Recursive Markov Decision Processes and Recursive Stochastic Games

We introduce Recursive Markov Decision Processes (RMDPs) and Recursive Simple Stochastic Games (RSSGs), which are classes of (finitely presented) countable-state MDPs and zero-sum turn-based (perfect information) stochastic games. They extend standard finite-state MDPs and stochastic games with a recursion feature. We study the decidability and computational complexity of these games under termination objectives for the two players: one player's goal is to maximize the probability of termination at a given exit, while the other player's goal is to minimize this probability. In the quantitative termination problems , given an RMDP (or RSSG) and probability p , we wish to decide whether the value of such a termination game is at least p (or at most p ); in the qualitative termination problem we wish to decide whether the value is 1. The important 1-exit subclasses of these models, 1-RMDPs and 1-RSSGs, correspond in a precise sense to controlled and game versions of classic stochastic models, including multitype Branching Processes and Stochastic Context-Free Grammars, where the objective of the players is to maximize or minimize the probability of termination (extinction). We provide a number of upper and lower bounds for qualitative and quantitative termination problems for RMDPs and RSSGs. We show both problems are undecidable for multi-exit RMDPs, but are decidable for 1-RMDPs and 1-RSSGs. Specifically, the quantitative termination problem is decidable in PSPACE for both 1-RMDPs and 1-RSSGs, and is at least as hard as the square root sum problem, a well-known open problem in numerical computation. We show that the qualitative termination problem for 1-RMDPs (i.e., a controlled version of branching processes) can be solved in polynomial time both for maximizing and minimizing 1-RMDPs. The qualitative problem for 1-RSSGs is in NP ∩ coNP, and is at least as hard as the quantitative termination problem for Condon's finite-state simple stochastic games, whose complexity remains a well known open problem. Finally, we show that even for 1-RMDPs, more general (qualitative and quantitative) model-checking problems with respect to linear-time temporal properties are undecidable even for a fixed property.