[Paper Review] Provably Stable Interpretable Encodings of Context Free Grammars in RNNs with a Differentiable Stack.

John Stogin, Ankur Mali+1 more|arXiv (Cornell University)|Jun 5, 2020

Machine Learning and Algorithms66 references4 citations

TL;DR

This paper proposes a differentiable, provably orbitally stable RNN with a specialized stack architecture that directly encodes context-free grammar (CFG) rules, enabling the model to learn and approximate pushdown automata (PDA) behavior for grammatical inference. The method ensures stable long-sequence processing and allows for interpretable extraction of PDA transition and stack operations from learned weights.

ABSTRACT

Given a collection of strings belonging to a context free grammar (CFG) and another collection of strings not belonging to the CFG, how might one infer the grammar? This is the problem of grammatical inference. Since CFGs are the languages recognized by pushdown automata (PDA), it suffices to determine the state transition rules and stack action rules of the corresponding PDA. An approach would be to train a recurrent neural network (RNN) to classify the sample data and attempt to extract these PDA rules. But neural networks are not a priori aware of the structure of a PDA and would likely require many samples to infer this structure. Furthermore, extracting the PDA rules from the RNN is nontrivial. We build a RNN specifically structured like a PDA, where weights correspond directly to the PDA rules. This requires a stack architecture that is somehow differentiable (to enable gradient-based learning) and stable (an unstable stack will show deteriorating performance with longer strings). We propose a stack architecture that is differentiable and that provably exhibits orbital stability. Using this stack, we construct a neural network that provably approximates a PDA for strings of arbitrary length. Moreover, our model and method of proof can easily be generalized to other state machines, such as a Turing Machine.

Motivation & Objective

• To address the challenge of grammatical inference from positive and negative string samples by learning the underlying context-free grammar (CFG).
• To design a recurrent neural network (RNN) architecture that explicitly models the structure of a pushdown automaton (PDA), ensuring interpretability of learned rules.
• To develop a stack mechanism that is both differentiable for gradient-based training and provably orbitally stable to maintain performance on long sequences.
• To enable the extraction of interpretable PDA transition and stack operation rules from trained network weights.

Proposed method

• The model uses a custom stack architecture with differentiable operations that maintain state stability through orbital stability, ensuring bounded error growth over time.
• The RNN is structured to directly emulate a PDA, where network weights correspond to PDA transition and stack operation rules.
• The stack employs a continuous, differentiable update rule that avoids discrete, non-differentiable operations common in standard RNNs.
• Orbital stability is proven via Lyapunov analysis, guaranteeing that small perturbations in input or hidden states do not lead to divergent behavior.
• The architecture is trained using standard backpropagation to classify strings as belonging or not belonging to a target CFG.
• The method generalizes beyond CFGs to other state machines, such as Turing machines, by extending the same stack and stability principles.

Experimental results

Research questions

• RQ1 Can a differentiable RNN with a stable stack architecture learn and represent the transition and stack operations of a pushdown automaton (PDA) for grammatical inference?
• RQ2 How can a neural network be designed so that its weights directly correspond to interpretable PDA rules?
• RQ3 What conditions ensure that the stack mechanism remains stable during long sequence processing, especially for strings longer than seen during training?
• RQ4 Can the model’s internal structure allow for the extraction of interpretable grammar rules from trained weights?
• RQ5 To what extent can this framework be generalized to other computational models, such as Turing machines?

Key findings

• The proposed stack architecture is provably orbitally stable, ensuring that the model maintains consistent behavior even with increasing sequence length.
• The RNN with the differentiable stack can approximate a PDA for strings of arbitrary length, overcoming the typical generalization limits of standard RNNs.
• The model’s weights directly encode interpretable PDA transition and stack operation rules, enabling rule extraction from trained networks.
• The method allows for effective grammatical inference from positive and negative string samples, learning the underlying CFG structure.
• The framework is generalizable to other state machines, such as Turing machines, by extending the stack and stability mechanisms.