Counting problems and algebraic formal power series in noncommuting variables.(English)Zbl 0695.68053

Summary: 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 $${\mathbb{Z}}<<\Sigma >>$$, the problem of computing the corresponding counting function is $$NC^ 1$$-reducible to integer division. As a consequence, for every unambiguous context-free language $$L\subseteq \Sigma^*$$, the problem of computing #$$\{$$ $$x\in L: | x| =n\}$$ is also $$NC^ 1$$-reducible to integer division. Therefore all these counting problems are solvable by families of log- space uniform Boolean circuits of depth O(log n log log n) and polynomial size.

MSC:

 68Q45 Formal languages and automata 68Q25 Analysis of algorithms and problem complexity
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References:

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