Counting Problems and Algebraic Formal Power Series in Noncommuting Variables

Abstract In this work we study the complexity of certain counting functions related to formal power series in noncommuting variables. We prove that, for every algebraic formal power series in Z 《∑》, the problem of computing the corresponding counting function is NC1-reducible to integer division. As a consequence, for every unambiguous context-free language L⊆∑*, the problem of computing #{xϵL:|x|=n} is also NC1-reducible to integer division. Therefore all t hese counting problems are solvable by families of log-space uniform boolean circuits of depth O(log n log log n) and polynomial size.