NONCOMMUTATIVE ARITHMETIC CIRCUITS - DEPTH REDUCTION AND SIZE LOWER BOUNDS

We investigate the phenomenon of depth-reduction in commutative and no n-commutative arithmetic circuits. We prove that in the commutative se tting, 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 se tting. This also provides a unified proof of the circuit characterizat ions of the class LOGCFL and its counting variant #LOGCFL. We show tha t 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 sire drawn between TC1 and arithmetic circuits of polyn omial size and degree. Then we consider non-commutative computation. W e show that over the algebra (Sigma, max, concat), arithmetic circuit s of polynomial size and polynomial degree can be reduced to O(log(2) n) depth land even to O(log n) depth if unbounded-fanin gates are allo wed). This establishes that OptLOGCFL is in AC(1). This is the first d epth-reduction result for arithmetic circuits over a non-commutative s emiring, and it complements the lower bounds of Kosaraju and Nisan sho wing that depth reduction cannot be done in the general non-commutativ e setting. We define new notions called ''short-left-paths'' and ''sho rt-right-paths'' and we show that these notions provide a characteriza tion of the classes of arithmetic circuits for which optimal depth red uction is possible. This class also can be characterized using the Aux PDA model. Finally, we characterize the languages generated by efficie nt circuits over the semiring (2(Sigma), union, concat) in terms of s imple one-way machines, and we investigate and extend earlier lower bo unds on non-commutative circuits. (C) 1998-Elsevier Science B.V. All r ights reserved.