research-article

Proving the unique fixed-point principle correct: an adventure with category theory

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Daniel W.H. JamesAuthors Info & Claims

ACM SIGPLAN Notices, Volume 46, Issue 9

Pages 359 - 371

https://doi.org/10.1145/2034574.2034821

Published: 19 September 2011 Publication History

Abstract

Say you want to prove something about an infinite data-structure, such as a stream or an infinite tree, but you would rather not subject yourself to coinduction. The unique fixed-point principle is an easy-to-use, calculational alternative. The proof technique rests on the fact that certain recursion equations have unique solutions; if two elements of a coinductive type satisfy the same equation of this kind, then they are equal. In this paper we precisely characterize the conditions that guarantee a unique solution. Significantly, we do so not with a syntactic criterion, but with a semantic one that stems from the categorical notion of naturality. Our development is based on distributive laws and bialgebras, and draws heavily on Turi and Plotkin's pioneering work on mathematical operational semantics. Along the way, we break down the design space in two dimensions, leading to a total of nine points. Each gives rise to varying degrees of expressiveness, and we will discuss three in depth. Furthermore, our development is generic in the syntax of equations and in the behaviour they encode - we are not caged in the world of streams.

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References

[1]

A. Abel. MiniAgda: Integrating Sized and Dependent Types. Electronic Proceedings in Theoretical Computer Science, 43: 14--28, 2010.

[2]

A. Abel and T. Altenkirch. A predicative analysis of structural recursion. JFP, 12 (1): 1--41, 2002.

Digital Library

[3]

H. Applegate. Acyclic models and resolvent functors. PhD thesis, Columbia University, 1965.

[4]

F. Bartels. Generalised coinduction. Mathematical Structures in Computer Science, 13 (2): 321--348, 2003.

Digital Library

[5]

R. S. Bird and O. De Moor. Algebra of Programming, volume 100 of International Series in Computing Science. Prentice Hall, 1997.

Digital Library

[6]

E. W. Dijkstra. Hamming's exercise in SASL. Personal Note EWD792, 1981.

[7]

J. Endrullis, C. Grabmayer, and D. Hendriks. Data-oblivious stream productivity. In Logic for Programming, Artificial Intelligence, and Reasoning, volume 5330 of LNCS, pages 79--96. Springer, 2008.

Digital Library

[8]

E. Giménez. Codifying guarded definitions with recursive schemes. In Types for Proofs and Programs, volume 996 of LNCS, pages 39--59. Springer, 1995.

[9]

R. Hinze. The Bird tree. JFP, 19 (5): 491--508, 2009.

Digital Library

[10]

R. Hinze. Concrete stream calculus - an extended study. JFP, 20 (5-6): 463--535, 2010.

Digital Library

[11]

R. Hinze and D. W. H. James. Proving the Unique-Fixed Point Principle Correct. Technical Report RR-11-03, Department of Computer Science, University of Oxford, 2011.

[12]

J. Hughes, L. Pareto, and A. Sabry. Proving the correctness of reactive systems using sized types. In POPL, pages 410--423. ACM, 1996.

Digital Library

[13]

M. Lenisa, J. Power, and H. Watanabe. Distributivity for endofunctors, pointed and co-pointed endofunctors, monads and comonads. Electronic Notes in Theoretical Computer Science, 33: 230--260, 2000.

[14]

C. McBride and R. Paterson. Applicative programming with effects. JFP, 18 (1): 1--13, 2008.

Digital Library

[15]

K. Mehltretter. Termination checking for a dependently typed language. Master's thesis, LMU Munich, 2007.

[16]

M. Niqui and J. J. M. M. Rutten. Sampling, splitting and merging in coinductive stream calculus. In MPC, volume 6120 of LNCS, pages 310--330. Springer, 2010.

Digital Library

[17]

J. J. M. M. Rutten. Fundamental study: Behavioural differential equations: A coinductive calculus of streams, automata, and power series. Theoretical Computer Science, 308: 1--53, 2003.

Digital Library

[18]

B. A. Sijtsma. On the productivity of recursive list definitions. ACM Trans. Program. Lang. Syst., 11 (4): 633--649, 1989.

Digital Library

[19]

A. Silva and J. J. M. M. Rutten. A coinductive calculus of binary trees. Information and Computation, 208: 578--593, 2010.

Digital Library

[20]

W. Swierstra. Data types à la carte. JFP, 18 (04): 423--436, 2008.

Digital Library

[21]

D. Turi and G. Plotkin. Towards a mathematical operational semantics. In Logic in Computer Science, pages 280--291. IEEE, 1997.

