Skip to main content
SpringerLink
Log in

Optimization in temporal qualitative constraint networks

  • Original Article
  • Published:
Acta Informatica Aims and scope Submit manuscript

Abstract

Various formalisms for representing and reasoning about temporal information with qualitative constraints have been studied in the past three decades. The most known are definitely the Point Algebra \((\mathsf {PA})\) and the Interval Algebra \(({\mathsf {IA}})\) proposed by Allen. In this paper, for both calculi, we study a problem that we call the minimal consistency problem \((\mathsf {MinCons})\). Given a temporal qualitative constraint network \((\mathsf {TQCN})\) and a positive integer \(k\), this problem consists in deciding whether or not this \(\mathsf {TQCN}\) admits a solution using at most \(k\) distinct points on the line.We show that \(\mathsf {MinCons}\) for \(\mathsf {PA}\) can be encoded into the finitary versions of Gödel logic. Furthermore, we prove that the \(\mathsf {MinCons}\) problem is \(\mathsf {NP}\)-complete for both \(\mathsf {PA}\) and \({\mathsf {IA}}\), in the general case. However, we show that for \(\mathsf {TQCN}\)s defined on the convex relations, \(\mathsf {MinCons}\) is polynomial. For such \(\mathsf {TQCN}\)s, we give a polynomial method that allows one to obtain compact scenarios.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Notes

  1. Given a lattice \((E,\le )\), an interval is either the empty set or a set \(\{e \in E: {\textit{min}} \le e \le {\textit{max}}\}\) for some \({\textit{min}}, {\textit{max}} \in E\) with \({\textit{min}}\le {\textit{max}}\).

References

  1. Allen, J.F.: An interval-based representation of temporal knowledge. In: IJCAI (1981)

  2. Allen, J.F.: Maintaining knowledge about temporal intervals. Commun. ACM 26(11), 832–843 (1983)

    Article  MATH  Google Scholar 

  3. Amaneddine, N., Condotta, J.-F., Sioutis, M.: Efficient approach to solve the minimal labeling problem of temporal and spatial qualitative constraints. In: Proceedings of the 23rd International Joint Conference on Artificial Intelligence (IJCAI’13), Beijing, China, 3–9 Aug (2013)

  4. Avron, A.: A tableau system for Gödel–Dummett logic based on a hypersequent calculus. In: TABLEAUX, pp. 98–111 (2000)

  5. Baaz, M., Ciabattoni, A., Fermüller, C.G.: Cut elimination for first order Gödel logic by hyperclause resolution. In: LPAR, pp. 451–466 (2008)

  6. Beeson, M.J.: Foundations of Constructive Mathematics. Modern Surveys in Mathematics. Springer, Berlin (1984)

    Google Scholar 

  7. Boutilier, C.: Toward a logic for qualitative decision theory. In: KR’94, pp. 75–86 (1994)

  8. Condotta, J.-F., Kaci, S.: Compiling preference queries in qualitative constraint problems. (à paraitre). In: Proceedings of the 26th International FLAIRS Conference (FLAIRS’13) (to appear) (2013)

  9. Drakengren, T., Jonsson, P.: A complete classification of tractability in Allen’s algebra relative to subsets of basic relations. Artif. Intell. 106(2), 205–219 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dummett, M.: A propositional calculus with denumerable matrix. J. Symb. Log. 24(2), 97–106 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  11. Garey, M.R., Johnson, D.S.: Computers and Intractibility, A Guide to the Theory of NP-Completeness. Freeman, New York (1979)

    MATH  Google Scholar 

  12. Hajek, P.: The Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)

    Book  MATH  Google Scholar 

  13. Huang, J., Li, J.J., Renz, J.: Decomposition and tractability in qualitative spatial and temporal reasoning. Artif. Intell. 195, 140–164 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kaci, S., van der Torre, L.: Reasoning with various kinds of preferences: logic, non-monotonicity and algorithms. Ann. Oper. Res. 163(1), 89–114 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Larchey-Wendling, D.: Counter-model search in Gödel–dummett logics. In: IJCAR, pp. 274–288 (2004)

