Polynomial Closure and Unambiguous Product

Pin, Jean-Éric; Weil, Pascal

doi:10.1007/s002240000058

Cited by 62 publications

(139 citation statements)

References 3 publications

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“…Moreover, as far as we know, the question on whether a regular tree language given by its minimal automaton can be accepted by a partially ordered tree automata is still open and, in that case, how to compute such an automaton. Notice that this question requires strong arguments to be solved in the word automaton case [1,13].…”

Section: Resultsmentioning

confidence: 99%

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A note on partially ordered tree automata

Héam¹

2008

Information Processing Letters

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A recent paper by Bouajjani, Muscholl and Touili shows that the class of languages accepted by partially ordered word automata (or equivalently accepted by Σ 2 -formulae) is closed under semi-commutation and it suggested the following open question: can we extend this result to tree languages? This problem can be addressed by proving (1) that the class of tree regular languages accepted by Σ 2 formulae is strictly included in the class of languages accepted by partially ordered automata, and (2) that Bouajjani and the others results cannot be extended to tree.We assume the reader is familiar with the basic notions on terms and tree automata. For general reference see [6].

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Section: Resultsmentioning

confidence: 99%

“…Indeed, this notion cannot be easily extended to tree languages and is out of the scope of this paper. See [13] for more details and E-mail address: heampc@lifc.univ-fcomte.fr.…”

Section: Proposition 1 the Following Propositions Are Equivalentmentioning

confidence: 99%

A note on partially ordered tree automata

Héam¹

2008

Information Processing Letters

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“…B 0 is decidable for trivial reasons. Pin and Weil [29] proved the decidability of B 1/2 , Knast [20] proved the decidability of B 1 , and Glaßer and Schmitz [14,15] proved the decidability of B 3/2 . Other levels are not known to be decidable, but it is widely believed that they are.…”

Section: Dot-depth Hierarchymentioning

confidence: 98%

“…The reason for this kind of definition is of historic nature: Originally, Brzozowski and Cohen were interested in the full levels B n and therefore, defined the dot-depth hierarchy in this way. Later Pin and Weil [29] considered both, the levels B n and their polynomial closure. To be consistent with Brzozowski and Cohen, they extended the dot-depth hierarchy by the half levels B n+1/2 .…”

Section: Dot-depth Hierarchymentioning

confidence: 99%

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Languages polylog-time reducible to dot-depth 1/2

Glaßer

2007

Journal of Computer and System Sciences

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We study (i) regular languages that are polylog-time reducible to languages of dot-depth 1/2 and (ii) regular languages that are polylog-time decidable. For both classes we provide• forbidden-pattern characterizations, and • characterizations in terms of regular expressions.This implies that both classes are decidable. In addition, we show that a language is in class (ii) if and only if the language and its complement are in class (i). Our observations have three consequences.(1) Gap theorems for balanced regular-leaf-language definable classes C and D:(a) Either C is contained in NP, or C contains coUP. (b) Either D is contained in P, or D contains UP or coUP. We also extend both theorems such that no promise classes are involved. Formerly, such gap theorems were known only for the unbalanced approach. (2) Polylog-time reductions can tremendously decrease dot-depth complexity (despite that these reductions cannot count). We construct languages of arbitrary dot-depth that are reducible to languages of dot-depth 1/2. (3) Unbalanced star-free leaf languages can be much stronger than balanced ones. We construct star-free regular languages L n such that L n 's balanced leaf-language class is NP, but the unbalanced leaf-language class of L n contains level n of the unambiguous alternation hierarchy. This demonstrates the power of unbalanced computations.

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Relating Automata-Theoretic Hierarchies to Complexity-Theoretic Hierarchies

Selivanov

2001

Fundamentals of Computation Theory

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Polynomial Closure and Unambiguous Product

Cited by 62 publications

References 3 publications

A note on partially ordered tree automata

A note on partially ordered tree automata

Languages polylog-time reducible to dot-depth 1/2

Relating Automata-Theoretic Hierarchies to Complexity-Theoretic Hierarchies

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