One-Unambiguous Regular Languages

Abstract

The ISO standard for the Standard Generalized Markup Language (SGML) provides a syntactic metalanguage for the de nition of textual markup systems. In the standard, the right-hand sides of productions are based on regular expressions; although only expressions that denote words unambiguously are allowed. In general, the fact that a word is denoted by an expression is witnessed by a sequence of occurrences of symbols in the expression that matches the word. In an unambiguous expression as de ned by Book, Even, Greibach, and Ott, each word has at most one witness. But the SGML standard also requires that a witness can be computed incrementally from the word with a one-symbol lookahead; we call such expressions 1-unambiguous. A regular language is 1-unambiguous if it is denoted by some 1-unambiguous expression. We give a Kleene theorem for 1-unambiguous languages and characterize them in terms of structural properties of the minimal deterministic automata that recognize them. This result enables us to decide whether a given regular expression denotes a 1-unambiguous language; if it does, then we can construct an equivalent 1-unambiguous expression in worst-case optimal time.

Deterministic regular expressions with back-references

“…[42]). Following the original paper [9], deterministic regular expressions have been studied extensively. Aspects include computing the Glushkov automaton and deciding the membership problem (e. g. [8,31,44]), static analysis (cf.…”

“…We also use these markings in the ref-words: For example, [ y (1) a (2) a (2) x (3) a (2) ] y (6) y (7) ∈ R(α). Before we explain this definition and use it to define deterministic regex, we first discuss the special case of deterministic regular expressions: A proper regular expression α is not deterministic if there exist words u, v 1 , v 2 ∈Σ * , a terminal a ∈ Σ and positions i = j such that ua (i) v 1 and ua (j) v 2 are elements of L(α) (see e. g. [9,31]). Otherwise, it is a deterministic proper regular expression (or, for short, just deterministic regular expression).…”

“…The language L from Lemma 5 is also known to be a non-deterministic regular language (see e. g. [9]). Our proof can be seen as taking the idea behind the characterization of deterministic regular languages from [9], applying it to the specific language L, and also taking variables into account.…”

“…The language L from Lemma 5 is also known to be a non-deterministic regular language (see e. g. [9]). Our proof can be seen as taking the idea behind the characterization of deterministic regular languages from [9], applying it to the specific language L, and also taking variables into account. While this accomplishes the task of proving that deterministic regex share some of the limitations of deterministic regular expressions, the approach does not generalize (at least not in a straightforward manner).…”

Practical Study of Deterministic Regular Expressions from Large-scale XML and Schema Data

“…Grijzenhout et al [18] studied the the quality of the XML Web in 2011. Björklund et al [10,14] focused on the usage of the counting operator in REs in 2015. Li et al [24] researched the complexity and usage subclasses of REs in DTDs and XSDs in 2016.…”