On the entropy of regular languages

Authors:
Tullio Ceccherini-Silberstein;Antonio Machì;Fabio Scarabotti
Affiliations:
Dipartimento di Ingegneria, Università del Sannio, Corso Garibaldi 107, I-82100 Benevento, Italy;Dipartimento di Matematica "G. Castelnuovo", Università degli Studi di Roma "La Sapienza", P.le A. Moro 5, 00185 Roma, Italy;Dipartimento MeMoMat, Università degli Studi di Roma "La Sapienza", Via A. Scarpa 16, 00161 Roma, Italy
Venue:
Theoretical Computer Science - WORDS
Year:
2003

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Abstract

Let L be an irreducible regular language. Let W be a non-empty set of words (or sub-words) of L and denote by LW = {v ∈ L:w ⊏ v, ∀w ∈ W} the language obtained from L by forbidding all the words w in W. Then the entropy decreases strictly: ent(LW) L). In this note we present a new proof of this fact, based on a method of Gromov, which avoids the Perron-Frobenius theory. This result applies to the regular languages of finitely generated free groups and an additional application is presented.