The paper discusses the use of finite-state transducers, particularly sequential transducers, in language and speech processing. It reviews classical theorems and introduces new characterizations of sequential string-to-string transducers. The paper also explores string-to-weight transducers, which output weights, and provides algorithms for determinizing and minimizing these transducers efficiently. These algorithms are applied to speech recognition, demonstrating their efficiency in reducing the size of word lattices and improving computational efficiency. The paper highlights the importance of sequential transducers due to their linear time complexity and space efficiency, making them suitable for handling large datasets in natural language processing.The paper discusses the use of finite-state transducers, particularly sequential transducers, in language and speech processing. It reviews classical theorems and introduces new characterizations of sequential string-to-string transducers. The paper also explores string-to-weight transducers, which output weights, and provides algorithms for determinizing and minimizing these transducers efficiently. These algorithms are applied to speech recognition, demonstrating their efficiency in reducing the size of word lattices and improving computational efficiency. The paper highlights the importance of sequential transducers due to their linear time complexity and space efficiency, making them suitable for handling large datasets in natural language processing.