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Entropy 2017, 19(7), 299;

Critical Behavior in Physics and Probabilistic Formal Languages

Department of Physics, Harvard University, Cambridge, MA 02138, USA
Department of Physics & MIT Kavli Institute, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Author to whom correspondence should be addressed.
Received: 15 December 2016 / Revised: 12 June 2017 / Accepted: 12 June 2017 / Published: 23 June 2017
(This article belongs to the Section Information Theory, Probability and Statistics)
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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. View Full-Text
Keywords: formal languages; statistical mechanics; criticality formal languages; statistical mechanics; criticality

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Lin, H.W.; Tegmark, M. Critical Behavior in Physics and Probabilistic Formal Languages. Entropy 2017, 19, 299.

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