On the Intersection of Context-Free and Regular Languages
This addresses a technical limitation in formal language theory for researchers and practitioners, but it is incremental as it extends an existing method.
The paper tackles the problem that the classic Bar-Hillel construction cannot handle finite-state automata with ε-arcs when intersecting context-free and regular languages, and presents a generalized construction that works with ε-arcs while maintaining asymptotic size.
The Bar-Hillel construction is a classic result in formal language theory. It shows, by a simple construction, that the intersection of a context-free language and a regular language is itself context-free. In the construction, the regular language is specified by a finite-state automaton. However, neither the original construction (Bar-Hillel et al., 1961) nor its weighted extension (Nederhof and Satta, 2003) can handle finite-state automata with $\varepsilon$-arcs. While it is possible to remove $\varepsilon$-arcs from a finite-state automaton efficiently without modifying the language, such an operation modifies the automaton's set of paths. We give a construction that generalizes the Bar-Hillel in the case where the desired automaton has $\varepsilon$-arcs, and further prove that our generalized construction leads to a grammar that encodes the structure of both the input automaton and grammar while retaining the asymptotic size of the original construction.