The complexity of bounded register and skew arithmetic computation

Abstract

We study two register arithmetic computation and skew arithmetic circuits. Our main results are the following: For commutative computations, we present an exponential circuit size lower bound for a model of 2-register straight-line programs (SLPs) which is a universal model of computation (unlike width-2 algebraic branching programs that are not universal [AW11]). For noncommutative computations, we show that Coppersmith's 2-register SLP model [BOC88], which can efficiently simulate arithmetic formulas in the commutative setting, is not universal. However, assuming the underlying noncommutative ring has quaternions, Coppersmith's 2-register model can simulate noncommutative formulas efficiently. We consider skew noncommutative arithmetic circuits and show: An exponential separation between noncommutative monotone circuits and noncommutative monotone skew circuits. We define k-regular skew circuits and show that (k+1)-regular skew circuits are strictly powerful than k-regular skew circuits, where(equation present). © 2014 Springer International Publishing Switzerland.