Skip to main content
SpringerLink
Log in

A Hoare logic for linear systems

  • Original Article
  • Published:
Formal Aspects of Computing

Abstract

We consider reasoning about linear systems expressed as block diagrams that give a graphical representation of a system of differential equations or recurrence equations. We use the notion of additive relation borrowed from homological algebra to give a convenient framework in which all diagrams have a semantic value. We give a sound system of Hoare-style rules for the block diagram constructors that singles out a tractable subset of the block diagram language in which all diagrams represent total functions. We show these rules in action on some simple examples from a variety of applications domains.

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. Abrial J-R, Börger E, Langmaack H (eds) (1996) Formal methods for industrial applications, specifying and programming the steam boiler control. Lecture notes in computer science, vol 1165. Springer, Berlin

  2. Arthan R, Caseley P, O’Halloran C, Smith A (2000) ClawZ: control laws in Z. In: 3rd international conference on formal engineering methods (ICFEM 2000)

  3. Abramsky S, Gay SJ, Nagarajan R (1995) Specification structures and propositions-as-types for concurrency. In: Moller F, Birtwistle G (eds) Logics for concurrency: structure vs. automata—proceedings of the 8th Banff higher order workshop. Springer, Berlin

  4. Arthan R, Martin U, Mathiesen EA, Oliva P (2007) Reasoning about linear systems. In: 5th IEEE international conference on software engineering and formal methods SEFM 2007, pp 123–134. IEEE Press

  5. Arthan R, Martin U, Mathiesen EA, Oliva P (2009) A general framework for sound and complete Hoare logics. ACM Trans Comput Logic 11(1): 1–31

    Article  MathSciNet  Google Scholar 

  6. Bloom SL, Ésik Z (1993) Iteration theories: the equational logic of iterative processes. Springer, Berlin

    MATH  Google Scholar 

  7. Boulton RJ, Hardy R, Martin U (2003) A Hoare logic for single-input single-output continuous-time control systems. In: Proceedings 6th international workshop on hybrid systems, computation and control. LNCS, vol 2623, pp 113–125. Springer, Berlin

  8. Cavalcanti ALC, Clayton P, O’Halloran C (2005) Control law diagrams in Circus. In: Fitzgerald J, Hayes IJ, Tarlecki A (eds) FM 2005: formal methods. Lecture notes in computer science, vol 3582, pp 253–268. Springer, Berlin

  9. Gottliebsen H, Kelsey T, Martin U (2005) Hidden verification for computational mathematics. J Symb Comput 39: 539–567

    Article  MathSciNet  MATH  Google Scholar 

  10. Hoare CAR (1969) An axiomatic basis for computer programming. Commun ACM 12(10)

  11. Haghverdi E, Scott P (2005) Towards a typed geometry of interaction. In: Ong L (ed) CSL’05. LNCS, vol 3634, pp 216–231. Springer, Berlin

  12. Jones CB (2003) The early search for tractable ways of reasoning about programs. Ann Hist Comput 25(2)

  13. Joyal A, Street R, Verity D (1996) Traced monoidal categories. Math Proc Cambridge Philos Soc 119: 447–468

    Article  MathSciNet  MATH  Google Scholar 

  14. Kozen D (1997) Kleene algebra with tests. ACM Trans Program Lang Syst 19(3): 427–443

    Article  Google Scholar 

  15. Leitner F (2008) Evaluation of the Matlab Simulink Design Verifier versus the model checker SPIN. Technical Report soft-08-05, University of Konstanz

  16. Mac Lane S (1975) Homology. In: Der Grundlehren der mathematischen Wissenschaften, vol 114. Springer, Berlin

  17. Martin U, Mathiesen EA, Oliva P (2006) Abstract Hoare logic. In: Proceedings of CSL’2006. LNCS, vol 4207, pp 501–515

  18. Platzer A (2010) Logical analysis of hybrid systems: proving theorems for complex dynamics. Springer, Heidelberg

    Book  MATH  Google Scholar 

  19. Polderman JW, Willems JC (1998) Introduction to mathematical systems theory: a behavioral approach. Springer, New York

    Book  Google Scholar 

  20. Rutten JJMM (2005) A tutorial on coinductive stream calculus and signal flow graphs. Theor Comput Sci 343(3): 443–481

    Article  MathSciNet  MATH  Google Scholar 

  21. Sheeran M (2005) Hardware design and functional programming: a perfect match. J Univ Comput Sci 11(7): 1135–1158

    Google Scholar 

  22. Sontag ED (1998) Mathematical control theory: deterministic finite dimensional systems, 2nd edn. Springer, New York

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Electronic Engineering and Computer Science, Queen Mary, University of London, London, E1 4NS, UK

    Rob Arthan, Ursula Martin & Paulo Oliva

Authors
  1. Rob Arthan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ursula Martin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Paulo Oliva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rob Arthan.

Additional information

Jonathan P. Bowen, Michael Butler, Mike Hinchey, Steve Reeves and Jim Woodcock

Rights and permissions

Reprints and permissions

About this article

Cite this article

Arthan, R., Martin, U. & Oliva, P. A Hoare logic for linear systems. Form Asp Comp 25, 345–363 (2013). https://doi.org/10.1007/s00165-011-0180-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00165-011-0180-9

Keywords

Search

Navigation