Abstract
In this paper we lift the result of Hashiguchi of decidability of the restricted star-height problem for words to the level of finite trees. Formally, we show that it is decidable, given a regular tree language L and a natural number k whether L can be described by a disjunctive μ-calculus formula with at most k nesting of fixpoints. We show the same result for disjunctive μ-formulas allowing substitution. The latter result is equivalent to deciding if the language is definable by a regular expression with nesting depth at most k of Kleene-stars.
The proof, following the approach of Kirsten in the word case, goes by reduction to the decidability of the limitedness problem for non-deterministic nested distance desert automata over trees. We solve this problem in the more general framework of alternating tree automata.
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Colcombet, T., Löding, C. (2008). The Nesting-Depth of Disjunctive μ-Calculus for Tree Languages and the Limitedness Problem. In: Kaminski, M., Martini, S. (eds) Computer Science Logic. CSL 2008. Lecture Notes in Computer Science, vol 5213. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87531-4_30
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DOI: https://doi.org/10.1007/978-3-540-87531-4_30
Publisher Name: Springer, Berlin, Heidelberg
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