In this paper we obtain complexity bounds for computational problems on algebraic power series over several commuting variables. The power series are specified by systems of polynomial equations: a formalism closely related to weighted context-free grammars. We focus on three problems -- decide whether a given algebraic series is identically zero, determine whether all but finitely many coefficients are zero, and compute the coefficient of a specific monomial. We relate these questions to well-known computational problems on arithmetic circuits and thereby show that all three problems lie in the counting hierarchy. Our main result improves the best known complexity bound on deciding zeroness of an algebraic series. This problem is known to lie in PSPACE by reduction to the decision problem for the existential fragment of the theory of real closed fields. Here we show that the problem lies in the counting hierarchy by reduction to the problem of computing the degree of a polynomial given by an arithmetic circuit. As a corollary we obtain new complexity bounds on multiplicity equivalence of context-free grammars restricted to a bounded language, language inclusion of a nondeterministic finite automaton in an unambiguous context-free grammar, and language inclusion of a non-deterministic context-free grammar in an unambiguous finite automaton.

翻译：本文研究了多个交换变量上代数形式幂级数的计算问题的复杂度上界。这些形式幂级数由多项式方程组指定，该形式化方法与加权上下文无关文法密切相关。我们聚焦于三个问题：判断给定代数级数是否恒为零、确定是否仅有有限个系数非零，以及计算特定单项式的系数。我们将这些问题与算术电路上的经典计算问题关联，从而证明这三个问题均属于计数层次。我们的主要结果改进了代数级数零性判定问题的最佳已知复杂度上界。该问题已知可通过归约到实闭域理论存在性片段的判定问题而属于PSPACE。本文通过将其归约到算术电路给定多项式次数的计算问题，证明该问题属于计数层次。作为推论，我们获得了有界语言限制下上下文无关文法的多重等价性、非确定性有限自动机语言包含于无歧义上下文无关文法、以及非确定性上下文无关文法语言包含于无歧义有限自动机这三大问题的新复杂度上界。