Introduction to Computational Linguistics

15. Finite State Transducers 57
And for a set S ⊆ A ∗
(151) R T [S ] := {⃗y : exists ⃗x ∈ S : ⃗x R T ⃗y}
This is the set of all strings over B that are the result of translation via T of a string
in S . In the present case, notice that every string over A has a translation, but not
every string over B is the translation of a string. This is the case if and only if it
has even length.
The translation need not be unique. Here is a finite state machine that translates
a into bc ∗ .
(152)
A := {a}, B := {b, c}, Q := {0, 1}, i 0 := 0, F := {1},
δ := {〈0, a, 1, b〉, 〈1, ε, 1, c〉}
This automaton takes as only input the word a. However, it outputs any of b, bc,
bcc and so on. Thus, the translation of a given input can be highly underdetermined.
Notice also the following. For a language S ⊆ A ∗ , we have the following.
⎧
⎪⎨ bc ∗
(153) R T [S ] = ⎪⎩ ∅
if a ∈ S
otherwise.
This is because only the string a has a translation. For all words ⃗x a we have
R T (⃗x) = ∅.
The transducer can also be used the other way: then it translates words over B
into words over A. We use the same machine, but now we look at the relation
(154) R ⌣ T (⃗y) := {⃗x : ⃗x R T ⃗y}
(This is the converse of the relation R T .) Similarly we define
(155)
(156)
R ⌣ T (⃗y) := {⃗x : ⃗x R T ⃗y}
R ⌣ T [S ] := {⃗x : exists ⃗y ∈ S : ⃗x R T ⃗y}
Notice that there is a transducer U such that R U = R ⌣ T .
We quote the following theorem.
Theorem 21 (Transducer Theorem) Let T be a transducer from A to B, and let
L ⊆ A ∗ be a regular language. Then R T [L] is regular as well.