Aperiodic String Transducers

TL;DR

This paper explores the relationships and transformations between different models of aperiodic string transducers, providing complexity results and preserving aperiodicity, thereby extending the understanding of FO definable string functions.

Contribution

It introduces three direct transformations between models of aperiodic string transducers and proves they preserve aperiodicity, extending the characterization of FO definable functions.

Findings

Transformations between SST and 2DFT models with complexity analysis

Preservation of aperiodicity in all transformations

Equivalence of FO definable functions to aperiodic SSTs

Abstract

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