A formal series in noncommuting variables $\Sigma$ over the rationals is a mapping $\Sigma^* \to \mathbb Q$. We say that a series is commutative if the value in the output does not depend on the order of the symbols in the input. The commutativity problem for a class of series takes as input a (finite presentation of) a series from the class and amounts to establishing whether it is commutative. This is a very natural, albeit nontrivial problem, which has not been considered before from an algorithmic perspective. We show that commutativity is decidable for all classes of series that constitute a so-called effective prevariety, a notion generalising Reutenauer's varieties of formal series. For example, the class of rational series, introduced by Sch\"utzenberger in the 1960's, is well-known to be an effective (pre)variety, and thus commutativity is decidable for it. In order to showcase the applicability of our result, we consider classes of formal series generalising the rational ones. We consider polynomial automata, shuffle automata, and infiltration automata, and we show that each of these models recognises an effective prevariety of formal series. Consequently, their commutativity problem is decidable, which is a novel result. We find it remarkable that commutativity can be decided in a uniform way for such disparate computation models. Finally, we present applications of commutativity outside the theory of formal series. We show that we can decide solvability in sequences and in power series for restricted classes of algebraic difference and differential equations, for which such problems are undecidable in full generality. Thanks to this, we can prove that the syntaxes of multivariate polynomial recursive sequences and of constructible differentially algebraic power series are effective, which are new results which were left open in previous work.

翻译：非交换变量上的形式级数是映射 $\Sigma^* \to \mathbb Q$。若级数的输出值不依赖于输入符号的顺序，则称其为交换级数。对于某类级数的可交换性问题，其输入为该类级数的一个（有限表示），目标是判定其是否具有交换性。这是一个非常自然但非平凡的问题，此前尚未从算法角度进行过研究。我们证明，对于所有构成所谓有效预簇的级数类，可交换性是可判定的；有效预簇这一概念推广了Reutenauer的形式级数簇。例如，由Schützenberger于20世纪60年代引入的有理级数类已知是一个有效（预）簇，因此其可交换性是可判定的。为展示我们结果的适用性，我们考虑推广有理级数的形式级数类。我们研究了多项式自动机、shuffle自动机和infiltration自动机，并证明这些模型各自识别一个有效的形式级数预簇。因此，它们的可交换性问题是可判定的，这是一个新的结果。我们发现，对于如此不同的计算模型，可交换性能以统一的方式判定，这一点值得关注。最后，我们展示了可交换性在形式级数理论之外的应用。我们证明，对于受限的代数差分方程和微分方程类，可以判定其在序列和幂级数中的可解性；而这类问题在一般情况下是不可判定的。基于此，我们可以证明多元多项式递归序列的语法以及可构造微分代数幂级数的语法是有效的，这些是先前工作中悬而未决的新结果。