| Summary: | 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 starfree 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.
|
|---|