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Combining Interval and Probabilistic Uncertainty: What Is Computable?

  • Vladik KreinovichEmail author
  • Andrzej Pownuk
  • Olga Kosheleva
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 107)

Abstract

In many practical problems, we need to process measurement results. For example, we need such data processing to predict future values of physical quantities. In these computations, it is important to take into account that measurement results are never absolutely exact, that there is always measurement uncertainty, because of which the measurement results are, in general, somewhat different from the actual (unknown) values of the corresponding quantities. In some cases, all we know about measurement uncertainty is an upper bound; in this case, we have an interval uncertainty, meaning that all we know about the actual value is that is belongs to a certain interval. In other cases, we have some information—usually partial—about the corresponding probability distribution. New data processing challenges appear all the time; in many of these cases, it is important to come up with appropriate algorithms for taking uncertainty into account.

Before we concentrate our efforts on designing such algorithms, it is important to make sure that such an algorithm is possible in the first place, i.e., that the corresponding problem is algorithmically computable. In this paper, we analyze the computability of such uncertainty-related problems. It turns out that in a naive (straightforward) formulation, many such problems are not computable, but they become computable if we reformulate them in appropriate practice-related terms.

Keywords

Probabilistic uncertainty Interval uncertainty Combining different types of uncertainty Computability Constructive mathematics Computable analysis 

Notes

Acknowledgements

This work was supported in part by the National Science Foundation grants HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and DUE-0926721. The authors are thankful to Walid Taha and to all the participants of the Second Hybrid Modeling Languages Meeting HyMC (Houston, Texas, May 7–8, 2015) for valuable discussions, and to the anonymous referees for useful suggestions.

References

  1. 1.
    Bishop, E., Bridges, D.: Constructive Analysis. Springer, Heidelberg (1985)CrossRefzbMATHGoogle Scholar
  2. 2.
    Fishburn, P.C.: Utility Theory for Decision Making. Wiley, New York (1969)zbMATHGoogle Scholar
  3. 3.
    Kreinovich, V., Lakeyev, A., Rohn, J., Kahl, P.: Computational Complexity and Feasibility of Data Processing and Interval Computations. Kluwer, Dordrecht (1997)zbMATHGoogle Scholar
  4. 4.
    Luce, R.D., Raiffa, R.: Games and Decisions: Introduction and Critical Survey. Dover, New York (1989)zbMATHGoogle Scholar
  5. 5.
    Nguyen, H.T., Kosheleva, O., Kreinovich, V.: Decision making beyond Arrow’s ‘impossibility theorem’, with the analysis of effects of collusion and mutual attraction. Int. J. Intell. Syst. 24 (1), 27–47 (2009)CrossRefzbMATHGoogle Scholar
  6. 6.
    Rabinovich, S.: Measurement Errors and Uncertainties: Theory and Practice. Springer, New York (2005)zbMATHGoogle Scholar
  7. 7.
    Raiffa, H.: Decision Analysis. Addison-Wesley, Reading, MA (1970)zbMATHGoogle Scholar
  8. 8.
    Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Vladik Kreinovich
    • 1
    Email author
  • Andrzej Pownuk
    • 1
  • Olga Kosheleva
    • 1
  1. 1.University of Texas at El PasoEl PasoUSA

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