The Inclusion Problem for Regular Expressions

This paper presents a polynomial-time algorithm for the inclusion problem for a large class of regular expressions. The algorithm is not based on construction of finite automata, and can therefore be faster than the lower bound implied by the Myhill-Nerode theorem. The algorithm automatically discards irrelevant parts of the right-hand expression. The irrelevant parts of the right-hand expression might even be 1-ambiguous. For example, if r is a regular expression such that any DFA recognizing r is very large, the algorithm can still, in time independent of r, decide that the language of ab is included in that of (a + r)b. The algorithm is based on a syntax-directed inference system. It takes arbitrary regular expressions as input. If the 1-ambiguity of the right-hand expression becomes a problem, the algorithm will report this. Otherwise, it will decide the inclusion problem for the input.