Critical Behavior in Physics and Probabilistic Formal Languages
Name
entropy-19-00299-v2.pdf
Size
1.47 MB
Format
Adobe PDF
Checksum (MD5)
7ba1070f8a3cd571b5afb0a47fab37f7
Author(s)
Lin, Henry
Tegmark, Max
Tegmark, Max Erik
Date Issued
June 2017
Journal
Entropy
Publisher
MDPI AG
Citation
Lin, Henry, and Max Tegmark. “Critical Behavior in Physics and Probabilistic Formal Languages.” Entropy 19, 12 (June 2017): 299 © 2017 The Author(s)
Version
Final published version
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. Keywords: formal languages; statistical mechanics; criticality
MIT Department
Massachusetts Institute of Technology. Department of Physics
MIT Kavli Institute for Astrophysics and Space Research
Terms of Use
Attribution 4.0 International (CC BY 4.0)
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.3390/E19070299
