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：循环神经级联（RNC）是指循环神经元间不存在循环依赖的循环神经网络类别。其子类RNC+（具有正循环权重）已被证明与无星号正则语言密切相关，后者是许多成熟时序逻辑的表达能力载体。现有表达能力结果表明，RNC+所捕获的正则语言正是无星号正则语言，但尚未排除RNC+可能捕获超越正则语言的可能性。针对包含恒等元（即出现任意次数均不影响输出的输入元素）的语言，我们排除了这种可能性。具体而言，在存在恒等元的条件下，我们证明RNC+捕获的语言恰好就是无星号正则语言。恒等元在时序模式中普遍存在，因此我们的结果适用于大量实际应用场景。本研究的价值不仅限于表达能力分析。我们在核心层面建立了RNC+与半自动机级联之间的紧密结构对应关系，证明每个神经元都可由三状态半自动机等价实现。这一结果的重要推论是：RNC+在表达简洁性上并不优于三状态半自动机级联。