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Counting Hierarchies: Polynomial Time And Constant Depth Circuits
, 1990
"... In the spring of 1989, Seinosuke Toda of the University of ElectroCommunications in Tokyo, Japan, proved that the polynomial hierarchy is contained in P PP [To89]. In this Structural Complexity Column, we will briefly review Toda's result, and explore how it relates to other topics of interest i ..."
Abstract

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In the spring of 1989, Seinosuke Toda of the University of ElectroCommunications in Tokyo, Japan, proved that the polynomial hierarchy is contained in P PP [To89]. In this Structural Complexity Column, we will briefly review Toda's result, and explore how it relates to other topics of interest in computer science. In particular, we will introduce the reader to The Counting Hierarchy: a hierarchy of complexity classes contained in PSPACE and containing the Polynomial Hierarchy. Threshold Circuits: circuits constructed of MAJORITY gates; this notion of circuit is being studied not only by complexity theoreticians, but also by researchers in an active subfield of AI studying "neural networks". Along the way, we'll review the important notion of an operator on a complexity class. 1. The Counting Hierarchy, and Operators on Complexity Classes The counting hierarchy was defined in [Wa86] and independently by Parberry and Schnitger in [PS88]. (The motivation for [Wa86] was the desir...