Skip to main content
SpringerLink
Log in

Generalized Definite Set Constraints

  • Published:
Constraints Aims and scope Submit manuscript

Abstract

Set constraints (SC) are logical formulae in which atoms are inclusions between set expressions. Those set expressions are built over a signature Σ, variables and various set operators. On a semantical point of view, the set constraints are interpreted over sets of trees built from Σ and the inclusion symbol is interpreted as the subset relation over those sets. By restricting the syntax of those formulae and/or the set of operators that can occur in set expressions, different classes of set constraints are obtained. Several classes have been proposed and studied for some problems such as satisfiability and entailment. Among those classes, we focus in this article on the class of definite SC's introduced by Heintze and Jaffar, and the class of co-definite SC's studied by Charatonik and Podelski. In spite of their name, those two classes are not dual from each other, neither through inclusion inversion nor through complementation. In this article, we propose an extension for each of those two classes by means of an intentional set construction, so called membership expression. A membership expression is an expression {x| Φ(x)}. The formula Φ(x) is a positive first-order formula built from membership atomst in S in which S is a set expression. We name those two classes respectively generalized definite and generalized co-definite set constraints. One of the main point concerning those so-extended classes is that the two generalized classes turn out to be dual through complementation. First, we prove in this article that generalized definite set constraints is a proper extension of the definite class, as it is more expressive in terms of sets of solutions. But we show also that those extensions preserve some main properties of the definite and co-definite class. Hence for instance, as definite set constraints, generalized definite SC's have a least solution whereas the generalized co-definite SC's have a greatest solution, just as co-definite ones. Furthermore, we devise an algorithm based on tree automata that solves the satisfiability problem for generalized definite set constraints. Due to the dualization, the algorithm solves the satisfiability problem for generalized co-definite set constraints as well. This algorithm proves first that for those generalized classes, the satisfiability problem remains DEXPTIME-complete. It provides also a proof for regularity of the least solution of generalized definite constraints and so, by dualization for the greatest solution for the generalized co-definite SC's.

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. Aiken, A., Kozen, D., Vardi, M., and Wimmers, E. (1993). The Complexity of Set Constraints. In Proceedings of Computer Science Logic, pages 1-17.

  2. Aiken, A., Kozen, D., and Wimmers, E. (1995). Decidability of Systems of Set Constraints with Negative Constraints. Information and Computation, 1(122):30-44.

    Google Scholar 

  3. Aiken, A., and Wimmers, E. L. (1992). Solving Systems of Set Constraints. In Proceedings of the 7 th IEEE Symposium on Logic in Computer Science, pages 329-340.

  4. Aiken, A., and Wimmers, E. L. (1993). Type Inclusion Constraints and Type Inference. In Proceedings of the Conference on Functional Programming Languages and Computer Architecture, pages 31-41. ACM Press.

  5. Aiken, A., Wimmers, E. L. and Lakshman, T. K. (1994). Soft typing with conditional types. In Principles of Programming Languages (POPL-94): 21st ACM SIGPLAN-SIGACT Symposium; proceedings, pages 163-173. ACM Press.

  6. Bachmair, L., Ganzinger, H., andWaldmann, U. (1993). Set Constraints are the Monadic Class. In Proceedings of the 8 th IEEE Symposium on Logic in Computer Science, pages 75-83.

  7. Chandra, A., Kozen, D., and Stockmeyer, L. (1981). Alternation. Journal of ACM, 28:114-133.

    Google Scholar 

  8. Charatonik, W. (1998). Set Constraints in Some Equational Theories. Information and Computation, 142(1):40-75.

    Google Scholar 

  9. Charatonik, W., McAllester, D., Niwi´nski, D., Podelski, A., and Walukiewicz, I. (1998). The Horn Mu-Calculus. In Proceedings of the 13th IEEE Symposium on Logic in Computer Science, pages 58-69.

  10. Charatonik, W., and Pacholski, L. (1994). Negative Set Constraints with Equality. In IEEE, editor, Proceedings of the 9 th IEEE Symposium on Logic in Computer Science, pages 128-136.

  11. Charatonik, W., and Pacholski, L. (1994). Set Constraints with Projections are in NEXPTIME. In Proceedings of the 35 th Symposium on Foundations of Computer Science, pages 642-653.

  12. Charatonik,W., and Podelski, A. (1997). Set Constraints with Intersection. In Proceedings of the 12th IEEE Symposium on Logic in Computer Science.

  13. Charatonik, W., and Podelski, A. (1997). Solving Set Constraints for Greatest Models. Technical Report MPI-I-97-2-004, Max-Planck-Institut für Informatik.

  14. Charatonik,W. and Podelski, A. (1998). Co-definite Set Constraints. In Proceedings of the 9th International Conference on Rewriting Techniques and Applications, LNCS.

  15. Devienne, P., Talbot, J-M., and Tison, S. (1997). Set-based Analysis for Logic Programming and Tree Automata. In P. Van Hentenryck, editor, Proceedings of the 4th International Static Analysis Symposium, LNCS 1302, pages 127-140. Springer.

