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; earlier proofs did not work in the uniform setting. This also provides a unified proof of the circuit characterizations of LOGCFL and \#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, and show that some depth reduction is possible over the algebra ($\Sigma^*,$ max, concat), thus establishing that OptLOGCFL is in AC$^1$. This is the first depth-reduction result for arithmetic circuits over a noncommutative semiring, and it complements the lower bounds of \cite{nisan,kosaraju} showing that depth reduction cannot be done in the general noncommutative 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 (union, concat) semiring in terms of simple one-way machines, and we investigate and extend earlier lower bounds on non-commutative circuits.