Tight Bounds on the Descriptional Complexity of Regular Expressions

Abstract

Abstract. We improve on some recent results on lower bounds for conversion problems for regular expressions. In particular we consider the conversion of planar deterministic finite automata to regular expressions, study the effect of the complementation operation on the descriptional complexity of regular expressions, and the conversion of regular expressions extended by adding intersection or interleaving to ordinary regular expressions. Almost all obtained lower bounds are optimal, and the presented examples are over a binary alphabet, which is best possible.

Succinctness of Regular Expressions with Interleaving, Intersection and Counting

“…At present these algorithms do not take into account the interleaving operator, but for Relax NG this would be wise as this would allow to learn significantly smaller expressions. It should be noted here that Gruber and Holzer independently obtained a similar result [18]. They show that any regular expression defining the language (a 1 b 1 ) * & (a 2 b 2 ) * & · · · & (a n b n ) * must be of size at least double exponential in n. Compared to the result in this paper, this gives a tighter bound (2 2 Ω(n) instead of 2 2 Ω( √ n) ), and shows that the double exponential size increase already occurs for very simple expressions.…”

“…2 Ω(n) [22] 2 2 Ω(n) (Proposition 7) 2 θ(n) [22] RE(∩) 2 Ω(n) (Proposition 5) 2 2 Ω(n) (Theorem 8) 2 2 Ω(n) [18] RE(&) 2 Ω(n) [ from Relax NG (which allows interleaving) to XML Schema Definitions (which does not). However, as XML Schema is the widespread W3C standard, and Relax NG is a more flexible alternative, such a translation would be more than desirable.…”

“…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.…”

The Complexity of Regular(-Like) Expressions

“…Preliminary results were reported in [17] were a stunningly large gap between the lower and upper bound on the effect of complementation remained. Finally, the problem was solved in [22], and the above mentioned upper bound turned out to be tight already for binary alphabets [21,30]. The complementation problem for regular expressions over unary alphabets was already settled in [17].…”

“…Nevertheless, together with clever encodings by star-height preserving homomorphisms as demonstrated in [30] makes the above theorem to a general purpose tool for proving lower bounds on alphabetic width for regular expressions. Let's briefly illustrate the application on a simple example: we aim to prove a lower bound for the intersection of languages K m = { w ∈ {a, b} * | |w| a ≡ 0 mod m } and L n = { w ∈ {a, b} * | |w| b ≡ 0 mod n }.…”