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. It is known that every class of NDAs with an integer as the discount factor 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. Extending the integer discount factor to an arbitrary rational number, loses most of these good properties. We define and analyze nondeterministic 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 and that their containment problem is undecidable. 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. Among their special cases are NMDAs that correlate discount factors to actions (alphabet letters) or to the elapsed time. We show that for every function $\theta$ that defines the choice of discount factors, the class of $\theta$-NMDAs enjoys all of the above good properties of NDAs with a single integral discount factor, as well as the same complexity of the required decision problems. Tidy NMDAs are also as expressive as deterministic integral NMDAs with an arbitrary choice of discount factors.

翻译：对未来事件影响进行折现是经济学中的关键范式，并广泛应用于计算机科学模型，如博弈、马尔可夫决策过程（MDPs）、强化学习和自动机。虽然单个博弈或MDP可能允许多种不同的折扣因子，但折扣求和自动机（NDAs）此前仅针对单一折扣因子进行研究。已知具有整数折扣因子的每一类NDA均具有良好的计算性质：其在确定化及代数运算（最小值、最大值、加法与减法）下封闭，且存在算法可解决其基本判定问题，如自动机等价性与包含性。将整数折扣因子扩展至任意有理数时，这些优良性质大多会丧失。本文定义并分析了一类非确定性折扣求和自动机，其中每个转移可具有不同的整数折扣因子（整数NMDAs）。我们证明，具有任意折扣因子选择的整数NMDAs在确定化与代数运算下不封闭，且其包含性问题是不可判定的。随后，我们定义并分析了一类受限的整数NMDAs，称为整洁NMDAs，其中折扣因子的选择取决于当前已读词的前缀。其特例包括将折扣因子与动作（字母表符号）或已流逝时间相关联的NMDAs。我们证明，对于每个定义折扣因子选择的函数θ，θ-NMDAs类均具有上述单一整数折扣因子NDAs的所有优良性质，且所需判定问题的复杂度相同。整洁NMDAs的表达能力亦与具有任意折扣因子选择的确定性整数NMDAs等价。