Non-commutative computation, depth reduction, and skew circuits (extended abstract)

Abstract

It is known that polynomial size polynomial degree Boolean circuits have equivalent semi-unbounded logarithmic depth circuits. For arithmetic circuits, the additional property of commutativity is sufficient for a similar depth reduction result.

We explore conditions under which depth reduction is possible even in non-commutative settings. Known results so far are of two types: specific operations for which depth reduction is possible, and specific instances where linear lower bounds on circuit depth exist. We show that any polynomial size polynomial degree circuit which has short-right-paths or short-left-paths — a purely syntactic notion which generalizes the notion of pushdown height — has an equivalent depth-reduced circuit. Our proof is a direct circuit construction which we also relate to generalized auxiliary pushdown machines.

Further, we extend the lower bounds on left-skew circuit size in a specific non-commutative setting, {UNION, CONCAT}, to more generalized skew circuits. We also show that the generalization is proper; there are problems hard for left-skew circuits which are easy in this model.

Work done while the author was with The Institute of Mathematical Sciences, Madras 600 113, India.