Confluence Modulo Equivalence with Invariants in Constraint Handling Rules

  • Daniel GallEmail author
  • Thom Frühwirth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10818)


Confluence denotes the property of a state transition system that states can be rewritten in more than one way yielding the same result. Although it is a desirable property, confluence is often too strict in practical applications because it also considers states that can never be reached in practice. Additionally, sometimes states that have the same semantics in the practical context are considered as different states due to different syntactic representations. By introducing suitable invariants and equivalence relations on the states, programs may have the property to be confluent modulo the equivalence relation w.r.t. the invariant which often is desirable in practice.

In this paper, a sufficient and necessary criterion for confluence modulo equivalence w.r.t. an invariant for Constraint Handling Rules (CHR) is presented. It is the first approach that covers invariant-based confluence modulo equivalence for the de facto standard semantics of CHR. There is a trade-off between practical applicability and the simplicity of proving a confluence property. Therefore, a better manageable subset of equivalence relations has been identified that allows for the proposed confluence criterion and simplifies the confluence proofs by using well established CHR analysis methods.



The authors would like to thank Henning Christiansen and Maja H. Kirkeby for the valuable discussions and ideas for future work. Additionally, the authors would like to thank the anonymous reviewers for their valuable comments.


  1. 1.
    Frühwirth, T.: Constraint Handling Rules. Cambridge University Press, New York (2009)CrossRefGoogle Scholar
  2. 2.
    Abdennadher, S., Frühwirth, T., Meuss, H.: On confluence of Constraint Handling Rules. In: Freuder, E.C. (ed.) CP 1996. LNCS, vol. 1118, pp. 1–15. Springer, Heidelberg (1996). Scholar
  3. 3.
    Abdennadher, S.: Operational semantics and confluence of constraint propagation rules. In: Smolka, G. (ed.) CP 1997. LNCS, vol. 1330, pp. 252–266. Springer, Heidelberg (1997). Scholar
  4. 4.
    Abdennadher, S., Frühwirth, T., Meuss, H.: Confluence and semantics of constraint simplification rules. Constraints 4(2), 133–165 (1999)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Duck, G.J., Stuckey, P.J., Sulzmann, M.: Observable confluence for Constraint Handling Rules. In: Dahl, V., Niemelä, I. (eds.) ICLP 2007. LNCS, vol. 4670, pp. 224–239. Springer, Heidelberg (2007). Scholar
  6. 6.
    Duck, G.J., Stuckey, P.J., Sulzmann, M.: Observable confluence for Constraint Handling Rules. In: Schrijvers, T., Frühwirth, T. (eds.) CHR 2006. K.U. Leuven, Department of Computer Science, Technical report CW 452, pp. 61–76, July 2006Google Scholar
  7. 7.
    Raiser, F.: Graph transformation systems in constraint handling rules: improved methods for program analysis. Ph.D. thesis, Ulm University, Germany (2010)Google Scholar
  8. 8.
    Christiansen, H., Kirkeby, M.H.: Confluence modulo equivalence in Constraint Handling Rules. In: Proietti, M., Seki, H. (eds.) LOPSTR 2014. LNCS, vol. 8981, pp. 41–58. Springer, Cham (2015). Scholar
  9. 9.
    Christiansen, H., Kirkeby, M.H.: On proving confluence modulo equivalence for Constraint Handling Rules. Formal Aspects Comput. 29(1), 57–95 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Raiser, F., Betz, H., Frühwirth, T.: Equivalence of CHR states revisited. In: Raiser, F., Sneyers, J. (eds.) 6th International Workshop on Constraint Handling Rules (CHR), KULCW, Technical report CW 555, pp. 33–48, July 2009Google Scholar
  11. 11.
    Huet, G.: Confluent reductions: abstract properties and applications to term rewriting systems. J. ACM (JACM) 27(4), 797–821 (1980)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ohlebusch, E.: Church-Rosser theorems for abstract reduction modulo an equivalence relation. In: Nipkow, T. (ed.) RTA 1998. LNCS, vol. 1379, pp. 17–31. Springer, Heidelberg (1998). Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Software Engineering and Programming LanguagesUlm UniversityUlmGermany

Personalised recommendations