Skip to main content
SpringerLink
Log in

In praise of algebra

  • Original Article
  • Published:
Formal Aspects of Computing

Abstract

We survey the well-known algebraic laws of sequential programming, and extend them with some less familiar laws for concurrent programming. We give an algebraic definition of the Hoare triple, and algebraic proofs of all the relevant laws for concurrent separation logic. We give the provable concurrency laws for Milner transitions, for the Back/Morgan refinement calculus, and for Dijkstra’s weakest preconditions. We end with a section in praise of algebra, of which Carroll Morgan is such a master.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Back R-J (1978) On the correctness of refinement steps in program development. PhD thesis, A^bo Akademi, Department of Computer Science. Helsinki, Finland. Report A-1978-4

  2. Back R-J (1979) On the notion of correct refinement of programs. In: 5th Scandinavian logic symposium, Aalborg, Denmark. Aalborg University Press, Denmark

  3. Back R-J (1980) Correctness preserving program refinements: proof theory and applications. Mathematical center tracts, vol 131. Mathematical Centre, Amsterdam

    Google Scholar 

  4. Back R-J (1981) Proving total correctness of nondeterministic programs in infinitary logic. Acta Informatica 15: 233–249

    Article  MathSciNet  MATH  Google Scholar 

  5. Böhme S, Nipkow T (2010) Sledgehammer: judgement day. In: Proceedings of the 5th international conference on automated reasoning, IJCAR’10. Springer, Berlin, pp 107–121

  6. Dijkstra EW (1976) A discipline of programming. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  7. Foster S, Struth G, Weber T (2011) Automated engineering of relational and algebraic methods in Isabelle/HOL. In: Proceedings of the 12th international conference on relational and algebraic methods in computer science, RAMICS’11. Springer, Berlin, pp 52–67

  8. Hoare CAR, Hayes IJ, Jifeng He, Morgan CC, Roscoe AW, Sanders JW, Sorensen IH, Spivey JM, Sufrin BA (1987) Laws of programming. Commun ACM 30: 672–686

    Article  MATH  Google Scholar 

  9. Hoare CAR, Hussain A, Möller B, O’Hearn P, Petersen R, Struth G (2011) On locality and the exchange law for concurrent processes. In: Joost-Pieter K, Barbara König (eds) CONCUR 2011 concurrency theory. Lecture Notes in Computer Science, vol 6901. Springer, Berlin, pp 250–264

    Chapter  Google Scholar 

  10. Hoare C, Möller B, Struth G, Wehrman I (2009) Concurrent Kleene algebra. In: Mario B, Gianluigi Z (eds) CONCUR 2009—concurrency theory. Lecture Notes in Computer Science, vol 5710. Springer, Berlin, pp 399–414

    Google Scholar 

  11. Hoare T, Möller B, Struth G, Wehrman I (2011) Concurrent Kleene algebra and its foundations. J Logic Algebraic Program, 80(6): 266–296

    Article  MATH  Google Scholar 

  12. Hoare CAR (1969) An axiomatic basis for computer programming. Commun ACM 12: 576–580

    Article  MATH  Google Scholar 

  13. Hoare CAR, Wehrman I, O’Hearn PW (2009) Graphical models of separation logic. In: Manfred B, Wassiou S, Tony H (eds) Proceedings of the 2008 Marktoberdorf Summer School on engineering methods and tools for software safety and security. IOS Press, Amsterdam

  14. Kozen D (1994) A completeness theorem for Kleene algebras and thealgebra of regular events. Inf Comput 110: 366–390

    Article  MathSciNet  MATH  Google Scholar 

  15. Milner R (1980) A calculus of communicating systems. Lecture Notes in Computer Science, vol 92. Springer, Berlin

    Google Scholar 

  16. Morgan C (1988) The specification statement. ACM Trans Program Lang Syst 10: 403–419

    Article  MATH  Google Scholar 

  17. Morgan C (1994) Programming from specifications, 2nd edn. Prentice Hall International (UK) Ltd, Hertfordshire

    MATH  Google Scholar 

  18. Nipkow T, Wenzel M, Paulson LC (2002) Isabelle/HOL: a proof assistant for higher-order logic. Springer, Berlin

    MATH  Google Scholar 

  19. O’Hearn P (2004) Resources, concurrency and local reasoning. In: Philippa G, Nobuko Y (eds) CONCUR 2004—concurrency theory. Lecture Notes in Computer Science. Springer, Berlin, pp 49–67

    Google Scholar 

  20. O’Hearn PW, Reynolds JC, Yang H (2001) Local reasoning about programs that alter data structures. In: CSL ’01, of LNCS, vol 2142. Springer, Berlin, pp 1–19

  21. Isabelle/HOL (2011) Proofs. http://se.inf.ethz.ch/people/vanstaden/InPraiseOfAlgebra.thy,

  22. Wehrman I, Hoare CAR, O’Hearn PW (2009) Graphical models of separation logic. Inf Process Lett 109(17): 1001–1004

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Microsoft Research, Cambridge, UK

    Tony Hoare

  2. ETH Zurich, Zurich, Switzerland

    Stephan van Staden

Authors
  1. Tony Hoare
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stephan van Staden
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stephan van Staden.

Additional information

Peter Höfner, Robert van Glabbeek and Ian Hayes

Dedication: to Carroll Morgan, whose use of algebra has so often given delight to so many of his audience.

Van Staden was supported by ETH Research Grant ETH-15 10-1.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hoare, T., van Staden, S. In praise of algebra. Form Asp Comp 24, 423–431 (2012). https://doi.org/10.1007/s00165-012-0249-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00165-012-0249-0

Keywords

Search

Navigation