For example I have 2 simple FSM. I will use regular expression for clarity.
FSM1 = 1+ = [1,11,111,...] FSM2 = 2+ = [2,22,222,...] FST = 1/2 = [1->2,11->22,...]
What is the general way to prove that there exists finite-state transducer that maps each string from the set defined by FSM1 to a string defined by FSM2? What is the algorithm to construct such transducer?
I want to have the next properties from mapping:
𝑇(𝐿(𝐹1))=𝐿(𝐹2), where 𝐿(𝐹1) is the regular language accepted by the first FSM, 𝐿(𝐹2) the one for the second FSM, 𝑇 is some transducer
𝑇 is bijective mapping from 𝐿(𝐹1) to 𝐿(𝐹2)
It have to specifically map the 𝑖th string from 𝐿(𝐹1) to the 𝑖th string from 𝐿(𝐹2) using lexicographic order