# Büchi Objectives in Countable MDPs

Kiefer, Stefan, Mayr, Richard, Shirmohammadi, Mahsa and Totzke, Patrick ORCID: 0000-0001-5274-8190
(2019) Büchi Objectives in Countable MDPs. .

We study countably infinite Markov decision processes with B\"uchi objectives, which ask to visit a given subset of states infinitely often. A question left open by T.P. Hill in 1979 is whether there always exist $\varepsilon$-optimal Markov strategies, i.e., strategies that base decisions only on the current state and the number of steps taken so far. We provide a negative answer to this question by constructing a non-trivial counterexample. On the other hand, we show that Markov strategies with only 1 bit of extra memory are sufficient.