Skip to main content
SpringerLink
Log in

Orders on intervals over partially ordered sets: extending Allen’s algebra and interval graph results

  • Focus
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

To make a decision, we need to compare the values of quantities. In many practical situations, we know the values with interval uncertainty. In such situations, we need to compare intervals. Allen’s algebra describes all possible relations between intervals on the real line which are generated by the ordering of endpoints; ordering relations between such intervals have also been well studied. In this paper, we extend this description to intervals in an arbitrary partially ordered set (poset). In particular, we explicitly describe ordering relations between intervals that generalize relation between points. As auxiliary results, we provide a logical interpretation of the relation between intervals, and extend the results about interval graphs to intervals over posets.

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

Similar content being viewed by others

Notes

  1. Note that this is almost always true, in that endpoint equality also has to be taken into account, yielding intervals which are equal at one endpoint comparable in both orders.

  2. The authors are thankful to an anonymous referee for this interesting suggestion.

  3. The authors are thankful to an anonymous referee for this interesting suggestion.

References

  • Allen JF (1983) Maintaining knowledge about temporal intervals. Commun ACM 26(11):832–843

    Article  MATH  Google Scholar 

  • Cormen TH, Leiserson CE, Rivest RL, Stein C (2009) Introduction to algorithms, MIT Press, Cambridge

  • Davey BA, Priestly HA (1990) Introduction to lattices and order, Cambridge University Press, Cambridge

  • Feynman RR, Leighton RR, Sands M (2005) The Feynman Lectures on Physics, Addison Wesley, Boston

  • Fishburn PC (1985) Interval orders and interval graphs: a study of partially ordered sets, Wiley, New York

  • Kosheleva OM, Kreinovich V, Nguyen HT, Bouchon-Meunier B (1998) How to describe partially ordered preferences: mathematical foundations, In: Proceedings Vietnam-Japan bilateral symposium on fuzzy systems and applications VJFUZZY98, HaLong Bay, Vietnam, pp 269–278

  • Kronheimer EH, Penrose R (1967) On the structure of causal spaces. Proc Cambr Phil Soc 63(2):481–501

    Article  MathSciNet  MATH  Google Scholar 

  • Mandoiu I, Zelikovsky A (2008) Bioinformatics algorithms: techniques and applications. Wiley, New York

  • Misner CW, Thorne KS, Wheeler JA (1973) Gravitation. Freeman, San Francisco

  • Moore RM, Kearfott RB, Cloud MJ (2009) Introduction to interval analysis. SIAM Press, Philadelphia

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

    Article  MATH  Google Scholar 

  • Papadakis SE, Kaburlasos VG (2010) Piecewise-linear approximation of non-linear models based on probabilistically/possibilistically interpreted interval numbers (INs). Info Sci 180:5060–5076

    Article  MATH  Google Scholar 

  • Tanenbaum PJ (1996) Simultaneous represention of interval and interval-containment orders. Order 13:339–350

    Article  MathSciNet  MATH  Google Scholar 

  • Tanenbaum PJ, de la Mora C, Wojciechowski P, Kosheleva O, Kreinovich V, Starks SA, Kuzminykh A (2004) Robust methodology for characterizing system response to damage: approach based on partial order. In: Lirkov I, Margenov S, Wasniewski J, Yalamov P (eds), Large-scale scientific computing, Springer Lecture Notes in Computer Science. Springer, Berlin 2907:276–283

  • Xu Y, Liu J, Zhong X, Chen S (2012) Lattice-valued matrix game with mixed strategies for intelligent decision support. J Know Based Syst 32:56–64

    Article  Google Scholar 

  • Zapata F, Kreinovich V (2012) Reconstructing an open order from its closure, with applications to space-time physics and to logic. Studia Logica 100(1–2):419–435

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was partly supported by a CONACyT scholarship, by the National Science Foundation grants HRD-0734825 and DUE-0926721, and by Grant 1 T36 GM078000-01 from the National Institutes of Health. The authors are thankful to all the participants of the Dagstuhl 2011 seminar Uncertainty Modeling and Analysis with Intervals: Foundations, Tools, Applications for valuable discussions, and to the anonymous referees for useful suggestions.

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Texas at El Paso, 500 W. University, El Paso, TX, 79968, USA

    Francisco Zapata & Vladik Kreinovich

  2. Pacific Northwest National Laboratory, National Security Directorate, PNNL/Battelle Suite 400, 1100 Dexter Ave. N, Seattle, WA, 98109, USA

    Cliff Joslyn

  3. Pacific Northwest National Laboratory, Fundamental and Computational Sciences Directorate, 902 Battelle Boulevard, MSIN K7-90, P.O. Box 999, Richland, WA, 99352, USA

    Emilie Hogan

Authors
  1. Francisco Zapata
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vladik Kreinovich
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Cliff Joslyn
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Emilie Hogan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladik Kreinovich.

Additional information

Communicated by V. Kreinovich.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zapata, F., Kreinovich, V., Joslyn, C. et al. Orders on intervals over partially ordered sets: extending Allen’s algebra and interval graph results. Soft Comput 17, 1379–1391 (2013). https://doi.org/10.1007/s00500-013-1010-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-013-1010-1

Keywords

Search

Navigation