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 $$\mathrm {2DFT}$$, iii¾?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 $$\mathrm {SST}$$.