On Topological Completeness of Regular Tree Languages

Measure Properties of Game Tree Languages

“…Also, our Theorem 1 can be easily inferred for k = 1. The case of k = 2 follows from the theorem of Burgess in conjunction with[18]. Our proof of Theorem 1 yields an independent and formal argument which backs the above statement of Burgess' theorem.…”

“…A game-theoretic approach to R-sets, closely related to this work, is developed by Burgess in [8] where the following characterization is stated as a remark without a formal proof: (1) every set A ⊆ X belongs to a finite level of the hierarchy of R-sets if and only if it is of the form A = (K), for some set K ⊆ ω ω which is a Boolean combination of F σ sets, and (2) the levels of the hierarchy of R-sets are in correspondence with the levels of the difference hierarchy (see [13, §22.E]) of F σ sets. The operation is the so-called game quantifier (see [13, §20.D] and [6,7,12,18]). Admittedly, our characterization of R-sets in terms of game tree languages W i,k , can be considered as a modern variant of the result of Burgess.…”

An upper bound on the complexity of recognizable tree languages

“…Moreover, the referee indicated us that one can actually verify that the language given in S. Hummel's paper [Hum12] is reducible to W 1,3 and is not complete for the class of Σ 1 1 -inductive sets. In another recent paper [GMMS14] T. Gogacz, H. Michalewski, M. Mio and M. Skrzypczak show a one-to-one correspondence between the levels of the hierarchy of Kolmogorov R-sets and parity index of regular languages which extends the theorem from [MN12]. Since due to a theorem of Burgess the R-sets are known to be in correspondence with the game quantifier, on a technical level this covers Theorem 4.4.…”

“…In J. Bradfield's paper [Bra03], and in [BDQ05] by J. Bradfield, J. Duparc and S. Quickert, a link between the game quantifier and the µ-calculus is described, and Corollary 11 of [Bra03] explicitely states an upper bound in a style similar to the one presented in our paper. In [MN12], D. Niwinski and H. Michalewski are also interested in the problem of finding upper bounds for the class of regular tree languages. Using a method developed by J. Saint Raymond in [SR06], they prove that the game tree language W 1,3 is complete for the class of Σ 1 1 -inductive sets.…”