By Michal Skrzypczak
The e-book is predicated at the PhD thesis “Descriptive Set Theoretic tools in Automata Theory,” presented the E.W. Beth Prize in 2015 for impressive dissertations within the fields of good judgment, language, and data. The thesis finds unforeseen connections among complicated options in good judgment, descriptive set idea, topology, and automata concept and gives many deep insights into the interaction among those fields. It opens new views on imperative difficulties within the conception of automata on limitless phrases and timber and gives very striking advances during this idea from the viewpoint of topology.
"…the thesis of Michał Skrzypczak bargains definitely what we think from very good arithmetic: new unforeseen connections among a priori unique suggestions, and proofs related to enlightening ideas.” Thomas Colcombet.
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Extra resources for Descriptive Set Theoretic Methods in Automata Theory: Decidability and Topological Complexity
Usually, exactly one class from a pair of dual classes of sets has the separation property. 2 Overview of the Parts The preliminary Chap. 1 introduces basic notions and known results that will be used later. The rest of the thesis is divided into three parts, each part has three chapters. 34 2 Introduction All the presented results study related problems of descriptive complexity. The respective parts group results of similar type. Most of the chapters present results that are technically independent, in particular they can be read separately.
N are pairwise disjoint. 28 from page 25 implies that for every pair of transitions δi = δ j there is an Comp( alt j−1 )-language that separates L δi from L δ j . Since Comp( alt )-languages are closed under Boolean combinations, we can find j−1 Comp(0, j−1)-automata Cδk for k = 1, 2, . . , n such that: – for k = 1, 2, . . ,n L(Cδk ) equals Tr A . These automata will be crucial ingredients of the construction. The following lemma formalizes the notion of the unique runs. 3. Let t ∈ Tr A be a tree.
The assignment of the symbols is opposite to the one from [FMS13]. 7 Regular Languages 23 – Output The minimal class of the alternating (non-deterministic) index hierarchy that contains L(A). Both problems were solved for deterministic automata [NW05, NW03], see Sect. 6. They are both open for general automata. Colcombet and Löding [CL08] have proposed a reduction of the non-deterministic index problem to a boundedness problem for a specific class of tree automata with counters. However, the latter problem is not known to be decidable.