Regular Expressions with Backreferences on Multiple Context-Free Languages, and the Closed-Star Condition

Abstract

Backreference is a well-known practical extension of regular expressions and is supported by the regular expression engines in the standard libraries of most modern programming languages, such as Java, Python, JavaScript and more. A difficulty of backreference is non-regularity: backreference strictly enhances the expressive power of regular expressions to the point that regular expressions with backreferences (rewbs) can describe non-regular (in fact, even non-context-free) languages. In this paper, we investigate the expressive power of rewbs by comparing rewbs to multiple context-free languages (MCFL) and parallel multiple context-free languages (PMCFL). First, we prove that the language class of rewbs is a proper subclass of unary-PMCFLs, which coincide with the EDT0L languages. Our result strictly improves the known (non-trivial) upper bound of rewbs, because the best-known bound was the intersection of the class of nondeterministic logspace languages and that of indexed languages, and, as we shall show in this paper, the class of EDT0L languages is a proper subclass of the intersection. Additionally, we show that, however, the language class of rewbs is not contained in that of MCFLs even when restricted to rewbs with only one capturing group and no captured references. Therefore, in general, the parallelism seems essential for rewbs. Backed by these results, we define a novel syntactic condition on rewbs that we call closed-star and observe that it provides an upper bound on the number of times a rewb references the same captured string. The closed-star condition allows dispensing with the parallelism: we prove that the language class of closed-star rewbs falls inside the class of unary-MCFLs, which is equivalent to that of EDT0L systems of finite index. Furthermore, we show that the language class of closed-star rewbs also falls inside the class of nonerasing stack languages.