The Current State of Circuit Lower

1. Abstract We review the existing results showing languages to be outside of complexity classes deened by tight constraints on boolean circuits, using the new characterizations of these classes in terms of automata theory BT88] and formal logic BIS88]. We outline the methods and results in the case of three types of classes deened by circuits of constant depth, polynomial size, and unbounded fan-in, for three diierent types of gates. These are AND and OR gates FSS84], AND, OR, and MOD-p gates Ra87, Sm87], and modular gates alone with certain other restrictions BST90]. Most of this survey was prepared for the McGill University Workshop in Theoretical Computer Science, held in February 1989. 2. Introduction It would be nice to show that the CLIQUE problem is not solved by a family of boolean circuits where circuit size grows polynomially with input size. This would not only show P 6 = NP, but also would demonstrate our \understanding" of polynomial-size circuits in a particular sense. We would then be provably able to separate languages computable by such circuits from languages not computable by them but near them in the complexity hierarchy. Less ambitiously, it would be nice to show this sort of understanding of the class NC 1 (languages computable by boolean circuits of depth O(log n) and fan-in two, or by boolean formulas of polynomial length), a class of \easy" problems in our usual reckoning. So far, however, we cannot show a language to be outside of NC 1 unless it is very complex indeed | we cannot rule out the possibility NC 1 = NP. One approach to this goal has been to move even lower in the complexity hierarchy, to classes which we do \understand". In this survey we will examine the techniques currently known for showing languages to be outside of circuit complexity classes. Our 1