Aperiodic String Transducers

Abstract

Regular string-to-string functions enjoy a nice triple characterization through deterministic two-way transducers ($\mathrm {2DFT}$), streaming string transducers ($\mathrm {SST}$) and $\mathrm {MSO}$ definable functions. This result has recently been lifted to $\mathrm {FO}$ definable functions, with equivalent representations by means of aperiodic $\mathrm {2DFT}$ and aperiodic 1-bounded $\mathrm {SST}$, extending a well-known result on regular languages. In this paper, we give three direct transformations: (i) from 1-bounded $\mathrm {SST}$ to $\mathrm {2DFT}$, (ii) from $\mathrm {2DFT}$ to copyless $\mathrm {SST}$, and (iii) from k-bounded to 1-bounded $\mathrm {SST}$. We give the complexity of each construction and also prove that they preserve the aperiodicity of transducers. As corollaries, we obtain that $\mathrm {FO}$ definable string-to-string functions are equivalent to $\mathrm {SST}$ whose transition monoid is finite and aperiodic, and to aperiodic copyless $\mathrm {SST}$.

This work is supported by the ARC project Transform (French speaking community of Belgium), the Belgian FNRS PDR project Flare, and the PHC project VAST (35961QJ) funded by Campus France and WBI.