Finite-state transducers are widely used in natural language processing for efficient string-to-string and string-to-weight transformations. This paper presents theoretical and algorithmic foundations for sequential transducers, which are deterministic and efficient in both time and space. Sequential transducers are characterized by their ability to process input strings in linear time and minimize their size through efficient algorithms. The paper discusses classical and new theorems characterizing these transducers, including their closure under composition and union. It also introduces algorithms for determining and minimizing sequential transducers, which are crucial for applications in speech recognition and language processing. The paper extends these concepts to string-to-weight transducers, which are used in language modeling and speech processing. It provides algorithms for determining and minimizing these transducers, as well as characterizations of their properties. The paper also discusses the use of these transducers in various applications, including the representation of large dictionaries, compilation of morphological and phonological rules, and syntax processing. The theoretical and algorithmic results presented here contribute to the understanding and application of finite-state transducers in natural language processing and speech recognition.Finite-state transducers are widely used in natural language processing for efficient string-to-string and string-to-weight transformations. This paper presents theoretical and algorithmic foundations for sequential transducers, which are deterministic and efficient in both time and space. Sequential transducers are characterized by their ability to process input strings in linear time and minimize their size through efficient algorithms. The paper discusses classical and new theorems characterizing these transducers, including their closure under composition and union. It also introduces algorithms for determining and minimizing sequential transducers, which are crucial for applications in speech recognition and language processing. The paper extends these concepts to string-to-weight transducers, which are used in language modeling and speech processing. It provides algorithms for determining and minimizing these transducers, as well as characterizations of their properties. The paper also discusses the use of these transducers in various applications, including the representation of large dictionaries, compilation of morphological and phonological rules, and syntax processing. The theoretical and algorithmic results presented here contribute to the understanding and application of finite-state transducers in natural language processing and speech recognition.