# Circuit complexity and the expressive power of generalized first-order formulas

## Abstract

The circuit complexity classes *AC*^{0}, *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 *AC*^{0}, *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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