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Regular Language Distance and Entropy

Authors Austin J. Parker, Kelly B. Yancey, Matthew P. Yancey

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Austin J. Parker
Kelly B. Yancey
Matthew P. Yancey

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Austin J. Parker, Kelly B. Yancey, and Matthew P. Yancey. Regular Language Distance and Entropy. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.MFCS.2017.3

Abstract

This paper addresses the problem of determining the distance between two regular languages. It will show how to expand Jaccard distance, which works on finite sets, to potentially-infinite regular languages. The entropy of a regular language plays a large role in the extension. Much of the paper is spent investigating the entropy of a regular language. This includes addressing issues that have required previous authors to rely on the upper limit of Shannon's traditional formulation of channel capacity, because its limit does not always exist. The paper also includes proposing a new limit based formulation for the entropy of a regular language and proves that formulation to both exist and be equivalent to Shannon's original formulation (when it exists). Additionally, the proposed formulation is shown to equal an analogous but formally quite different notion of topological entropy from Symbolic Dynamics -- consequently also showing Shannon's original formulation to be equivalent to topological entropy. Surprisingly, the natural Jaccard-like entropy distance is trivial in most cases. Instead, the entropy sum distance metric is suggested, and shown to be granular in certain situations.
Keywords
  • regular languages
  • channel capacity
  • entropy
  • Jaccard
  • symbolic dynamics

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