We investigate computational complexity of questions of various problems for simple recurrent neural networks (RNNs) as formal models for recognizing weighted languages. We focus on the single-layer, ReLU-activation, rational-weight RNNs with softmax, which are commonly used in natural language processing applications. We show that most problems for such RNNs are undecidable, including consistency, equivalence, minimization, and finding the highest-weighted string. However, for consistent RNNs the last problem becomes decidable, although the solution can be exponentially long. If additionally the string is limited to polynomial length, the problem becomes NP-complete and APX-hard. In summary, this shows that approximations and heuristic algorithms are necessary in practical applications of those RNNs. We also consider RNNs as unweighted language recognizers and situate RNNs between Turing Machines and Random-Access Machines regarding their real-time recognition powers.
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