On the Problem of Computing the Probability of Regular Sets of Trees

We consider the problem of computing the probability of regular languages of infinite trees with respect to the natural coin-flipping measure. We propose an algorithm which computes the probability of languages recognizable by \emph{game automata}. In particular this algorithm is applicable to all deterministic automata. We then use the algorithm to prove through examples three properties of measure: (1) there exist regular sets having irrational probability, (2) there exist comeager regular sets having probability $0$ and (3) the probability of \emph{game languages} $W_{i,k}$, from automata theory, is $0$ if $k$ is odd and is $1$ otherwise

On the Borel Inseparability of Game Tree Languages

The game tree languages can be viewed as an automata-theoretic counterpart of parity games on graphs. They witness the strictness of the index hierarchy of alternating tree automata, as well as the fixed-point hierarchy over binary trees. We consider a game tree language of the first non-trivial level, where Eve can force that 0 repeats from some moment on, and its dual, where Adam can force that 1 repeats from some moment on. Both these sets (which amount to one up to an obvious renaming) are complete in the class of co-analytic sets. We show that they cannot be separated by any Borel set, hence {\em a fortiori} by any weakly definable set of trees. This settles a case left open by L.Santocanale and A.Arnold, who have thoroughly investigated the separation property within the $\mu$-calculus and the automata index hierarchies. They showed that separability fails in general for non-deterministic automata of type $\Sigma^{\mu}_{n}$, starting from level $n=3$, while our result settles the missing case $n=2$

On the separation question for tree languages

We show that the separation property fails for the classes Sigma_n of the Rabin-Mostowski index hierarchy of alternating automata on infinite trees. This extends our previous result (obtained with Szczepan Hummel) on the failure of the separation property for the class Sigma_2 (i.e., for co-Buchi sets). It remains open whether the separation property does hold for the classes Pi_n of the index hierarchy. To prove our result, we first consider the Rabin-Mostowski index hierarchy of deterministic automata on infinite words, for which we give a complete answer (generalizing previous results of Selivanov): the separation property holds for Pi_n and fails for Sigma_n-classes. The construction invented for words turns out to be useful for trees via a suitable game

Atari games and Intel processors

The asynchronous nature of the state-of-the-art reinforcement learning algorithms such as the Asynchronous Advantage Actor-Critic algorithm, makes them exceptionally suitable for CPU computations. However, given the fact that deep reinforcement learning often deals with interpreting visual information, a large part of the train and inference time is spent performing convolutions. In this work we present our results on learning strategies in Atari games using a Convolutional Neural Network, the Math Kernel Library and TensorFlow 0.11rc0 machine learning framework. We also analyze effects of asynchronous computations on the convergence of reinforcement learning algorithms