Regular Expressions with Backreferences and Lookaheads Capture NLOG

TL;DR

This paper characterizes the expressive power of regular expressions with backreferences and lookaheads, showing they match the complexity class NLOG and establishing their membership problem as PSPACE-complete.

Contribution

It provides the first precise complexity classification of REGEX with backreferences and lookaheads, connecting them to NLOG and PSPACE-complete problems.

Findings

REWBLk equals NLOG in expressive power

Membership problem for REWBLk is PSPACE-complete

Addresses the complexity of combining backreferences and lookaheads

Abstract

Backreferences and lookaheads are vital features to make classical regular expressions (REGEX) practical. Although these features have been widely used, understanding of the unrestricted combination of them has been limited. Practically, most likely no implementation fully supports them. Theoretically, while some studies have addressed these features separately, few have dared to combine them. In those few studies, it has been made clear that the amalgamation of these features renders REGEX significantly expressive. However, no acceptable expressivity bound for REWBLk$\unicode{x2014}$REGEX with backreferences and lookaheads$\unicode{x2014}$has been established. We elucidate this by establishing that REWBLk coincides with NLOG, the class of languages accepted by log-space nondeterministic Turing machines (NTMs). In translating REWBLk to log-space NTMs, negative lookaheads are the most challenging part since it essentially requires complementing log-space NTMs in nondeterministic log-space. To address this problem, we revisit Immerman$\unicode{x2013}$Szelepcs\'enyi theorem. In addition, we employ log-space nested-oracles NTMs to naturally handle nested lookaheads of REWBLk. Utilizing such oracle machines, we also present the new result that the membership problem of REWBLk is PSPACE-complete.