Criticality in Formal Languages and Statistical Physics

TL;DR

This paper explores how mutual information decay patterns in formal languages relate to physical phenomena and discusses implications for machine learning, revealing fundamental differences between regular and context-free grammars.

Contribution

It establishes a connection between mutual information decay in formal languages and physical phase transitions, introducing the rational mutual information and analyzing complex Bayesian networks.

Findings

Mutual information decays exponentially in probabilistic regular grammars.

Mutual information can decay as a power law in context-free grammars.

Connections to phase transitions and power-law correlations in physics.

Abstract

We show that the mutual information between two symbols, as a function of the number of symbols between the two, decays exponentially in any probabilistic regular grammar, but can decay like a power law for a context-free grammar. This result about formal languages is closely related to a well-known result in classical statistical mechanics that there are no phase transitions in dimensions fewer than two. It is also related to the emergence of power-law correlations in turbulence and cosmological inflation through recursive generative processes. We elucidate these physics connections and comment on potential applications of our results to machine learning tasks like training artificial recurrent neural networks. Along the way, we introduce a useful quantity which we dub the rational mutual information and discuss generalizations of our claims involving more complicated Bayesian networks.