola10.1007/978-3-031-73709-1_5(62-74)Online publication date: 27-Oct-2024
https://dl.acm.org/doi/10.1007/978-3-031-73709-1_5
Muroya KSanada TUrabe N(2024)Preorder-Constrained Simulations for Program Refinement with EffectsCoalgebraic Methods in Computer Science10.1007/978-3-031-66438-0_3(44-64)Online publication date: 6-Apr-2024
https://dl.acm.org/doi/10.1007/978-3-031-66438-0_3
Sangiorgi D(2022)From enhanced coinduction towards enhanced inductionProceedings of the ACM on Programming Languages10.1145/34986796:POPL(1-29)Online publication date: 12-Jan-2022
https://dl.acm.org/doi/10.1145/3498679
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Index Terms

Equations, Contractions, and Unique Solutions
1. Theory of computation
  1. Logic
    1. Programming logic
  2. Semantics and reasoning
    1. Program reasoning
    2. Program semantics
      1. Operational semantics

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Published In

cover image ACM Transactions on Computational Logic

ACM Transactions on Computational Logic Volume 18, Issue 1

January 2017

277 pages

ISSN:1529-3785

EISSN:1557-945X

DOI:10.1145/3041822

Editor:
Orna Kupferman
School of Computer Science and Engineering, Hebrew University, Israel

Issue’s Table of Contents

Copyright © 2017 ACM.

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Publication History

Published: 16 March 2017

Accepted: 01 June 2016

Received: 01 January 2016

Published in TOCL Volume 18, Issue 1

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Cited By

Sangiorgi D(2024)An Abstract Account of Up-to Techniques for Inductive Behavioural RelationsLeveraging Applications of Formal Methods, Verification and Validation. REoCAS Colloquium in Honor of Rocce De Nicola10.1007/978-3-031-73709-1_5(62-74)Online publication date: 27-Oct-2024
https://dl.acm.org/doi/10.1007/978-3-031-73709-1_5
Muroya KSanada TUrabe N(2024)Preorder-Constrained Simulations for Program Refinement with EffectsCoalgebraic Methods in Computer Science10.1007/978-3-031-66438-0_3(44-64)Online publication date: 6-Apr-2024
https://dl.acm.org/doi/10.1007/978-3-031-66438-0_3
Sangiorgi D(2022)From enhanced coinduction towards enhanced inductionProceedings of the ACM on Programming Languages10.1145/34986796:POPL(1-29)Online publication date: 12-Jan-2022
https://dl.acm.org/doi/10.1145/3498679
Tian CSangiorgi D(2020)Unique solutions of contractions, CCS, and their HOL formalisationInformation and Computation10.1016/j.ic.2020.104606275(104606)Online publication date: Dec-2020
https://doi.org/10.1016/j.ic.2020.104606
Abbasi KKhan THaq I(2019)Hierarchical Modeling of Complex Internet of Things Systems Using Conceptual Modeling ApproachesIEEE Access10.1109/ACCESS.2019.29309337(102772-102791)Online publication date: 2019
https://doi.org/10.1109/ACCESS.2019.2930933
Tian CSangiorgi D(2018)Unique Solutions of Contractions, CCS, and their HOL FormalisationElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.276.10276(122-139)Online publication date: 24-Aug-2018
https://doi.org/10.4204/EPTCS.276.10
Danielsson N(2017)Up-to techniques using sized typesProceedings of the ACM on Programming Languages10.1145/31581312:POPL(1-28)Online publication date: 27-Dec-2017
https://dl.acm.org/doi/10.1145/3158131

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