Proving Properties of Fibonacci Representations via Automata Theory

Abstract

In this work, we introduce a novel framework for mechanically testing the completeness and unambiguity of Fibonacci-based representations via automata theory. We call a representation (or a number system) complete and unambiguous when it provides one and only one representation for each number in the range covered by the representation. Many commonly used representations are complete and unambiguous: consider the familiar binary number system—each natural number has a unique representation up to leading zeros. Additionally, if a representation is complete, we describe an algorithm, of O(log n) complexity, to find a representation for any particular number n.