Greatest Fixed Points of Probabilistic Min/Max Polynomial Equations, and Reachability for Branching Markov Decision Processes

We give polynomial time algorithms for quantitative (and qualitative) reachability analysis for Branching Markov Decision Processes (BMDPs). Specifically, given a BMDP, and given an initial population, where the objective of the controller is to maximize (or minimize) the probability of eventually reaching a population that contains an object of a desired (or undesired) type, we give algorithms for approximating the supremum (infimum) reachability probability, within desired precision ǫ > 0, in time polynomial in the encoding size of the BMDP and in log(1/ǫ). We furthermore give P-time algorithms for computing ǫ-optimal strategies for both maximization and minimization of reachability probabilities. We also give P-time algorithms for all associated qualitative analysis problems, namely: deciding whether the optimal (supremum or infimum) reachability probabilities are 0 or 1. Prior to this paper, approximation of optimal reachability probabilities for BMDPs was not even known to be decidable. Our algorithms exploit the following basic fact: we show that for any BMDP, its maximum (minimum) non-reachability probabilities are given by the greatest fixed point (GFP) solution g∗ ∈ [0, 1]n of a corresponding monotone max (min) Probabilistic Polynomial System of equations (max/minPPS), x = P (x), which are the Bellman optimality equations for a BMDP with non-reachability objectives. We show how to compute the GFP of max/minPPSs to desired precision in P-time. We also study more general branching simple stochastic games (BSSGs) with (non-)reachability objectives. We show that: (1) the value of these games is captured by the GFP, g∗, of a corresponding max-minPPS, x = P (x); (2) the quantitative problem of approximating the value is in TFNP; and (3) the qualitative problems associated with the value are all solvable in P-time.