  16. Devienne, P., Talbot, J-M., and Tison, S. (1997). Solving Classes of Set Constraints with Tree Automata. In G. Smolka, editor, Proceedings of the 3rd International Conference on Principles and Practice of Constraint Programming, LNCS 1330, pages 62-76.

  17. Devienne, P., Talbot, J-M., and Tison, S. (1998). Co-definite Set Constraints with Membership Expressions. In Joxan Jaffar, editor, Proceedings of the 1998 Joint International Conference and Symposium on Logic Programming, pages 25-39. MIT-Press.

  18. Frühwirth, T., Shapiro, E., Vardi, M. Y., and Yardeni, E. (1991). Logic Programs as Types for Logic Programs. In Proceedings of the 6th IEEE Symposium on Logic in Computer Science, pages 300-309.

  19. Gécseg, F., and Steinby, M. (1984). Tree Automata. Akadémiai Kiadó, Budapest.

    Google Scholar 

  20. Gilleron, R., Tison, S., and Tommasi, M. (1993). Solving System of Set Constraints using Tree Automata. In Proceedings of the 10th Symposium on Theoretical Aspects of Computer Science, LNCS 665, pages 505-514.

  21. Gilleron, R., Tison, S., and Tommasi, M. (1993). Solving Systems of Set Constraints with Negated Subset Relationships. In Proceedings of the 34th Symposium on Foundations of Computer Science, pages 372-380.

  22. Gilleron, R., Tison, S., and Tommasi, M. (1993). Solving systems of set constraints with negated subset relationships. In Proceedings of the 34th Symp. on Foundations of Computer Science, pages 372-380.

  23. Givan, R., Kozen, D., McAllester, D., and Witty, C. (1996). Tarskian Set constraints. In Proceedings of the 11th IEEE Symposium on Logic in Computer Science, pages 138-147.

  24. Heintze, N. (1992). Practical Aspects of Set Based Analysis. In Proceedings of the Joint International Conference and Symposium on Logic Programming. MIT-Press.

  25. Heintze, N. (1992). Set Based Program Analysis. PhD thesis, Carnegie Mellon University.

  26. Heintze, N. (1994). Set-based Analysis of ML Programs. In Lisp and Functional Programming, pages 306-317. ACM.

  27. Heintze, N., and Jaffar, J. (1990). A Decision Procedure for a Class of Herbrand Set Constraints. In Proceedings of the 5th IEEE Symposium on Logic in Computer Science, pages 42-51.

  28. Heintze, N., and Jaffar, J. (1990). A Finite Presentation Theorem for Approximating Logic Programs. In Proceedings of the 17th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pages 197-209.

  29. Kozen, D. (1998). Set Constraints and Logic Programming. Information and Computation, 142(1):2-25.

    Google Scholar 

  30. Müller, M., and Niehren, J. (1997). Entailment of Set Constraints is not Feasible. Technical report, Universität des Saarlandes, Programming Systems Lab. Available at http://www.ps.unisb.de/mmueller/papers/conp.ps.Z.

  31. Müller, M., Niehren, J., and Podelski, A. (1997). Inclusion Constraints over Non-Empty Sets of Trees. In Proceedings of 7th International Joint Conference CAAP/FASE-(TAPSOFT'97), LNCS 1214, pages 345-356.

  32. Müller, M. (1998). Set-based Failure Diagnosis for Concurrent Constraint Programming. PhD thesis, Universität des Saarlandes, Technische Fakultät, D-66041 Saarbrücken.

    Google Scholar 

  33. Niwínski, D. (1997). Fixed Point Characterization of Infinite Behaviour of Finite State System. Theoretical Computer Science, 189(1-2):1-69.

    Google Scholar 

  34. Post, E. L. (1944). Recursively Enumerable Sets of Positive Integers and their Decision Problems. Bulletion of the American Mathematical Society, 50:284-316.

    Google Scholar 

  35. Slutzki, G. (1985). Alternating Tree Automata. Theoretical Computer Science, 41(1):305-318.

    Google Scholar 

  36. Stefansson, K. (1994). Systems of Set Constraints with Negative Constraints are NEXPTIME-Complete. In Proceedings of the 9 th IEEE Symposium on Logic in Computer Science.

  37. Talbot, J-M. (1998). Contraintes Ensemblistes Définies et Co-définies: Extensions et Applications. PhD thesis, Université des Sciences et Technologies de Lille.

  38. Tommasi, M. (1994). Automates et Contraintes Ensemblistes. PhD thesis, Université des Sciences et Technologies de Lille.

Download references

Author information

Authors and Affiliations

  1. Max-Planck-Institut für Informatik, Im Stadtwald, 66213, Saarbrücken -, GERMANY

    Jean-Marc Talbot

  2. Laboratoire d'Informatique Fondamentale de Lille, Université des Sciences et Technologies de Lille, 59655, Villeneuve d'Ascq Cedex -, FRANCE

    Philippe Devienne & Sophie Tison

Authors
  1. Jean-Marc Talbot
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Philippe Devienne
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sophie Tison
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Talbot, JM., Devienne, P. & Tison, S. Generalized Definite Set Constraints. Constraints 5, 161–202 (2000). https://doi.org/10.1023/A:1009826603139

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1009826603139

Search

Navigation