Critical Behavior in Physics and Probabilistic Formal Languages
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...
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| Format: | Article |
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MDPI AG
2018
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| Online Access: | http://hdl.handle.net/1721.1/114221 https://orcid.org/0000-0001-7670-7190 |
| _version_ | 1826199442769838080 |
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| author | Lin, Henry Tegmark, Max Tegmark, Max Erik |
| author2 | Massachusetts Institute of Technology. Department of Physics |
| author_facet | Massachusetts Institute of Technology. Department of Physics Lin, Henry Tegmark, Max Tegmark, Max Erik |
| author_sort | Lin, Henry |
| collection | MIT |
| description | 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 |
| first_indexed | 2024-09-23T11:20:18Z |
| format | Article |
| id | mit-1721.1/114221 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T11:20:18Z |
| publishDate | 2018 |
| publisher | MDPI AG |
| record_format | dspace |
| spelling | mit-1721.1/1142212022-09-27T18:49:37Z Critical Behavior in Physics and Probabilistic Formal Languages Lin, Henry Tegmark, Max Tegmark, Max Erik Massachusetts Institute of Technology. Department of Physics MIT Kavli Institute for Astrophysics and Space Research Tegmark, Max Erik 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 2018-03-19T18:54:50Z 2018-03-19T18:54:50Z 2017-06 2017-06 2018-02-16T19:12:38Z Article http://purl.org/eprint/type/JournalArticle 1099-4300 http://hdl.handle.net/1721.1/114221 Lin, Henry, and Max Tegmark. “Critical Behavior in Physics and Probabilistic Formal Languages.” Entropy 19, 12 (June 2017): 299 © 2017 The Author(s) https://orcid.org/0000-0001-7670-7190 http://dx.doi.org/10.3390/E19070299 Entropy Attribution 4.0 International (CC BY 4.0) https://creativecommons.org/licenses/by/4.0/ application/pdf MDPI AG MDPI |
| spellingShingle | Lin, Henry Tegmark, Max Tegmark, Max Erik Critical Behavior in Physics and Probabilistic Formal Languages |
| title | Critical Behavior in Physics and Probabilistic Formal Languages |
| title_full | Critical Behavior in Physics and Probabilistic Formal Languages |
| title_fullStr | Critical Behavior in Physics and Probabilistic Formal Languages |
| title_full_unstemmed | Critical Behavior in Physics and Probabilistic Formal Languages |
| title_short | Critical Behavior in Physics and Probabilistic Formal Languages |
| title_sort | critical behavior in physics and probabilistic formal languages |
| url | http://hdl.handle.net/1721.1/114221 https://orcid.org/0000-0001-7670-7190 |
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