The Logarithmic Alternation Hierarchy Collapses: A∑2L = A∏2L

Abstract We show that AΣ 2 L = AΣ k L , k ≥ 2, by proving that AΣ 2 L coincides with AΠ 2 L . Essentially this is done by reducing the AΣ 2 L -complete set (GAP¢CoGap) (ℶ) to the question whether of two vectors A and B of n components, A contains more “solvable” components, i.e., components which are contained in GAP, than B . Moreover, using a similar technique we show AΣ 2 L = L hd (NL) . Finally, we consider the relevance of our proof technique for polynomial time classes, e.g., the Boolean NP -Hierarchy.

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