Succinctness of the Complement and Intersection of Regular Expressions

Abstract

Abstract. We study the succinctness of the complement and intersection of regular expressions. In particular, we show that when constructing a regular expression defining the complement of a given regular expression, a double exponential size increase cannot be avoided. Similarly, when constructing a regular expression defining the intersection of a fixed and an arbitrary number of regular expressions, an exponential and double exponential size increase, respectively, can in worst-case not be avoided. All mentioned lower bounds improve the existing ones by one exponential and are tight in the sense that the target expression can be constructed in the corresponding time class, i.e., exponential or double exponential time. As a by-product, we generalize a theorem by Ehrenfeucht and Zeiger stating that there is a class of DFAs which are exponentially more succinct than regular expressions, to a fixed four-letter alphabet. When the given regular expressions are one-unambiguous, as for instance required by the XML Schema specification, the complement can be computed in polynomial time whereas the bounds concerning intersection continue to hold. For the subclass of single-occurrence regular expressions, we prove a tight exponential lower bound for intersection.

Simplifying XML Schema: Single-type approximations of regular tree languages

“…Again, Lemma 3.5 becomes pspace-complete. Since the smallest expression for the intersection of two regular expressions can be exponential, and since complementing a regular expression can cause a double-exponential blow-up [11], we have an (optimal) exponential upper bound for Theorem 3.7 and an optimal double exponential upper bound for Theorem 3.10. For deterministic regular expression the complexity of all decision problems remains the same as there is an efficient translation to DFAs.…”

Succinctness of Regular Expressions with Interleaving, Intersection and Counting

“…Also the classes RE(#) [22,27] and RE(&) [12,24] have received interest. Succinctness of regular expressions has been studied by Ehrenfeucht and Zeiger [8] and, more recently, by Ellul et al [9], Gelade and Neven [13], Gruber and Holzer [15][16][17][18], and Gruber and Johannsen [19]. See the Ph.D. theses of Gruber [14] and Gelade [11] for an overview of many of the recently obtained results.…”

“…For RE(#) and RE(∩) the complexity of this translation has already been settled and is exponential [22] and double exponential [13], respectively. We show that also in constructing an expression for the interleaving of a set of expressions (an hence also for an RE(&)) a double exponential size increase cannot be avoided.…”

Deciding Definability by Deterministic Regular Expressions

“…Furthermore, Brüggemann-Klein and Wood showed that it is decidable whether a given regular language is definable by a DRE. Since then, DREs have been studied in the context of language approximations [3], learning [4], descriptional complexity [13,21] and static analysis [8,9]. Recently, it was shown that testing if a regular expression is deterministic can be done in linear time [14].…”