One-Unambiguous Regular Languages

Abstract

The ISO standard for the Standard Generalized Markup Language (SGML) provides a syntactic metalanguage for the definition of textual markup systems. In the standard, the right-hand sides of productions are based on regular expressions, although only regular expressions that denote words unambiguously, in the sense of the ISO standard, are allowed. In general, a word that is denoted by a regular expression is witnessed by a sequence of occurrences of symbols in the regular expression that match the word. In an unambiguous regular expression as defined by Book et al. (1971, IEEE Trans. Comput. C-20(2), 149 153), each word has at most one witness. But the SGML standard also requires that a witness be computed incrementally from the word with a onesymbol lookahead; we call such regular expressions 1-unambiguous. A regular language is a 1-unambiguous language if it is denoted by some 1-unambiguous regular expression. We give a Kleene theorem for 1-unambiguous languages and characterize 1-unambiguous regular languages in terms of structural properties of the minimal deterministic automata that recognize them. As a result we are able to prove the decidability of whether a given regular expression denotes a 1-unambiguous language; if it does, then we can construct an equivalent 1-unambiguous regular expression in worst-case optimal time.

Deciding Definability by Deterministic Regular Expressions

“…The best known upper bound is from Brüggeman-Klein and Wood, who showed that the problem is in EXPTIME (by exhibiting an algorithm that works in polynomial time on the minimal DFA [7]) and the best known lower bound is PSPACE-hardness [3]. The main result of this paper settles this question and proves that this problem is PSPACE-complete.…”

“…Deterministic regular expressions or DREs have therefore been a subject of research since their foundations were laid in a seminal paper by Brüggemann-Klein and Wood [6,7]. The most important contribution of this paper is a characterization of languages definable by DREs in terms of structural properties on the minimal DFA.…”

XML schema validation using parsing expression grammars

“…In addition, DTDs disallow non-deterministic regular expressions, such as (a|(a, b)) and (a * , a). These properties are also called one-unambiguous regular expressionsBrüggemann- Klein and Wood (1998). More importantly, these restrictions are intended to avoid backtracking and the underlying exponential time cost.…”

Which XML Schemas are Streaming Bounded Repairable?

“…t(⃗ ı ⋅ n) ∈ L(t(⃗ ı)), where n is the number of children of ⃗ ı. We denote by L (D) the language of trees satisfying the DTD D. In accordance with the XML standards, DTDs are often assumed to be deterministic, that is, to use only rules with one-unambiguous regular expressions [9] and a singleton set of initial symbols. We will often omit the alphabet and the set of initial symbols from the definition of a DTD, since these can be easily understood from the context (e.g.…”