Abstract
This paper treates classes in the polynomial hierarchy of type two,\(\left\{ {\square _n^{0,p} ,\Delta _n^{0,p} ,\Sigma _n^{0,p} \Pi _n^{0,p} |n \geqslant 0} \right\}\), that were first developed by Townsend as a natural extension of the Meyer-Stockmeyer polynomial hierarchy in complexity theory. For these classes, it is discussed whether each of them has the extension property and the three recursion-theoretic properties: separation, reduction, and pre-wellordering. This paper shows that every\(\Pi _n^{0,p} ,n > 0\), lacks the pre-wellordering property by using a probabilistic argument on constant-depth Boolean circuits. From the assumption NP = coNP it follows by a pruning argument that\(\Sigma _n^{0,p} \) has the separation and extension properties.
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Yamakami, T. Structural properties for feasibly computable classes of type two. Math. Systems Theory 25, 177–201 (1992). https://doi.org/10.1007/BF01374524
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DOI: https://doi.org/10.1007/BF01374524