Deterministic regular languages

Abstract

The ISO standard for Standard Generalized Markup Language (SGML) provides a syntactic meta-language for the definition of textual markup systems. In the standard the right hand sides of productions are called content models and they are based on regular expressions. The allowable regular expressions are those that are “unambiguous” as defined by the standard. Unfortunately, the standard's use of the term “unambiguous” does not correspond to the two well known notions, since not all regular languages are denoted by “unambiguous” expressions. Furthermore, the standard's definition of “unambiguous” is somewhat vague. Therefore, we provide a precise definition of “unambiguous expressions” and rename them deterministic regular expressions to avoid any confusion. A regular expression E is deterministic if the canonical ε-free finite automaton M _e recognizing L(E) is deterministic. A regular language is deterministic if there is a deterministic expression that denotes it. We give a Kleene-like theorem for deterministic regular languages and we characterize them in terms of the structural properties of the minimal deterministic automata recognizing them. The latter result enables us to decide if a given regular expression denotes a deterministic regular language and, if so, to construct an equivalent deterministic expression.