Weak monadic second-order logic (wMSO) is a foundational tool for specifying regular properties. Traditional decision procedures for this logic typically translate wMSO formulas into finite automata. Although the logic is decidable, this approach incurs non-elementary complexity in the worst-case. Nearly thirty years ago, the state-of-the-art MONA tool showed that, despite these theoretical limits, wMSO can be decided efficiently in practice through carefully optimized automata constructions. We revisit wMSO from an algebraic perspective by introducing Extended Regular Expressions with Quantifiers (EREQ). Instead of relying on automata determinization, EREQ employs symbolic derivatives to perform incremental quantifier elimination, providing a compositional and symbolic alternative to classical automata-based approaches. We present a linear-time translation of wMSO into EREQ and a derivative-based decision procedure for EREQ. We prove the correctness of the translation and of the derivative construction in the Lean proof assistant. We implement our approach in Rust and evaluate it on a set of established MONA benchmarks, demonstrating competitive performance with state-of-the-art tools. Our results demonstrate the potential of derivative-based methods, opening new avenues for efficient decision procedures in EREQ.

Abstract We develop decision procedures for extended regular expressions in the new $$\textbf{ERE} \texttt {\#}$$ ERE # framework that uses span semantics , utilizing the power of symbolic derivatives . We prove a normal form theorem in Lean for $$\textbf{ERE} \texttt {\#}$$ ERE # that is closed under all Boolean operations and provides the basis for the given decision procedures. The tool is evaluated on existing SMT benchmarks for regexes that shows it to be the fastest solver to date – often orders of magnitude faster than state-of-the-art – albeit specialized for the single-variable fragment of string theory.

We present a tool and theory RE # for regular expression matching that is built on symbolic derivatives, does not use backtracking, and, in addition to the classical operators, also supports complement, intersection and restricted lookarounds. We develop the theory formally and show that the main matching algorithm has input-linear complexity both in theory as well as experimentally. We apply thorough evaluation on popular benchmarks that show that RE # is over 71% faster than the next fastest regex engine in Rust on the baseline, and outperforms all state-of-the-art engines on extensions of the benchmarks often by several orders of magnitude.