This paper studies the algorithms for the minimisation of weighted automata. It starts with the definition of morphisms-which generalises and unifies the notion of bisimulation to the whole class of weighted automata-and the unicity of a minimal quotient for every automaton, obtained by partition refinement. From a general scheme for the refinement of partitions, two strategies are considered for the computation of the minimal quotient: the Domain Split and the Predecesor Class Split algorithms. They correspond respectivly to the classical Moore and Hopcroft algorithms for the computation of the minimal quotient of deterministic Boolean automata. We show that these two strategies yield algorithms with the same quadratic complexity and we study the cases when the second one can be improved in order to achieve a complexity similar to the one of Hopcroft algorithm.

For the past two decades, specialised events on finite-state methods have been successful in presenting interesting studies on natural language processing to the public through journals and collections. The FSMNLP workshops have become well-known among researchers and are now the main forum of the Association for Computational Linguistics' (ACL) Special Interest Group on Finite-State Methods (SIGFSM). The current issue on finite-state methods and models in natural language processing was planned in 2008 in this context as a response to a call for special issue proposals. In 2010, the issue received a total of sixteen submissions, some of which were extended and updated versions of workshop papers, and others which were completely new. The final selection, consisting of only seven papers that could fit into one issue, is not fully representative, but complements the prior special issues in a nice way. The selected papers showcase a few areas where finite-state methods have less than obvious and sometimes even groundbreaking relevance to natural language processing (NLP) applications.

We give a new, and hopefully more easily understandable, structural proof of the decomposition of a $k$-valued transducer into $k$ unambiguous functional ones, a result established by A. Weber in 1996. Our construction is based on a lexicographic ordering of computations of automata and on two coverings that can be build by means of this ordering. The complexity of the construction, measured as the number of states of the transducers involved in the decomposition, improves the original one by one exponential. Moreover, this method allows further generalisation that solves the problem of decomposition of rational relations with bounded length-degree, which was left open in Weber's paper.