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In this paper we study the parameterized complexity of approximating the
parameterized counting problems contained in the class ,
the parameterized analogue of . We prove a parameterized analogue of a
famous theorem of Stockmeyer claiming that approximate counting belongs to
the second level of the polynomial hierarchy.
We display a complexity notion based on the syntax of a tree
series which yields two distinct hierarchies, one within the class of recognizable tree series and another one in the class of
non-recognizable tree series.
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