Stochastic processes with expected stopping time - Chatterjee, Krishnendu; Doyen, Laurent; LICS: Symposium on Logic in Computer Science; Formal Methods for Stochastic Models: Algorithms and Applications (Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science, 2021)
Markov chains are the de facto finite-state model for stochastic dynamical systems, and Markov decision processes (MDPs) extend Markov chains by incorporating non-deterministic behaviors. Given an MDP and rewards on states, a classical optimization criterion is the maximal expected total reward where the MDP stops after T steps, which can be computed by a simple dynamic programming algorithm. We consider a natural generalization of the problem where the stopping times can be chosen according to a probability distribution, such that the expected stopping time is T, to optimize the expected total reward. Quite surprisingly we establish inter-reducibility of the expected stopping-time problem for Markov chains with the Positivity problem (which is related to the well-known Skolem problem), for which establishing either decidability or undecidability would be a major breakthrough. Given the hardness of the exact problem, we consider the approximate version of the problem: we show that it can be solved in exponential time for Markov chains and in exponential space for MDPs.
published
yes
Keywords: conference paper; Computer science -- Heuristic algorithms -- Memory management -- Automata -- Markov processes -- Probability distribution -- Complexity theory