Ranking and formal power series

Abstract

We prove that the ranking problem for unambiguous context-free languages is NC 1-reducible to the value problem for algebraic formal power series in noncommuting variables. In the particular case of regular languages we show that the problem of ranking is NC 1-reducible to the problem of counting the number of strings of given length in suitable regular languages. As a consequence ranking problems for regular languages are NC 1-reducible to integer division and hence computable by log-space uniform boolean circuits of polynomial size and depth O(log n log log n), or by P-uniform boolean circuits of polynomial size and depth O(log n).