An Effective Algorithm for the Membership Problem for Extended Regular Expressions

By adding the complement operator (\neg), extended regular expressions ({\ERE}) can encode regular languages non-elementarily more succinctly than regular expressions. The {\ERE} membership problem asks whether a word w of size n belongs to the language of an {\ERE} R of size m. Unfortunately, the best known membership algorithms are either non-elementary in m or otherwise require space \Omega(n^2) and time \Omega(n^3); since in many practical applications n can be very large (in the order of billions, e.g., in testing where w represents the execution trace of some system), these space and time requirements could be prohibitive. In this paper we present a simple to implement {\ERE} membership algorithm that runs in space O(n \cdot (m + \log n) \cdot 2^{m} \cdot k) and in time O(n^2 \cdot (m + \log n)^2 \cdot 2^m \cdot k), where k is the number of complement operators in R. The presented algorithm outperforms the best known algorithms when n is large