Extracting finite state automata (FSAs) fromblack-box models offers a powerful approachto gaining interpretable insights into complexmodel behaviors. To support this pursuit, wepresent a weighted variant of Angluin’s (1987)L* algorithm for learning FSAs. We stay faithful to the original formulation, devising a wayto exactly learn deterministic weighted FSAswhose weights support division. Furthermore,we formulate the learning process in a mannerthat highlights the connection with FSA minimization, showing how L* directly learns aminimal automaton for the target language.

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 ε-arcs. While it is possible to remove ε-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 the desired automaton has ε-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.