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.

Symbolic automata are finite state automata that support potentially infinite alphabets, such as the set of rational numbers, generally applied to regular expressions and languages over finite words. In symbolic automata (or automata modulo 𝒜), an alphabet is represented by an effective Boolean algebra 𝒜, supported by a decision procedure for satisfiability. Regular languages over infinite words (so called ω -regular languages) have a rich history paralleling that of regular languages over finite words, with well known applications to model checking via Büchi automata and temporal logics. We generalize symbolic automata to support ω -regular languages via transition terms and symbolic derivatives , bringing together a variety of classic automata and logics in a unified framework that provides all the necessary ingredients to support symbolic model checking modulo 𝒜. In particular, we define: (1) alternating Büchi automata modulo 𝒜( AB W 𝒜 ) as well (non-alternating) nondeterministic Büchi automata modulo 𝒜( NB W 𝒜 );(2) an alternation elimination algorithm Æ that incrementally constructs an NB W 𝒜 from an AB W 𝒜 , and can also be used for constructing the product of two NB W 𝒜 ; (3) a definition of linear temporal logic modulo 𝒜, LTL ⟨𝒜⟩, that generalizes Vardi's construction of alternating Büchi automata from LTL, using (2) to go from LTL modulo 𝒜 to NB W 𝒜 via AB W 𝒜 . Finally, we present RLTL ⟨ 𝒜 ⟩, a combination of LTL ⟨ 𝒜 ⟩ with extended regular expressions modulo 𝒜 that generalizes the Property Specification Language (PSL). Our combination allows regex complement , that is not supported in PSL but can be supported naturally by using transition terms. We formalize the semantics of RLTL ⟨ 𝒜 ⟩ using the Lean proof assistant and formally establish correctness of the main derivation theorem.

We present a formalization of a matching algorithm for extended regular expression matching based on locations and symbolic derivatives which supports intersection, complement and lookarounds and whose implementation mirrors an extension of the recent .NET NonBacktracking regular expression engine. The formalization of the algorithm and its semantics uses the Lean 4 proof assistant. The proof of its correctness is with respect to standard matching semantics.