On the Expressivity of Recurrent Neural Cascades with Identity

TL;DR

This paper characterizes the expressivity of Recurrent Neural Cascades with identity, showing they precisely capture star-free regular languages in the presence of an identity element, and establishes a structural link to semiautomata cascades.

Contribution

It proves that RNC+ with an identity element exactly captures star-free regular languages and relates RNC+ to three-state semiautomata, clarifying their expressivity and structural properties.

Findings

RNC+ with identity elements capture exactly star-free regular languages.

Every neuron in RNC+ can be represented by a three-state semiautomaton.

RNC+ are not more succinct than cascades of three-state semiautomata.

Abstract

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.