Effective optimization with weighted automata on decomposable trees

In this paper, we consider quantitative optimization problems on decomposable discrete systems. We restrict ourselves to labeled trees as the description of the systems and we use weighted automata on them as our computational model. We introduce a new kind of labeled decomposable trees, sum-like weighted labeled trees, and propose a method, which allows us to reduce the solution of an optimization problem, defined in a fragment of Weighted Monadic Second Order Logic, on such a tree to the solution of effectively derived problems, defined in the same logic, on its components with some additional post-processing. The approach originates from a method, proposed first by Feferman and Vaught in 1959, which we adapt and generalize. We adapt the method to the case of (fragments of) Weighted Monadic Second Order Logic and generalize it to the case of sum-like labeled trees rather than disjoint union and product. The main result of the paper may be applied in the wide range of optimization problems, such as critical path analysis or project planning, network optimization, generalized assignment problem, routing and scheduling as well as in the modern document languages like XML, image processing and compression, probabilistic systems or speech-to-text processing.