Criticality in Formal Languages and Statistical Physics
This work connects formal language theory to statistical physics, offering insights for machine learning tasks like training recurrent neural networks, though it is incremental in its theoretical extensions.
The paper demonstrates that mutual information between symbols decays exponentially in probabilistic regular grammars but can follow a power law in context-free grammars, linking this to statistical physics concepts like phase transitions and turbulence.
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.