Bimachines and Structurally-Reversed Automata

Although bimachines are not widely used in practice, they represent a central concept in the study of rational functions. Indeed, they are finite state machines specifically designed to implement rational word functions. Their modelling power is equal to that of single-valued finite transducers. From the theoretical point of view, bimachines reflect the decomposition of a rational function into a left and a right sequential function. In this paper we define three new types of bimachines, classified according to the scanning direction of their reading heads. Then we prove that these types of bimachines are equivalent to the classical one and for doing so, we define and use a new concept, of structurally-reversed automaton. Consequently, we prove that the scanning directions of bimachines are irrelevant from the point of view of their modelling power. This leads to a method of simulating a bimachine by a left sequential transducer (or generalized sequential machines GSM for short). Indeed, a preprocessing of the input word allows sequential transducers to realize the full range of rational functions. Remarkably enough, we basically show that the so versatile functional transducers nondeterministic and with λ-input transitions can successfully be replaced by a simple deterministic setup: a “trimmer” coupled with a GSM . Intuitively, this fact proves that sequential functions are not much weaker than rational functions.