On the Expressivity of Recurrent Neural Cascades with Identity
This work provides a theoretical limitation for RNC+ models in applications involving identity elements, such as temporal logic and pattern recognition, but it is incremental as it extends prior expressivity results.
The paper tackled the expressivity of Recurrent Neural Cascades with positive weights (RNC+) by proving that, in the presence of an identity element, they capture exactly the star-free regular languages, excluding the possibility of going beyond regular languages. This result applies broadly to temporal patterns, establishing a structural correspondence to semiautomata cascades and showing no greater succinctness than three-state semiautomata.
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.