arXiv subjects
Publications and source records attributed to Henryk Michalewski.
Subzero automata is a class of tree automata whose acceptance condition can express probabilistic constraints. Our main result is that the problem of determining if a subzero automaton accepts some regular tree is decidable.
A non-deterministic automaton running on infinite trees is unambiguous if it has at most one accepting run on every tree. The class of languages recognisable by unambiguous tree automata is still not well-understood. In particular, decidability of the problem whether a given language is recognisable by some unambiguous automaton is open. Moreover, there are no known upper bounds on the descriptive complexity of unambiguous languages among all regular tree languages. In this paper we show the following complexity collapse: if a non-deterministic parity tree automaton $A$ is unambiguous and its priorities are between $i$ and $2n$ then the language recognised by $A$ is in the class $Comp(i+1,2n)$. A particular case of this theorem is for $i=n=1$: if $A$ is an unambiguous Buchi tree automaton then $L(A)$ is recognisable by a weak alternating automaton (or equivalently definable in weak MSO). The main motivation for this result is a theorem by Finkel and Simonnet stating that every unambiguous Buchi automaton recognises a Borel language. The assumptions of the presented theorem are syntactic (we require one automaton to be both unambiguous and of particular parity index). However, to the authors' best knowledge this is the first theorem showing a collapse of the parity index that exploits the fact that a given automaton is unambiguous.
We train a number of neural networks to play games Bowling, Breakout and Seaquest using information stored in the memory of a video game console Atari 2600. We consider four models of neural networks which differ in size and architecture: two networks which use only information contained in the RAM and two mixed networks which use both information in the RAM and information from the screen. As the benchmark we used the convolutional model proposed in NIPS and received comparable results in all considered games. Quite surprisingly, in the case of Seaquest we were able to train RAM-only agents which behave better than the benchmark screen-only agent. Mixing screen and RAM did not lead to an improved performance comparing to screen-only and RAM-only agents.
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.
We study the strength of set-theoretic axioms needed to prove Rabin's theorem on the decidability of the MSO theory of the infinite binary tree. We first show that the complementation theorem for tree automata, which forms the technical core of typical proofs of Rabin's theorem, is equivalent over the moderately strong second-order arithmetic theory $\mathsf{ACA}_0$ to a determinacy principle implied by the positional determinacy of all parity games and implying the determinacy of all Gale-Stewart games given by boolean combinations of ${\bf Σ^0_2}$ sets. It follows that complementation for tree automata is provable from $Π^1_3$- but not $Δ^1_3$-comprehension. We then use results due to MedSalem-Tanaka, Möllerfeld and Heinatsch-Möllerfeld to prove that over $Π^1_2$-comprehension, the complementation theorem for tree automata, decidability of the MSO theory of the infinite binary tree, positional determinacy of parity games and determinacy of $\mathrm{Bool}({\bf Σ^0_2})$ Gale-Stewart games are all equivalent. Moreover, these statements are equivalent to the $Π^1_3$-reflection principle for $Π^1_2$-comprehension. It follows in particular that Rabin's decidability theorem is not provable in $Δ^1_3$-comprehension.
We show that it is decidable whether a given a regular tree language belongs to the class ${\bf Δ^0_2}$ of the Borel hierarchy, or equivalently whether the Wadge degree of a regular tree language is countable.
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 $μ$-calculus and the automata index hierarchies. They showed that separability fails in general for non-deterministic automata of type $Σ^μ_{n} $, starting from level $n=3$, while our result settles the missing case $n=2$.
We prove a preservation theorem for the class of Valdivia compact spaces, which involves inverse sequences of ``simple'' retractions. Consequently, a compact space of weight $\loe\aleph_1$ is Valdivia compact iff it is the limit of an inverse sequence of metric compacta whose bonding maps are retractions. As a corollary, we show that the class of Valdivia compacta of weight at most $\aleph_1$ is preserved both under retractions and under open 0-dimensional images. Finally, we characterize the class of all Valdivia compacta in the language of category theory, which implies that this class is preserved under all continuous weight preserving functors.