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...ision procedures determining an ω-regular language’s position in both hierarchies were given [4, 7, 15]. For tree automata the index problem is only solved when the input is a deterministic automaton =-=[9, 13]-=-. As for topological complexity of recognisable tree languages, it goes much . Indeed, co-Büchi automata over higher than that of ω-regular languages, which are all ∆0 3 trees may recognise Π1 1 -comp...

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...to be decidable in the Mostowski hierarchy of languages accepted by deterministic automata. The problem is decidable if both the input language and the output class are deterministic [13] 2(see also =-=[14]-=- for more details). The problem is also known to be decidable if the input language is deterministic, and the output class is non-deterministic [14] (which is a refinement of the case of a path langua...

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... admitted a number of techniques developed for ω-languages. Soon, the Borel hierarchy, the Wadge hierarchy, and the index hierarchy of deterministic tree languages were described and proved decidable =-=[7,8,9,10]-=-. The real challenge seems to be nondeterminism. The power it gives to tree automata makes them extremely difficult to tackle. Therefore, the investigation has basically concentrated on a very special...

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...arious versions of this pattern method were successfully applied to deterministic automata on trees, resulting in a full classification in terms of Wadge equivalence [16, 18], non-deterministic index =-=[19]-=-, and weak alternating index [17]. Owing to the elegant correspondence between certain set-theoretical and ordinal operations [4], the whole Wadge hierarchy of Borel sets of finite rank can be generat...

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...Commons Attribution-NoDerivs Licenses574 FILIP MURLAK remains unsolved. For deterministic languages, Niwiński and Walukiewicz gave algorithms to compute the deterministic and nondeterministic indices =-=[14, 16]-=-. The theoretical significance of the weak index is best reflected by its coincidence with the quantifier alternation depth in the weak monadic second order logic [9]. Further interesting facts are re...

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...ene star) for L which uses at most n nestings of Kleene-star operations? Colcombet and Löding [4] have used a similar approach over finite trees. The theory of regular cost functions over finite words [3] and finite trees [6] can be viewed as a unifying framework for these problems. It is desirable to extend the theory to infinite trees. For instance, the “parity-index problem” asks: given a regular language L of infinite trees and i < j, is there a parity automaton using only priorities {i, i+1, . . . , j}? This is known to be decidable in some special cases (e.g. for deterministic languages [11]), but a general decision procedure is not known. However, Colcombet and Löding [5] have reduced the parity-index problem to another open problem, namely the decidability of ≈ for regular cost functions over infinite trees. Weak cost automata (recently studied in [13]) are a natural starting point in this line of research on regular cost functions over infinite trees. In the classical setting, weak automata are a restricted form of alternating Büchi automata which have a fixed bound on the number of alternations between accepting and rejecting states across all runs. They were introduced in [1...

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... that quite recently Arnold and Santocanale has shown a surprising behaviour of diagonal classes [5].) Once the basic hierarchy questions are resolved, the next challenge is to provide algorithms for determining the level in the hierarchy of a given recognizable language. The first step was to give a polynomial time algorithm for computing the level in the hierarchy of deterministic automata [65]. Next, Urbanski [80] has shown that it is decidable if a deterministic Rabin tree automaton is equivalent to a nondeterministic Buchi one. Actually, the problem is also in Ptime [66]. More recently [67], the case of deterministic tree automata was completely solved. There are forbidden pattern characterizations for all the levels of the hierarchy of nondeterministic automata; that is, given a deterministic automaton one can tell by examining its structure to which level of the hierarchy of nondeterministic automata it belongs to. This also solves the problem for levels of alternating automata hierarchy as all deterministic languages are recognizable by co-Buchi alternating automata. The challenge for the future is to calculate hierarchy levels for nondeterministic automata. There are numero...

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...y often”). It is believed to reflect the inherent computational complexity of the language, and therefore has attracted a lot of attention encouraged by the expectations of the verification community =-=[3,4,10,17,18,19,20]-=-. The classical Borel hierarchy is based on the nesting of countable unions and negations in the set theoretic definition of the language, starting from the simplest (open) sets. It drew attention of ...