Counting Hierarchies: Polynomial Time and Constant Depth Circuits

In the spring of 1989, Seinosuke Toda of the University of Electro-Communications in Tokyo, Japan, proved that the polynomial hierarchy is contained in P PP To-89]. In this Structural Complexity Column, we will brieey 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. of circuit is being studied not only by complexity theoreticians, but also by researchers in an active subbeld of AI studying \neural networks". Along the way, we'll review the important notion of an operator on a complexity class. The counting hierarchy was deened in Wa-86] and independently by Parberry and Schnitger in PS-88]. (The motivation for Wa-86] was the desire to classify precisely the complexity of certain combinatorial problems on graphs with succinct descriptions. Parberry and Schnitger were studying \threshold Turing machines" in connection with parallel computation.) One way to deene the counting hierarchy is to take the usual deenition of the polynomial hierarchy:

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