Succinctness of Regular Expressions with Interleaving, Intersection and Counting

Abstract

a b s t r a c tIn this paper, we study the succinctness of regular expressions (REs) extended with interleaving, intersection and counting operators. We show that in a translation from REs with interleaving to standard regular expressions a double exponential size increase cannot be avoided. We also consider the complexity of translations to finite automata. We give a tight exponential lower bound on the translation of REs with intersection to NFAs, and, for each of the three classes of REs, we show that in a translation to a DFA a double exponential size increase cannot be avoided. Together with known results, this gives a complete picture of the complexity of translating REs extended with interleaving, intersection or counting into (standard) regular expressions, NFAs, and DFAs.

Tight Bounds on the Descriptional Complexity of Regular Expressions

“…For this case, we also devise a new set of encodings which are economic and, in some precise sense, robust with respect to both the Kleene star and the interleaving operation. This extends the outreach of known proof techniques, and allows us to give a definitive answer to some questions regarding the descriptional complexity of regular expressions that were not yet settled completely in previous works [5,7,8,10]. Our main results are summarized and compared to known results in Table 1.…”

“…Of course, it appears more natural to compare the gain in succinctness of such extended regular expressions to ordinary regular expressions rather than to finite automata. There a 2 2 O(n) doubly exponential upper bound readily follows by combining standard constructions [7]. Yet a roughly doubly-exponential lower bound of 2 2 Ω( √ n) , for alphabets of growing size, was found only recently in [8], and a follow-up paper [7] shows that this can be reached already for binary alphabets.…”

“…There a 2 2 O(n) doubly exponential upper bound readily follows by combining standard constructions [7]. Yet a roughly doubly-exponential lower bound of 2 2 Ω( √ n) , for alphabets of growing size, was found only recently in [8], and a follow-up paper [7] shows that this can be reached already for binary alphabets. Here we finally establish a tight doubly-exponential lower bound, which even holds for binary alphabets.…”

“…They also showed that regular expressions extended with an interleaving operator can be exponentially more succinct than nondeterministic finite automata [18]. Very recently, it was shown in [7] that regular expressions with interleaving can be roughly doubly-exponentially more succinct than regular expressions: Converting such expressions into ordinary regular expressions can cause a blow-up in required expression size of 2 2 Ω( √ n) , for constant alphabet size. This bound is close to an easy upper bound of 2 2 O(n) that follows from standard constructions, see, e.g., [7] for details.…”

“…Although this resulted in quite a few new insights into the nature of regular expressions,see also [7,11,12], proving tight lower bounds for small alphabets remains a challenging task, and not all bounds in the mentioned references are both tight and cover all alphabet sizes. In this work, we close some of the remaining gaps: In the case of converting planar finite automata to regular expressions, we reduce the alphabet size to binary while retaining the tight lower bound.…”