Computational Expressivity of Recurrent Neural Language Models

Core Concepts

Recurrent neural language models with rational weights and unbounded computation time can simulate any deterministic probabilistic Turing machine with rationally weighted transitions.

Abstract

This work investigates the computational expressivity of recurrent neural language models (RLMs). The authors first establish an upper bound on the expressive power of RLMs by showing that RLMs with rational weights and unbounded computation time can simulate any deterministic probabilistic Turing machine (PTM) with rationally weighted transitions.
To do this, the authors introduce a variant of probabilistic Turing machines called Rational-valued Probabilistic Turing Machines (QPTMs), which can have an arbitrary number of rationally weighted transition functions. They show that QPTMs are strongly equivalent to probabilistic two-stack pushdown automata (2PDAs).
The authors then review the classical construction by Siegelmann and Sontag (1992) for simulating unweighted Turing machines with recurrent neural networks (RNNs). They extend this construction to the probabilistic case, showing how a rationally weighted RLM with unbounded computation time (called an εRLM) can simulate any QPTM.
As a lower bound, the authors study a second type of RLMs restricted to operate in real-time, meaning they can only perform a constant number of computational steps per symbol. They show that these real-time RLMs can simulate deterministic real-time rational PTMs.
The authors conclude that their results provide a first step towards a comprehensive characterization of the expressivity of RLMs in terms of the classes of probability measures they can represent.

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Key Insights Distilled From

On the Representational Capacity of Recurrent Neural Language Models

by Franz Nowak,... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2310.12942.pdf

Deeper Inquiries

What is the exact computational power of a rationally weighted RLM without the assumption of unbounded computation time

The exact computational power of a rationally weighted Recurrent Neural Language Model (RLM) without the assumption of unbounded computation time is not definitively established in the context provided. The upper bound on the expressivity of RLMs is shown by their ability to simulate any deterministic probabilistic Turing machine with rationally weighted transitions. However, the lower bound on the computational power of RLMs is characterized by their ability to simulate real-time deterministic probabilistic two-stack pushdown automata (2PDA). This lower bound restricts the models to operate in real-time, processing a symbol at every time step, which is more aligned with practical implementations of RLMs. Further research may be needed to precisely determine the computational power of rationally weighted RLMs without the assumption of unbounded computation time.

Can the deterministic QPTMs introduced in this work be related to non-deterministic probabilistic finite-state automata, which cannot be represented by deterministic finite-state automata

Deterministic QPTMs introduced in this work can potentially be related to non-deterministic probabilistic finite-state automata that cannot be represented by deterministic finite-state automata. The deterministic QPTMs are a specific class of QPTMs that operate in real-time and have deterministic properties. Non-deterministic probabilistic finite-state automata, on the other hand, introduce non-determinism in their transitions, allowing for multiple possible paths of computation. Exploring the relationship between these deterministic QPTMs and non-deterministic probabilistic finite-state automata could provide insights into the computational power and expressive capabilities of these models.

Are the εRLMs introduced in this work weakly equivalent to non-deterministic QPTMs without the need to introduce two different types of ε symbols

The εRLMs introduced in this work are shown to be weakly equivalent to non-deterministic QPTMs without the need to introduce two different types of ε symbols. The εRLMs are a specific type of RLM that can output empty symbols (ε), allowing for an arbitrary number of computations per symbol of the string. By establishing the weak equivalence between εRLMs and non-deterministic QPTMs, the study shows that these models share similar computational capabilities without the requirement of additional symbols to store the direction of the head in the outputs. This equivalence highlights the flexibility and expressive power of εRLMs in representing probabilistic languages.