Non - Commutative Arithmetic Circuits : DepthReduction and Size Lower

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; earlier proofs did not work in the uniform setting. This also provides a uni ed proof of the circuit characterizations of LOGCFL and #LOGCFL. We show that AC1 has no more power than arithmetic circuits of polynomial size and degree nO(log logn) (improving the trivial bound of nO(logn)). Connections are drawn between TC1 and arithmetic circuits of polynomial size and degree. Then we consider non-commutative computation, and show that some depth reduction is possible over the algebra ( ; max, concat), thus establishing that OptLOGCFL is in AC1. This is the rst depth-reduction result for arithmetic circuits over a noncommutative semiring, and it complements the lower bounds of Kosaraju and Nisan showing that depth reduction cannot be done in the general noncommutative setting. We de ne 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 e cient circuits over the (union, concat) semiring in terms of simple one-way machines, and we investigate and extend earlier lower bounds on non-commutative circuits.1 1The results in this paper were originally announced in papers in Proc. 25th ACM Symposium on Theory of Computing and in Proc. 14th Conference on Foundations of Software Technology and Theoretical Computer Science, Lecture Notes in Computer Science 880.