Digital Library

[22]

H. Zantema. Well-definedness of streams by transformation and termination. Logical Methods in Computer Science, 6 (3:21), 2010.

Cited By

Blanchette JPopescu ATraytel D(2015)Foundational extensible corecursion: a proof assistant perspectiveACM SIGPLAN Notices10.1145/2858949.278473250:9(192-204)Online publication date: 29-Aug-2015
https://dl.acm.org/doi/10.1145/2858949.2784732
Blanchette JPopescu ATraytel DFisher KReppy J(2015)Foundational extensible corecursion: a proof assistant perspectiveProceedings of the 20th ACM SIGPLAN International Conference on Functional Programming10.1145/2784731.2784732(192-204)Online publication date: 29-Aug-2015
https://dl.acm.org/doi/10.1145/2784731.2784732
HINZE RWU N(2016)Unifying structured recursion schemesJournal of Functional Programming10.1017/S095679681500025826Online publication date: 3-Feb-2016
https://doi.org/10.1017/S0956796815000258
Show More Cited By

Index Terms

Proving the unique fixed-point principle correct: an adventure with category theory
1. Software and its engineering
  1. Software creation and management
    1. Software verification and validation
      1. Formal software verification
  2. Software organization and properties
    1. Software functional properties
      1. Correctness
      2. Formal methods
2. Theory of computation
  1. Logic
    1. Proof theory
  2. Semantics and reasoning
    1. Program reasoning
      1. Program specifications

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Published In

cover image ACM SIGPLAN Notices

ACM SIGPLAN Notices Volume 46, Issue 9

ICFP '11

September 2011

456 pages

ISSN:0362-1340

EISSN:1558-1160

DOI:10.1145/2034574

Issue’s Table of Contents

ICFP '11: Proceedings of the 16th ACM SIGPLAN international conference on Functional programming
September 2011
470 pages
ISBN:9781450308656
DOI:10.1145/2034773
General Chairs:
Manuel Chakravarty
University of New South Wales, Australia
,
Zhenjiang Hu
National Institute of Informatics, Japan
,
Program Chair:
Olivier Danvy
Aarhus University, Denmark

Copyright © 2011 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 19 September 2011

Published in SIGPLAN Volume 46, Issue 9

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Cited By

Blanchette JPopescu ATraytel D(2015)Foundational extensible corecursion: a proof assistant perspectiveACM SIGPLAN Notices10.1145/2858949.278473250:9(192-204)Online publication date: 29-Aug-2015
https://dl.acm.org/doi/10.1145/2858949.2784732
Blanchette JPopescu ATraytel DFisher KReppy J(2015)Foundational extensible corecursion: a proof assistant perspectiveProceedings of the 20th ACM SIGPLAN International Conference on Functional Programming10.1145/2784731.2784732(192-204)Online publication date: 29-Aug-2015
https://dl.acm.org/doi/10.1145/2784731.2784732
HINZE RWU N(2016)Unifying structured recursion schemesJournal of Functional Programming10.1017/S095679681500025826Online publication date: 3-Feb-2016
https://doi.org/10.1017/S0956796815000258
Hinze RWu NGibbons J(2013)Unifying structured recursion schemesACM SIGPLAN Notices10.1145/2544174.250057848:9(209-220)Online publication date: 25-Sep-2013
https://dl.acm.org/doi/10.1145/2544174.2500578
Gammie P(2013)Synchronous digital circuits as functional programsACM Computing Surveys10.1145/2543581.254358846:2(1-27)Online publication date: 1-Nov-2013
https://dl.acm.org/doi/10.1145/2543581.2543588
Hinze RWu NGibbons JMorrisett GUustalu T(2013)Unifying structured recursion schemesProceedings of the 18th ACM SIGPLAN international conference on Functional programming10.1145/2500365.2500578(209-220)Online publication date: 25-Sep-2013
https://dl.acm.org/doi/10.1145/2500365.2500578
Madlener KSmetsers S(2013)GSOS Formalized in CoqProceedings of the 2013 International Symposium on Theoretical Aspects of Software Engineering10.1109/TASE.2013.34(199-206)Online publication date: 1-Jul-2013
https://dl.acm.org/doi/10.1109/TASE.2013.34
Hinze RJames DHarper TWu NMagalhães JLöh AGarcia R(2012)Sorting with bialgebras and distributive lawsProceedings of the 8th ACM SIGPLAN workshop on Generic programming10.1145/2364394.2364405(69-80)Online publication date: 12-Sep-2012
https://dl.acm.org/doi/10.1145/2364394.2364405

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