A profinite approach to complete bifix decodings of recurrent languages

TL;DR

This paper uses profinite monoids to analyze complete bifix decodings of recurrent languages, proving that such decodings preserve recurrence properties, thus advancing the theoretical understanding of language decoding.

Contribution

It introduces a profinite monoid framework to study bifix decodings, demonstrating recurrence preservation in the process, which is a novel theoretical approach.

Findings

Complete bifix decoding of a uniformly recurrent language remains uniformly recurrent.

An analogous recurrence preservation result holds for recurrent languages.

The approach provides new insights into the structure of language decodings.

Abstract

We approach the study of complete bifix decodings of (uniformly) recurrent languages with the help of the free profinite monoid. We show that the complete bifix decoding of a uniformly recurrent language $F$ by an $F$-charged rational complete bifix code is uniformly recurrent. An analogous result is obtained for recurrent languages.