Modal Design Algebra

  • Walter Guttmann
  • Bernhard Möller
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4010)


We give an algebraic model of the designs of UTP based on a variant of modal semirings, hence generalising the original relational model. This is intended to exhibit more clearly the algebraic principles behind UTP and to provide deeper insight into the general properties of designs, the program and specification operators, and refinement. Moreover, we set up a formal connection with general and total correctness of programs as discussed by a number of authors. Finally we show that the designs form a left semiring and even a Kleene and omega algebra. This is used to calculate closed expressions for the least and greatest fixed-point semantics of the demonic while loop that are simpler than the ones obtained from standard UTP theory and previous algebraic approaches.


Boolean Algebra Great Element Formal Connection Galois Connection Order Isomorphism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Backhouse, R.C., van der Woude, J.: Demonic operators and monotype factors. Mathematical Structures in Computer Science 3, 417–433 (1993)CrossRefMATHGoogle Scholar
  2. 2.
    Berghammer, R., Zierer, H.: Relational algebraic semantics of deterministic and non-deterministic programs. Theoretical Computer Science 43, 123–147 (1986)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Broy, M., Gnatz, R., Wirsing, M.: Semantics of nondeterministic and non-continuous constructs. In: Gerhart, S.L., Pair, C., Pepper, P.A., Wössner, H., Dijkstra, E.W., Guttag, J.V., Owicki, S.S., Partsch, H., Bauer, F.L., Gries, D., Griffiths, M., Horning, J.J., Wirsing, M. (eds.) Program Construction. LNCS, vol. 69, pp. 553–592. Springer, Heidelberg (1979)CrossRefGoogle Scholar
  4. 4.
    Cohen, E.: Separation and reduction. In: Backhouse, R., Oliveira, J.N. (eds.) MPC 2000. LNCS, vol. 1837, pp. 45–59. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  5. 5.
    Desharnais, J., Belkhiter, N., Sghaier, S.B.M., Tchier, F., Jaoua, A., Mili, A., Zaguia, N.: Embedding a demonic semilattice in a relation algebra. Theoretical Computer Science 149, 333–360 (1995)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Desharnais, J., Mili, A., Nguyen, T.T.: Refinement and demonic semantics. In: Brink, C., Kahl, W., Schmidt, G. (eds.) Relational methods in computer science, ch. 11, pp. 166–183. Springer, Heidelberg (1997)Google Scholar
  7. 7.
    Desharnais, J., Möller, B., Struth, G.: Kleene algebra with domain. ACM TOCL (to appear)Google Scholar
  8. 8.
    Desharnais, J., Möller, B., Tchier, F.: Kleene under a modal demonic star. Journal of Logic and Algebraic Programming (Special Issue on Relation Algebra and Kleene Algebra) 66(2), 127–160 (2006)MATHGoogle Scholar
  9. 9.
    Doornbos, H.: A relational model of programs without the restriction to Egli-Milner-monotone constructs. In: Olderog, E.-R. (ed.) Programming concepts, methods and calculi, pp. 363–382. North-Holland, Amsterdam (1994)Google Scholar
  10. 10.
    Dunne, S.: Recasting Hoare and He’s unifying theory of programs in the context of general correctness. In: Butterfield, A., Strong, G., Pahl, C. (eds.) 5th Irish Workshop on Formal Methods. EWiC, The British Computer Society (2001)Google Scholar
  11. 11.
    Guttmann, W.: Non-termination in Unifying Theories of Programming. In: MacCaull, W., Winter, M., Düntsch, I. (eds.) RelMiCS 2005. LNCS, vol. 3929, pp. 108–120. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    Guttmann, W., Möller, B.: Modal design algebra. Institut für Informatik, Universität Augsburg, Report 2005-15Google Scholar
  13. 13.
    Hoare, C.A.R., He, J.: Unifying theories of programming. Prentice-Hall, Englewood Cliffs (1998)Google Scholar
  14. 14.
    Kozen, D.: A completeness theorem for Kleene algebras and the algebra of regular events. Information and Computation 110, 366–390 (1994)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Kozen, D.: Kleene algebra with tests. ACM TOPLAS 19, 427–443 (1997)CrossRefGoogle Scholar
  16. 16.
    Möller, B.: Lazy Kleene algebra. In: Kozen, D. (ed.) MPC 2004. LNCS, vol. 3125, pp. 252–273. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  17. 17.
    Möller, B., Struth, G.: Modal Kleene Algebra and Partial Correctness. In: Rattray, C., Maharaj, S., Shankland, C. (eds.) AMAST 2004. LNCS, vol. 3116, pp. 379–393. Springer, Heidelberg (2004); Revised and extended version: Möller, B., Struth, G.: Algebras of modal operators and partial correctness. Theoretical Computer Science 351, 221–239 (2006)CrossRefGoogle Scholar
  18. 18.
    Möller, B., Struth, G.: wp is wlp. In: MacCaull, W., Winter, M., Düntsch, I. (eds.) RelMiCS 2005. LNCS, vol. 3929, pp. 200–211. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  19. 19.
    Nelson, G.: A generalization of Dijkstra’s calculus. ACM TOPLAS 11, 517–561 (1989)CrossRefGoogle Scholar
  20. 20.
    Nguyen, T.T.: A relational model of nondeterministic programs. International J. Foundations Comp. Sci. 2, 101–131 (1991)CrossRefMATHGoogle Scholar
  21. 21.
    Parnas, D.: A generalized control structure and its formal definition. Commun. ACM 26, 572–581 (1983)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Walter Guttmann
    • 1
  • Bernhard Möller
    • 2
  1. 1.Abteilung Programmiermethodik und Compilerbau, Fakultät für InformatikUniversität UlmUlmGermany
  2. 2.Institut für InformatikUniversität AugsburgAugsburgGermany

Personalised recommendations