Language Operations with Regular Expressions of Polynomial Size

In the last 20 years, a large body of research on the descriptional complexity of finite automata has been developed. To the authors’ knowledge, the first systematic attempt to start a parallel development for the descriptional complexity of regular expressions was presented by Ellul et al. [4] at the workshop “Descriptional Complexity of Formal Systems” (DCFS), in 2002. In particular, they raised the question of determining how basic language operations such as complementation and intersection affect the required regular expression size. For the intersection and shuffle operation, exponential lower bounds are known, and complementation can even incur a doubly-exponential blow-up [5, 6]. In [6] it was shown that the star height of a regular language is at most logarithmic in the minimum regular expression size, and lower bounds are proved by finding families of languages for which the respective language operations give rise to a dramatic increase in star height. In contrast, it is well known that taking language quotients does not increase the star height [3]. This and similar language operations appear to be a natural testing ground for deepening our understanding of the descriptional complexity of regular expressions: Either one has to find some new lower bound techniques, or one has to find a nontrivial implementation of these operations on regular expressions, or both—a straightforward procedure would be to convert the expression into a finite automaton, implement the operation on a finite automaton, and convert back to a regular expression using state elimination. Yet that last step can incur an exponential blow-up in general, even over binary alphabets [6].