Stronger Separation of Analog Neuron Hierarchy by Deterministic Context-Free Languages

We analyze the computational power of discrete-time recurrent neural networks (NNs) with the saturated-linear activation function within the Chomsky hierarchy. This model restricted to integer weights coincides with binary-state NNs with the Heaviside activation function, which are equivalent to finite automata (Chomsky level 3) recognizing regular languages (REG), while rational weights make this model Turing-complete even for three analog-state units (Chomsky level 0). For the intermediate model $\alpha$ANN of a binary-state NN that is extended with $\alpha\geq 0$ extra analog-state neurons with rational weights, we have established the analog neuron hierarchy 0ANNs $\subset$ 1ANNs $\subset$ 2ANNs $\subseteq$ 3ANNs. The separation 1ANNs $\subsetneqq$ 2ANNs has been witnessed by the non-regular deterministic context-free language (DCFL) $L_\#=\{0^n1^n\mid n\geq 1\}$ which cannot be recognized by any 1ANN even with real weights, while any DCFL (Chomsky level 2) is accepted by a 2ANN with rational weights. In this paper, we strengthen this separation by showing that any non-regular DCFL cannot be recognized by 1ANNs with real weights, which means (DCFLs $\setminus$ REG) $\subset$ (2ANNs $\setminus$ 1ANNs), implying 1ANNs $\cap$ DCFLs = 0ANNs. For this purpose, we have shown that $L_\#$ is the simplest non-regular DCFL by reducing $L_\#$ to any language in this class, which is by itself an interesting achievement in computability theory.