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arXiv · 2209.06809

On the Intersection of Context-Free and Regular Languages

Clemente Pasti·Andreas Opedal·Tiago Pimentel·Tim Vieira·Jason Eisner·Ryan Cotterell
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Abstract

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.

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Clemente Pasti, Andreas Opedal, Tiago Pimentel, Tim Vieira, Jason Eisner, Ryan Cotterell. 2022-09-14. On the Intersection of Context-Free and Regular Languages. https://arxiv.org/abs/2209.06809

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