  16. Larchey-Wendling, D.: Bounding resource consumption with Gödel–Dummett logics. In: LPAR, pp. 682–696 (2005)

  17. Ligozat, G.F.: On generalized interval calculi. In: Proceedings of the Ninth National Conference on Artificial Intelligence (AAAI’91), pp. 234–240. American Association for Artificial Intelligence (July 1991)

  18. Mackworth, A.K.: Consistency in networks of relations. Artif. Intell. 8, 99–118 (1977)

    Article  MATH  Google Scholar 

  19. Nebel, B.: Solving hard qualitative temporal reasoning problems: evaluating the efficienty of using the ORD-Horn class. In: ECAI (1996)

  20. Nebel, B., Bürckert, H.-J.: Reasoning about temporal relations: a maximal tractable subclass of Allen’s interval algebra. JACM 42, 43–66 (1995)

    Article  MATH  Google Scholar 

  21. Nökel, K.: Temporally distributed symptoms in technical diagnosis. LNCS 517, 1–184 (1991)

    MATH  Google Scholar 

  22. Nökel, K.: Temporally Distributed Symptoms in Technical Diagnosis, vol. 517 of Lecture Notes in Computer Science. Springer, Berlin(1991)

  23. Renz, J., Ligozat, G.: Weak composition for qualitative spatial and temporal reasoning. In: Proceedings of the 11th International Conference Principles and Practice of Constraint Programming (CP’05), Sitges, Spain, vol. 3709 of Lecture Notes in Computer Science, pp. 534–548 (2005)

  24. The, A.N., Tsoukiàs, A.: Numerical representation of PQI interval orders. Discrete Appl. Math. 147(1), 125–146 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Troelstra, A.S., van Dalen, D.: Constructivism in Mathematics, an Introduction. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1988)

    MATH  Google Scholar 

  26. Van Beek, P., Cohen, R.: Exact and approximate reasoning about temporal relations. Comput. Intell. 6, 133–144 (1990)

    Article  Google Scholar 

  27. Vilain, M., Kautz, H.: Constraint Propagation Algorithms for Temporal Reasoning. In: Proceedings of the Fifth National Conference on Artificial Intelligence (AAAI’86), pp. 377–382 (1986)

  28. Vilain, M., Kautz, H., van Beek, P.: Constraint propagation algorithms for temporal reasoning: a revised report. In: Weld, D.S., de Kleer, J. (eds.) Readings in Qualitative Reasoning about Physical Systems, pp. 373–381. Morgan-Kaufman, San Mateo, CA (1989)

Download references

Author information

Authors and Affiliations

  1. CRIL CNRS, UMR 8188, Université Lille-Nord de France, Lens, France

    Jean-François Condotta & Yakoub Salhi

  2. LIRMM CNRS, UMR 5506, Université Montpellier, Montpellier, France

    Souhila Kaci

Authors
  1. Jean-François Condotta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Souhila Kaci
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yakoub Salhi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Souhila Kaci.

Appendix

Appendix

See Table 1.

Table 1 The table of composition of \({\mathsf {IA}}\) with \(\alpha =\{p,m,o\}, \beta =\{d,o,s\}, \gamma =\{d,oi,f\}, \delta =\{eq,f,fi\}, \theta =\{eq,s,si\}, \epsilon =\{p,m,o,di,fi\}, \phi =\{p,m,o,d,s\}, \uplambda =\{d,di,s,si,f,fi,o,oi,eq\}\)
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Condotta, JF., Kaci, S. & Salhi, Y. Optimization in temporal qualitative constraint networks. Acta Informatica 53, 149–170 (2016). https://doi.org/10.1007/s00236-015-0228-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00236-015-0228-z

Search

Navigation