The Rational Construction of a Wheeler DFA

Deterministic Finite Wheeler Automata are a natural generalisation to regular languages of the theory of compressed data structures originated by the introduction of the Burrows-Wheeler transform. Indeed, if we can find a Wheeler automaton recognizing a given language L, such automaton can be used to design time and space efficient algorithms for representing and searching L. In this paper we introduce an alternative representation of Deterministic Wheeler Automata by showing that a natural map between strings and rational numbers in ℚ [0,1) can be extended to represent the automaton’s states as intervals in ℚ [0,1). Using this representation it emerges a natural relationship between automata properties and some properties of real numbers. In addition, such representation enables us to formulate problems related to automata in a numerical setting. Although at the moment the numerical approach does not lead to time efficient algorithms, we believe this new perspective deserves further consideration. As a further demonstration of the convenience of this new representation, we use it to provide a simple proof of an unexpected result on regular languages. More precisely, we compare the size of the smallest Wheeler automaton recognizing a given language L with respect to the size of the smallest automaton, possibly non-Wheeler, recognizing the same language. We show settings in which there can be an exponential gap between the two sizes, and we discuss the implications of this result on the problem of representing regular languages.