“…The mathematical theory behind WFAs, that of rational power series, has been extensively studied in the past [28,53,38,14] and has been more recently the topic of a dedicated handbook [25]. WFAs are widely used in modern applications, perhaps most prominently in image processing and speech recognition where the terminology of weighted automata seems to have been first introduced and made popular [32,46,51,44,48], in several other speech processing applications such as speech synthesis [54,2], in phonological and morphological rule compilation [34,35,50], in parsing [47], machine translation [23], bioinformatics [27,3], sequence modeling and prediction [21], formal verification and model checking [5,4], in optical character recognition [17], and in many other areas.…”