Abstract
We provide a hierarchy of tree languages recognised by nondeterministic parity tree automata with priorities in \(\{0,1,2\}\), whose length exceeds the first fixed point of the \(\varepsilon \) operation (that itself enumerates the fixed points of \(x\mapsto \omega ^x\)). We conjecture that, up to Wadge equivalence, it exhibits all regular tree languages of index \([0,2]\).
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Notes
- 1.
We emphasize that this is done without any determinacy principle. In particular, we do not require \(\mathbf {\Delta }^{1}_{2}\)-determinacy.
- 2.
Not to be mistaken with an \(\varepsilon \)-move.
- 3.
A pointclass is a collections of subsets of a topological space that is closed under continuous preimages.
- 4.
This is in general stronger than the usual \(A<_{W}B\) if and only if \(A\le _{W}B\) and \(B\not \le _{W}A\), but the two definitions coincide when the classes considered are determined.
- 5.
A player in charge of
in a conciliatory game is like a player in charge of \(L(\mathcal {A})\), but with the extra possibility at any moment of the play to reach a definitively rejecting position. - 6.
\( \varepsilon = {\left\{ \begin{array}{ll} -1 &{}\text { if } d_{c}(L)<\omega ; \\ 0 &{}\text { if } d_{c}(L) = \beta + n \text { and } {{\mathrm{cof}}}(\beta ) = \omega _{1}; \\ 1 &{}\text { if } d_{c}(L) = \beta + n \text { and } {{\mathrm{cof}}}(\beta ) = \omega . \end{array}\right. } \).
- 7.
Another way to characterise \(\varphi _{2}(0)\) is to remember that an ordinal is the set of its predecessors and notice that a nonzero ordinal is of the form respectively \(\omega ^\alpha \) iff it is closed under addition and \(\varepsilon _\alpha \) iff it is closed under \(x\longmapsto \omega ^x\). Then \(\varphi _{2}(0)\) is the first non null ordinal closed under \(x\longmapsto \varepsilon _x\) as well as \(x\longmapsto \omega ^x\) and \(x,y\longmapsto x+y\).
- 8.
Notice that we have \({\alpha _i}={\left( \omega ^{\omega }\right) }^{\alpha _i}\iff {\alpha _i}={\omega ^{\alpha _i}}\).
in a conciliatory game is like a player in charge of References
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Duparc, J., Fournier, K. (2015). A Tentative Approach for the Wadge-Wagner Hierarchy of Regular Tree Languages of Index [0, 2]. In: Shallit, J., Okhotin, A. (eds) Descriptional Complexity of Formal Systems. DCFS 2015. Lecture Notes in Computer Science(), vol 9118. Springer, Cham. https://doi.org/10.1007/978-3-319-19225-3_7
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