文章基本信息

标题：Discounted-Sum Automata with Multiple Discount Factors
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作者：Udi Boker ; Guy Hefetz
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2021
卷号：183
页码：12:1-12:23
DOI：10.4230/LIPIcs.CSL.2021.12
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：Discounting the influence of future events is a key paradigm in economics and it is widely used in computer-science models, such as games, Markov decision processes (MDPs), reinforcement learning, and automata. While a single game or MDP may allow for several different discount factors, discounted-sum automata (NDAs) were only studied with respect to a single discount factor. For every integer Î» â^^ â".â§µ{0,1}, as opposed to every Î» â^^ â"Sâ§µâ"., the class of NDAs with discount factor Î» (Î»-NDAs) has good computational properties: it is closed under determinization and under the algebraic operations min, max, addition, and subtraction, and there are algorithms for its basic decision problems, such as automata equivalence and containment. We define and analyze discounted-sum automata in which each transition can have a different integral discount factor (integral NMDAs). We show that integral NMDAs with an arbitrary choice of discount factors are not closed under determinization and under algebraic operations. We then define and analyze a restricted class of integral NMDAs, which we call tidy NMDAs, in which the choice of discount factors depends on the prefix of the word read so far. Tidy NMDAs are as expressive as deterministic integral NMDAs with an arbitrary choice of discount factors, and some of their special cases are NMDAs in which the discount factor depends on the action (alphabet letter) or on the elapsed time. We show that for every function Î¸ that defines the choice of discount factors, the class of Î¸-NMDAs enjoys all of the above good properties of integral NDAs, as well as the same complexities of the required decision problems. To this end, we also improve the previously known complexities of the decision problems of integral NDAs, and present tight bounds on the size blow-up involved in algebraic operations on them. All our results hold equally for automata on finite words and for automata on infinite words.
关键词：Automata; Discounted-sum; Quantitative verification; NMDA; NDA