Circuit complexity and the expressive power of generalized first-order formulas
The circuit complexity classes AC0, ACC, and CC (also called pure-ACC) can be characterized as the classes of languages definable in certain extensions of first-order logic. All of the known and conjectured inclusions among these classes have been shown to be equivalent to a single conjecture concerning the form of the formulas required to define the regular languages they contain. (The conjecture states, roughly, that when a formula defines a regular language, predicates representing numerical relations on the positions in a string can be replaced by predicates computed by finite state automata.) Here this conjecture is established in a special case: It is shown that the conjecture holds for the subclasses of AC0, ACC, and CC defined by restricting all the numerical predicates occurring in the logical formulas to be either unary relations, or the order relation <.
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