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Discounting the Future in Systems Theory

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Automata, Languages and Programming (ICALP 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2719))

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Abstract

Discounting the future means that the value, today, of a unit payoffis 1 if the payoffo ccurs today, a if it occurs tomorrow, a 2 if it occurs the day after tomorrow, and so on, for some real-valued discount factor 0 < a < 1. Discounting (or inflation) is a key paradigm in economics and has been studied in Markov decision processes as well as game theory. We submit that discounting also has a natural place in systems engineering: for nonterminating systems, a potential bug in the far-away future is less troubling than a potential bug today. We therefore develop a systems theory with discounting. Our theory includes several basic elements: discounted versions of system properties that correspond to the ω-regular properties, fixpoint-based algorithms for checking discounted properties, and a quantitative notion of bisimilarity for capturing the difference between two states with respect to discounted properties. We present the theory in a general form that applies to probabilistic systems as well as multicomponent systems (games), but it readily specializes to classical transition systems. We show that discounting, besides its natural practical appeal, has also several mathematical benefits. First, the resulting theory is robust, in that small perturbations of a system can cause only small changes in the properties of the system. Second, the theory is computational, in that the values of discounted properties, as well as the discounted bisimilarity distance between states, can be computed to any desired degree of precision.

This research was supported in part by the NSF CAREER award CCR-0132780, the DARPA grant F33615-C-98-3614, the NSF grants CCR-9988172, CCR-0234690 and CCR-0225610, and the ONR grant N00014-02-1-0671.

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Authors and Affiliations

  1. Department of Computer Engineering, UC Santa Cruz, Santa Cruz

    Luca de Alfaro

  2. Department of Electrical Engineering and Computer Sciences, UC Berkeley, Berkeley

    Thomas A. Henzinger & Rupak Majumdar

Authors
  1. Luca de Alfaro
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  2. Thomas A. Henzinger
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  3. Rupak Majumdar
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Editor information

Editors and Affiliations

  1. Dept. of Mathematics and Computer Science, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Jos C. M. Baeten

  2. School of Industrial and Systems Engineering, Georgia Institute of Technology, 765 Ferst Drive, Atlanta, GA, 30332-0205, USA

    Jan Karel Lenstra

  3. Department of Information Technology, Uppsala University, P.O. Box 337, 75105, Uppsala, Sweden

    Joachim Parrow

  4. Faculty of Electrical Engineering, Mathematics and Computer Science, University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands

    Gerhard J. Woeginger

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de Alfaro, L., Henzinger, T.A., Majumdar, R. (2003). Discounting the Future in Systems Theory. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds) Automata, Languages and Programming. ICALP 2003. Lecture Notes in Computer Science, vol 2719. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45061-0_79

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  • DOI: https://doi.org/10.1007/3-540-45061-0_79

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