An improved algorithm for the membership problem for extended regular expressions

Extended regular expressions (ERE) define regular languages using union, concatenation, repetition, intersection, and complementation operators. The fact ERE allow intersection and complementation makes them exponentially more succinct than regular expressions. The membership problem for extended regular expressions is to decide, given an expression r and a word w, whether w belongs to the language defined by r. Since regular expressions are useful for describing patterns in strings, the membership problem has numerous applications. In many such applications, the words w are very long and patterns are conveniently described using ERE, making efficient solutions to the membership problem of great practical interest. In this paper we introduce alternating automata with synchronized universality and negation, and use them in order to obtain a simple and efficient algorithm for solving the membership problem for ERE. Our algorithm runs in time O(m ・ n²) and space O(m ・ n + k ・ n²), where m is the length of r, n is the length of w, and k is the number of intersection and complementation operators in r. This improves the best known algorithms for the problem.