Recurrent Neural Cascades (RNC) are the class of recurrent neural networks with no cyclic dependencies among recurrent neurons. Their subclass RNC+ with positive recurrent weights has been shown to be closely connected to the star-free regular languages, which are the expressivity of many well-established temporal logics. The existing expressivity results show that the regular languages captured by RNC+ are the star-free ones, and they leave open the possibility that RNC+ may capture languages beyond regular. We exclude this possibility for languages that include an identity element, i.e., an input that can occur an arbitrary number of times without affecting the output. Namely, in the presence of an identity element, we show that the languages captured by RNC+ are exactly the star-free regular languages. Identity elements are ubiquitous in temporal patterns, and hence our results apply to a large number of applications. The implications of our results go beyond expressivity. At their core, we establish a close structural correspondence between RNC+ and semiautomata cascades, showing that every neuron can be equivalently captured by a three-state semiautomaton. A notable consequence of this result is that RNC+ are no more succinct than cascades of three-state semiautomata.

翻译：递归神经级联（RNC）是递归神经元之间不存在循环依赖关系的递归神经网络类别。其子类RNC+（具有正递归权重）已被证明与星自由正则语言密切相关，而这类语言正是许多成熟时序逻辑的表达能力范畴。现有表达能力结果表明，RNC+捕获的正则语言即为星自由语言，但尚存RNC+可能捕获超越正则语言的可能性。我们排除了含恒等元语言（即存在一个可无限次出现且不影响输出的输入元素）的此类可能性：具体而言，在存在恒等元的情况下，我们证明RNC+捕获的语言恰好是星自由正则语言。恒等元在时序模式中普遍存在，因此我们的结论适用于大量应用场景。该成果的意义远不止于表达能力范畴。其核心在于，我们建立了RNC+与半自动机级联之间的紧密结构对应关系，证明每个神经元均可等价地由三态半自动机刻画。该结论的一个重要推论是：RNC+并不比三态半自动机级联更具简洁性。