  Finite state transducers, multitape automata and weighted automata have a lot
in common. By studying their universal foundations, one can discover some new
insights into all of them. The main result presented here is the introduction
of lexicographic finite state transducers, that could be seen as intermediate
model between multitape automata and weighted transducers. Their most
significant advantage is being equivalent, but often exponentially smaller than
even smallest nondeterministic automata without weights. Lexicographic
transducers were discovered by taking inspiration from Eilenberg's algebraic
approach to automata and Solomonoff's treatment of a priori probability.
Therefore, a quick and concise survey of those topics is presented, prior to
introducing lexicographic transducers.
