Advertisement

A Semiring Approach to Equivalences, Bisimulations and Control

  • Roland Glück
  • Bernhard Möller
  • Michel Sintzoff
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5827)

Abstract

Equivalences, partitions and (bi)simulations are usually tackled using concrete relations. There are only few treatments by abstract relation algebra or category theory. We give an approach based on the theory of semirings and quantales. This allows applying the results directly to structures such as path and tree algebras which is not as straightforward in the other approaches mentioned. Also, the amount of higher-order formulations used is low and only a one-sorted algebra is used. This makes the theory suitable for fully automated first-order proof systems. As a small application we show how to use the algebra to construct a simple control policy for infinite-state transition systems.

Keywords

Equivalence Class Control Policy Inverse Semigroup Control Objective Relation Algebra 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baier, C., Katoen, J.-P.: Principles of Model Checking. MIT Press, Cambridge (2008)zbMATHGoogle Scholar
  2. 2.
    Birkhoff, G.: Lattice Theory, 3rd edn., vol. XXV. Colloquium Publications, American Mathematical Society (1967)Google Scholar
  3. 3.
    Carré, B.: Graphs and Networks. Oxford Univ. Press, Oxford (1979)zbMATHGoogle Scholar
  4. 4.
    Desharnais, J., Möller, B., Struth, G.: Kleene algebra with domain. ACM Transactions on Computational Logic 7, 798–833 (2006)MathSciNetGoogle Scholar
  5. 5.
    Glück, R., Möller, B.: Circulations, fuzzy relations and semirings. In: Audebaud, P., Paulin-Mohring, C. (eds.) MPC 2008. LNCS, vol. 5133, pp. 134–152. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Kawahara, Y.: On the cardinality of relations. In: Schmidt, R.A. (ed.) RelMiCS/AKA 2006. LNCS, vol. 4136, pp. 251–265. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Manes, E., Benson, D.: The inverse semigroup of a sum-ordered semiring. Semigroup Forum 31, 129–152 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Pous, D.: Complete lattices and up-to techniques. In: Shao, Z. (ed.) APLAS 2007. LNCS, vol. 4807, pp. 351–366. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Schmidt, G., Ströhlein, T.: Relations and Graphs: Discrete Mathematics for Computer Scientists. Springer, Heidelberg (1993)zbMATHGoogle Scholar
  10. 10.
    Sintzoff, M.: Synthesis of optimal control policies for some infinite-state transition systems. In: Audebaud, P., Paulin-Mohring, C. (eds.) MPC 2008. LNCS, vol. 5133, pp. 336–359. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  11. 11.
    Winter, M.: A relation-algebraic theory of bisimulations. Fundam. Inf. 83(4), 429–449 (2008)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Roland Glück
    • 1
  • Bernhard Möller
    • 1
  • Michel Sintzoff
    • 2
  1. 1.Universität Augsburg 
  2. 2.Université catholique de Louvain 

Personalised recommendations