Classifying all transducer degrees below N3

Abstract

We answer an open question in the theory of transducer degrees initially posed in Endrullis et al. (2016, Proc. Conf. on Developments in Language Theory (DLT 2016), 164–176), on the structure of polynomial transducer degrees. Transducer degrees are the equivalence classes formed by word transformations which can be realized by a finite-state transducer (FST). While there are no general techniques to tell if a word $\sigma $ can be transformed into $\tau $ via an FST, the work of Endrullis et al. (2010, J. Int., 11B.A6, 164–176) provides a test for the class of streams determined by spiralling functions, which includes all streams determined by polynomials. We fully classify the degrees of all cubic polynomial streams which are below the stream corresponding to $n^3$, and many of the methods can also be used to classify the degrees of polynomial streams of higher orders.