I have posted a lightly-revised slide deck from a talk I gave at Johns Hopkins University here. In it, I give my most detailed-yet description of the weighted finite-state transducer formalism and describe two reasonably interesting algorithms, the optimization algorithm underlying Pynini’s optimize method and Thrax’s Optimize function, and a new A*-based single shortest string algorithm for non-idempotent semirings underlying BaumWelch’s baumwelchdecode CLI tool.