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.

We answer an open question in the theory of transducer degrees initially posed in [3], on the structure of polynomial transducer degrees, in particular the question of what degrees, if any, lie below the degree of n 3 . Transducer degrees are the equivalence classes formed by word transformations which can be realized by a finite-state transducer. While there are no general techniques to tell if a word w1 can be transformed into w2 via an FST, the work of Endrullis et al. in [2] provides a test for the class of spiralling functions, which includes all polynomials. We classify fully the degrees of all cubic polynomials which are below n 3 , and many of the methods can also be used to classify the degrees of polynomials of higher orders.

We answer an open question in the theory of transducer degrees initially posed in [1] on the existence of a diamond structure in the transducer hierarchy. Transducer degrees are the equivalence classes formed by word transformations which can be realized by a finite state transducer, which form an order based on which words can be transformed into other words. We provide a construction which proves the existence of a diamond structure, while also introducing a new function on streams which may be useful for proving more results about the transducer hierarchy.

No abstract