Discounting the future in systems theory

L. De Alfaro, T.A. Henzinger, R. Majumdar, in:, Springer, 2003, pp. 1022–1037.

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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.
Publishing Year
Date Published
2003-06-25
Acknowledgement
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.
Volume
2719
Page
1022 - 1037
Conference
ICALP: Automata, Languages and Programming
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De Alfaro L, Henzinger TA, Majumdar R. Discounting the future in systems theory. In: Vol 2719. Springer; 2003:1022-1037. doi:10.1007/3-540-45061-0_79
De Alfaro, L., Henzinger, T. A., & Majumdar, R. (2003). Discounting the future in systems theory (Vol. 2719, pp. 1022–1037). Presented at the ICALP: Automata, Languages and Programming, Springer. https://doi.org/10.1007/3-540-45061-0_79
De Alfaro, Luca, Thomas A Henzinger, and Ritankar Majumdar. “Discounting the Future in Systems Theory,” 2719:1022–37. Springer, 2003. https://doi.org/10.1007/3-540-45061-0_79.
L. De Alfaro, T. A. Henzinger, and R. Majumdar, “Discounting the future in systems theory,” presented at the ICALP: Automata, Languages and Programming, 2003, vol. 2719, pp. 1022–1037.
De Alfaro L, Henzinger TA, Majumdar R. 2003. Discounting the future in systems theory. ICALP: Automata, Languages and Programming, LNCS, vol. 2719. 1022–1037.
De Alfaro, Luca, et al. Discounting the Future in Systems Theory. Vol. 2719, Springer, 2003, pp. 1022–37, doi:10.1007/3-540-45061-0_79.

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