Non-commutative arithmetic circuits: depth reduction and size lower bounds

Abstract

We investigate the phenomenon of depth-reduction in commutative and non-commutative arithmetic circuits. We prove that in the commutative setting, uniform semi-unbounded arithmetic circuits of logarithmic depth are as powerful as uniform arithmetic circuits of polynomial degree (and unrestricted depth); earlier proofs did not work in the uniform setting. This also provides a unified proof of the circuit characterizations of the class LOGCFL and its counting variant #LOGCFL. We show that AC 1 has no more power than arithmetic circuits of polynomial size and degree n O( log log n) (improving the trivial bound of n O( log n) ). Connections are drawn between TC 1 and arithmetic circuits of polynomial size and degree. Then we consider non-commutative computation. We show that over the algebra ( ∑ ∗ , max, concat), arithmetic circuits of polynomial size and polynomial degree can be reduced to O( log 2 n) depth (and even to O( log n) depth if unbounded-fanin gates are allowed). This establishes that OptLOGCFL is in AC 1. This is the first depth-reduction result for arithmetic circuits over a non-commutative semiring, and it complements the lower bounds of Kosaraju and Nisan showing that depth reduction cannot be done in the general non-commutative setting. We define new notions called “short-left-paths” and “short-right-paths” and we show that these notions provide a characterization of the classes of arithmetic circuits for which optimal depth reduction is possible. This class also can be characterized using the AuxPDA model. Finally, we characterize the languages generated by efficient circuits over the semiring ( 2 ∑ ∗ , union, concat) in terms of simple one-way machines, and we investigate and extend earlier lower bounds on non-commutative circuits.