The $omega$-power of a finitary language L over a finite alphabet $Sigma$ is the language of infinite words over $Sigma$ defined by L $infty$ := {w 0 w 1. .. $in$ $Sigma$ $omega$ | $forall$i $in$ $omega$ w i $in$ L}. The $omega$-powers appear very na
turally in Theoretical Computer Science in the characterization of several classes of languages of infinite words accepted by various kinds of automata, like B{u}chi automata or B{u}chi pushdown automata. We survey some recent results about the links relating Descriptive Set Theory and $omega$-powers.
We prove that, for any natural number n $ge$ 1, we can find a finite alphabet $Sigma$ and a finitary language L over $Sigma$ accepted by a one-counter automaton, such that the $omega$-power L $infty$ := {w 0 w 1. .. $in$ $Sigma$ $omega$ | $forall$i $
in$ $omega$ w i $in$ L} is $Pi$ 0 n-complete. We prove a similar result for the class $Sigma$ 0 n .
We study the class of analytic binary relations on Polish spaces, compared with the notions of continuous reducibility or injective continuous reducibility. In particular, we characterize when a locally countable Borel relation is $Sigma$ 0 $xi$ (or
$Pi$ 0 $xi$), when $xi$ $ge$ 3, by providing a concrete finite antichain basis. We give a similar characterization for arbitrary relations when $xi$ = 1. When $xi$ = 2, we provide a concrete antichain of size continuum made of locally countable Borel relations minimal among non-$Sigma$ 0 2 (or non-$Pi$ 0 2) relations. The proof of this last result allows us to strengthen a result due to Baumgartner in topological Ramsey theory on the space of rational numbers. We prove that positive results hold when $xi$ = 2 in the acyclic case. We give a general positive result in the non-necessarily locally countable case, with another suitable acyclicity assumption. We provide a concrete finite antichain basis for the class of uncountable analytic relations. Finally, we deduce from our positive results some antichain basis for graphs, of small cardinality (most of the time 1 or 2).
We study the analytic digraphs of uncountable Borel chromatic number on Polish spaces, and compare them with the notion of injective Borel homomorphism. We provide some minimal digraphs incomparable with G 0. We also prove the existence of antichains
of size continuum, and that there is no finite basis. 2010 Mathematics Subject Classification. 03E15, 54H05
We study the class of Borel equivalence relations under continuous reducibility. In particular , we characterize when a Borel equivalence relation with countable equivalence classes is $Sigma$ 0 $xi$ (or $Pi$ 0 $xi$). We characterize when all the equ
ivalence classes of such a relation are $Sigma$ 0 $xi$ (or $Pi$ 0 $xi$). We prove analogous results for the Borel equivalence relations with countably many equivalence classes. We also completely solve these two problems for the first two ranks. In order to do this, we prove some extensions of the Louveau-Saint Raymond theorem which itself generalized the Hurewicz theorem characterizing when a Borel subset of a Polish space is G $delta$ .
We provide dichotomy results characterizing when two disjoint analytic binary relations can be separated by a countable union of ${bfSigma}^0_1 !times! {bfSigma}^0_xi$ sets, or by a ${bfPi}^0_1 !times! {bfPi}^0_xi$ set.
The third author noticed in his 1992 PhD Thesis [Sim92] that every regular tree language of infinite trees is in a class $Game (D_n({bfSigma}^0_2))$ for some natural number $ngeq 1$, where $Game$ is the game quantifier. We first give a detailed expos
ition of this result. Next, using an embedding of the Wadge hierarchy of non self-dual Borel subsets of the Cantor space $2^omega$ into the class ${bfDelta}^1_2$, and the notions of Wadge degree and Veblen function, we argue that this upper bound on the topological complexity of regular tree languages is much better than the usual ${bfDelta}^1_2$.
We prove that for every Borel equivalence relation $E$, either $E$ is Borel reducible to $mathbb{E}_0$, or the family of Borel equivalence relations incompatible with $E$ has cofinal essential complexity. It follows that if $F$ is a Borel equivalence
relation and $cal F$ is a family of Borel equivalence relations of non-cofinal essential complexity which together satisfy the dichotomy that for every Borel equivalence relation $E$, either $Ein {cal F}$ or $F$ is Borel reducible to $E$, then $cal F$ consists solely of smooth equivalence relations, thus the dichotomy is equivalent to a known theorem.
We study the extension of the Kechris-Solecki-Todorcevic dichotomy on analytic graphs to dimensions higher than 2. We prove that the extension is possible in any dimension, finite or infinite. The original proof works in the case of the finite dimens
ion. We first prove that the natural extension does not work in the case of the infinite dimension, for the notion of continuous homomorphism used in the original theorem. Then we solve the problem in the case of the infinite dimension. Finally, we prove that the natural extension works in the case of the infinite dimension, but for the notion of Baire-measurable homomorphism.
This is an extended abstract presenting new results on the topological complexity of omega-powers (which are included in a paper Classical and effective descriptive complexities of omega-powers available from arXiv:0708.4176) and reflecting also some
open questions which were discussed during the Dagstuhl seminar on Topological and Game-Theoretic Aspects of Infinite Computations 29.06.08 - 04.07.08.
