Paper Info
Title
Approximating the termination value of one-counter MDPs and stochastic games
Abstract
One-counter MDPs (OC-MDPs) and one-counter simple stochastic games (OC-SSGs) are 1-player, and 2-player turn-based zero-sum, stochastic games played on the transition graph of classic one-counter automata (equivalently, pushdown automata with a 1-letter stack alphabet). A key objective for the analysis and verification of these games is the termination objective, where the players aim to maximize (minimize, respectively) the probability of hitting counter value 0, starting at a given control state and given counter value. Recently, we studied qualitative decision problems (''is the optimal termination value equal to 1?'') for OC-MDPs (and OC-SSGs) and showed them to be decidable in polynomial time (in NP@?coNP, respectively). However, quantitative decision and approximation problems (''is the optimal termination value at least p'', or ''approximate the termination value within @e'') are far more challenging. This is so in part because optimal strategies may not exist, and because even when they do exist they can have a highly non-trivial structure. It thus remained open even whether any of these quantitative termination problems are computable. In this paper we show that all quantitative approximation problems for the termination value for OC-MDPs and OC-SSGs are computable. Specifically, given an OC-SSG, and given @e0, we can compute a value v that approximates the value of the OC-SSG termination game within additive error @e, and furthermore we can compute @e-optimal strategies for both players in the game. A key ingredient in our proofs is a subtle martingale, derived from solving certain linear programs that we can associate with a maximizing OC-MDP. An application of Azuma@?s inequality on these martingales yields a computable bound for the ''wealth'' at which a ''rich person@?s strategy'' becomes @e-optimal for OC-MDPs.
Year
DOI
Venue
2011
10.1016/j.ic.2012.01.008
international colloquium on automata, languages and programming
Keywords
DocType
Volume
stochastic game,quantitative termination problem,one-counter mdps,quantitative approximation problem,termination objective,counter value,optimal termination value,quantitative decision,oc-ssg termination game,optimal strategy,termination value,value v
Conference
222,
ISSN
Citations 
PageRank 
0890-5401
11
0.65
References 
Authors
10
4
Name
Order
Citations
PageRank
Tomáš Brázdil119413.11
Václav Brozek2342.23
Kousha Etessami34177210.71
Antonín Kučera426218.04