Weighted Automata Algorithms

This chapter presents several fundamental algorithms for weighted automata and transducers. While the mathematical counterparts of weighted transducers, rational power series , have been extensively studied in the past [22, 54, 13, 36], several essential weighted transducer algorithms, e.g., composition, determinization, minimization, have been devised only in the last decade [38, 43], in part motivated by novel applications in speech recognition, speech synthesis, machine translation, other areas of natural language processing, image processing, optical character recognition, and more recently machine learning. These algorithms can be viewed as the generalization to the weighted transducer case of the standard algorithms for unweighted acceptors. However, this generalization is often not straightforward and has required a number of specific studies either because the old schema could not be applied in the presence of weights and a novel technique was required, as in the case of composition [50, 46], or because of the analysis of the conditions of application of an algorithm as in the case of determinization [38, 3]. The chapter favors a presentation of weighted automata and transducers in terms of graphs, the natural concepts for an algorithmic description and complexity analysis. Also, while power series lead to more concise and rigorous proofs in most cases [36], proofs related to questions of ambiguity naturally require the introduction of paths and reasoning on graph concepts.