... efficiently restricted,manipulated or combined using rational operationsas before. An artificial example will appear in §2.The availability of toolkits for this weighted case(Mohri et al., 1998; ... regexp built up from atomic expressions a : b (for a ∈ Σ ∪ {}, b ∈ ∆ ∪ {}) using concatenation,probabilistic union +p, and probabilistic closure ∗p. For defining conditional relations, a good ... synchronous PCFGs built for translation. These are parameterized by the PCFG’sparameters, but add or remove strings of the PCFGto leave an improper probability distribution.Fortunately for those techniques,...