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Greatest fixed points of probabilistic min/max polynomial equations, and reachability for branching Markov decision processes. (English) Zbl 1394.60084
Halldórsson, Magnús M. (ed.) et al., Automata, languages, and programming. 42nd international colloquium, ICALP 2015, Kyoto, Japan, July 6–10, 2015. Proceedings. Part II. Berlin: Springer (ISBN 978-3-662-47665-9/pbk; 978-3-662-47666-6/ebook). Lecture Notes in Computer Science 9135, 184-196 (2015).
Summary: 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 $$\epsilon > 0$$, in time polynomial in the encoding size of the BMDP and in $$\log (1/\epsilon )$$. We furthermore give P-time algorithms for computing $$\epsilon$$-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 $$g^* \in [0,1]^n$$ of a corresponding monotone max (min) probabilistic polynomial system of equations (max/min-PPS), $$x=P(x)$$, which are the Bellman optimality equations for a BMDP with non-reachability objectives. We show how to compute the GFP of max/min PPSs to desired precision in P-time.
For the entire collection see [Zbl 1316.68013].

##### MSC:
 60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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