Semilinear real-time systolic trellis automata

Abstract

A real-time systolic trellis automaton/RTSTA/ will be called semilinear of degree (at most) k if its underlying trellis T is semilinear of degree (at most) k, i. e. the set of rows of T can be expressed as a finite union of languages of the form

$$\left\{ {u_0 v_1^i u_1 v_2^i u_2 ...u_{k - 1} v_k^i u_k ;{\text{ }}i{\text{ }} \geqq 1} \right\} .$$

It is proved that every semilinear RTSTA of arbitrary degree k is equivalent with a semilinear regular RTSTA of degree 3. If regularity is not requested the degree can be diminished to 2, but not to 1.