The Complexity of Two Register and Skew Arithmetic Computation

We study two register arithmetic computation and skew arithmetic circuits. Our main results are the following: • For commutative computations, we show that 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 k ≤ n ω(logn) . – We give a deterministic (white box) polynomial-time identity testing algorithm for noncommutative skew circuits. 1 Two register arithmetic computations An arithmetic circuit over a field F and indeterminates X = {x1, x2, · · · , xn} is a directed acyclic graph with each node of indegree zero labeled by a variable or a scalar constant. Each internal node g of the DAG is labeled by + or × (i.e. it is a plus or multiply gate) and is of indegree two. A node of the DAG is designated as the output gate. Each gate of the arithmetic circuit computes a polynomial, in the commutative ring F[X], by adding or multiplying its input polynomials. The polynomial computed at the output gate is the polynomial computed by the circuit. If the indeterminates X = {x1, x2, · · · , xn} are noncommuting with no relations between them, then the circuit is called a noncommutative circuit and it computes a polynomial in the free noncommutative ring F〈X〉. 1 ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 28 (2014)