Fibonacci representations and finite automata

Finite-state automata are used as a simple model of computation since only a finite memory is needed. The problem of passing from any representation to the normal representation of an integer within the Fibonacci numeration system, which is called the process of normalization, is addressed. It is shown that the normalization can be realized by means of infinite automata. More precisely, this function can be obtained by the composition of two subsequential transducers that are simply obtained from the linear recurrence definition of the basis of the Fibonacci system, one processing words from left to right and the other from right to left. The normalization, although not a sequential process, can be obtained in two sequential passes. It is proved that it is possible to add to integers written in the Fibonacci numeration system of order m by means of a finite-state automaton. The conversion from a Fibonacci representation to the standard binary representation cannot be realized by a finite-state automaton