Let T 1 : Σ * ×Ω * → K and T 2 : Ω * ×Γ * → K be two transducers defined over the same semiring K. Their composition T 1 • T 2 realizes the function T : Σ * ×Γ * → K and the theory has been described in detail in (Pereira and Riley, 1996).
in Section 4.1 we will compare the efficiency of different toolkits.