On the Expressivity of Recurrent Neural Cascades with Identity
Nadezda Alexandrovna Knorozova, Alessandro Ronca
TL;DR
This paper characterizes the expressivity of Recurrent Neural Cascades in the presence of an identity element. Building on the known result that $RNC^{+}$ realises exactly the star-free regular languages, it shows that when an identity element is present, the expressivity remains within the star-free (aperiodic) regime, and it establishes a structural equivalence to cascades of three-state semiautomata. The core technical contribution is a lemma proving that any $RNC^{+}$ implementing a function with an identity element can be represented by a cascade of three-state automata, yielding a $3^n$ state bound and implying no greater succinctness than such cascades. Consequently, languages with an identity element recognised by $RNC^{+}$ are star-free regular, completing the expressivity characterization in the identity-enabled setting. These results offer theoretical guarantees for learning temporal patterns with $RNC^{+}$ and set the stage for exploring non-identity, negative-weight, or alternative-activation regimes.
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.