Finite State Transducers

A relation is a set of pairs—in this paper, a subset of ⌃⇤ ⇥ ⇤, so it relates strings over an “input” alphabet ⌃ to strings over an “output” alphabet . A weighted relation is a function R that maps any string pair (x,y) to a weight in R 0. We say that the relation R is rational if R can be defined by some weighted finite-state transducer (FST) T . As formalized in Appendix A.3, this means thatR(x,y) is the total weight of all accepting paths in T that are labeled with (x,y) (which is 0 if there are no such accepting paths). The weight of each accepting path in T is given by the product of its arc weights, which fall in R>0. The set of pairs support(R) , {(x,y) : R(x,y) > 0} is then said to be a regular relation because it is recognized by the unweighted FST obtained by dropping the weights from T . In this paper, we are interested in defining non-rational weighting functions R with this same regular support set.