A Complete Propositional Dynamic Logic for Regular Expressions with Lookahead

We consider (logical) reasoning for regular expressions with lookahead (REwLA). In this paper, we give an axiomatic characterization for both the (match-)language equivalence and the largest substitution-closed equivalence that is sound for the (match-)language equivalence. To achieve this, we introduce a variant of propositional dynamic logic (PDL) on finite linear orders, extended with two operators: the restriction to the identity relation and the restriction to its complement. Our main contribution is a sound and complete Hilbert-style finite axiomatization for the logic, which captures the equivalences of REwLA. Using the extended operators, the completeness is established via a reduction into an identity-free variant of PDL on finite strict linear orders. Moreover, the extended PDL has the same computational complexity as REwLA.

💡 Research Summary

The paper tackles the logical foundations of regular expressions that incorporate look‑ahead operators (REwLA). Unlike classical regular expressions, REwLA’s language equivalence is not closed under substitution, which prevents the existence of a sound and complete axiomatization for the usual language equivalence. To address this, the authors introduce two notions of equivalence: (i) the standard match‑language equivalence, which compares the set of string‑position pairs matched by an expression, and (ii) the largest substitution‑closed equivalence that is sound for the match‑language equivalence. Both notions can be captured by relational semantics on finite linear orders, where a string is represented as a generalized structure whose underlying relation is the natural “≤” order on positions.

The central technical contribution is a new logic, PDL REwLA⁺, which extends propositional dynamic logic (PDL) on finite linear orders with two additional program operators: restriction to the identity relation (· ∩ 1) and restriction to its complement (· ∩ ¬1). These operators allow the encoding of the antidomain (negative look‑ahead) and domain (positive look‑ahead) constructs of REwLA directly inside the box modality