Bridging Neural and Symbolic Computation: A Learnability Study of RNNs on Counter and Dyck Languages

Neisarg Dave, Daniel Kifer, C. Lee Giles, Ankur Mali

Keywords: Neuro-Symbolic Learning, Recurrent Neural Networks, Formal Language Classification, Dyck Languages, Counter Languages, Finite Precision, Learnability, Automata Theory, Symbolic Memory, Stability and Generalization

Abstract: This work presents a neuro-symbolic analysis of the learnability of Recurrent Neural Networks (RNNs) in classifying structured formal languages—specifically, **counter languages** and **Dyck languages**, which serve as canonical examples of context-free and mildly context-sensitive grammars. While prior studies have highlighted the expressive power of first-order (LSTM) and second-order (O2RNN) architectures within the Chomsky hierarchy, we challenge this perspective by shifting the focus from theoretical expressivity to *practical learnability under finite precision constraints*. Our results suggest that RNNs function more as finite-state machines than stack-based automata when implemented with realistic training regimes and embedding representations. We show that classification performance degrades sharply as structural similarities between positive and negative sequences increase—highlighting a core limitation in the RNN's ability to internalize hierarchical structure without symbolic scaffolding. Interestingly, even simple linear classifiers built on top of RNN-derived embeddings outperform chance, underscoring the hidden representational capacity within learned states. To probe generalization, we train models on input lengths up to 40 and evaluate on lengths extending to 500, using 10 distinct seeds to measure statistical robustness. O2RNNs consistently demonstrate greater stability and generalization compared to LSTMs, particularly under varied initialization strategies. These findings expose the fragility of learned language representations and emphasize the role of architectural bias, initialization, and data sampling in determining what is truly learnable. Ultimately, our study reframes RNN learnability through the lens of *symbolic structure and computational constraints*, advocating for stronger formal criteria when assessing neural models’ capacity to reason over structured sequences. We argue that expressivity alone is insufficient—**stability, precision, and symbolic alignment** are essential for true neuro-symbolic generalization.

Track: Neurosymbolic Methods for Trustworthy and Interpretable AI

Paper Type: Long Paper

Resubmission: No

Submission Number: 58