Discounted-Sum Automata with Multiple Discount Factors

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.