Finite State Transducers

A Mealy machine is usually defined as a 6-tuple

M=(Q,A,B,δ,f,q0)

Where

Q = set of states
A = input alphabet
B = output alphabet
δ = transition function Q x A -> Q
f = output function Q x A -> B
q₀ = initial or start state

Matt Foley Pictorial Example:

                             ____ 
            ----   a/X     |      |    b/U    ---- 
           |    |--------->|      |--------->|    |
      ---->| s0 |          |  s1  |          | s2 |
            ----           |      |<--------- ----
            ^ |              ----       a/V     |
         a/X| |b/Y            |                 | b/T
            | V               |                 |
            ----              |a/S              V
           |    |_____        |               ---- _______
           | s4 |     | b/Y    ------------->|    |       | a/S,b/T
            ---- <----                       | s3 |<------
                                              ----

Q = {s₀, s₁, s₂, s₃, s₄}
A = {a,b}
B = {S,T,U,V,X,Y}
q₀ = s₀
δ: Q x A -> Q     f: Q x A -> B

 Q \ A |   a    |     b            Q \ A|   a    |     b    
-------|--------|----------      -------|--------|----------
   s0  |   s1   |     s4            s0  |   X    |     Y    
-------|--------|----------      -------|--------|----------
   s1  |   s3   |     s2            s1  |   S    |     U    
-------|--------|----------      -------|--------|----------
   s2  |   s1   |     s3            s2  |   V    |     T    
-------|--------|----------      -------|--------|----------
   s3  |   s3   |     s3            s3  |   S    |     T    
-------|--------|----------      -------|--------|----------
   s4  |   s1   |     s4            s4  |   X    |     Y    

Extending δ to operate on strings.

Given Mealy machine M with transition function δ, and output function f, define δ' and f' as:

δ' : Q x A* -> Q
δ'(q,λ) = q - on null input stays in original state
δ'(q,a) = δ(q,a) if a in A
δ'(q,w) = δ'(d(q,a),x) if w = ax for a in A, x in A*

And

f' : Q x A* -> B
f'(q,ε) = ε - on null input, no output
f'(q,a) = f(q,a) if a in A
f'(q,w) = f(q,a) . f'(δ(q,a),x) if w = ax for a in A, x in A*
(using '.' to denote concatenation.)

These are called the extended transition and output functions, respectively.

• combining transition and output function into one:

  δ : Q X A X B -> Q

  Rather than producing output, this FST considers what would have been output from a Mealy machine as a part of the input.

• including a set of final states.

We can then note that with this definition an FST can be viewed as:

1. recognizer - recognizes the language of corresponding strings in the input/output pair

   it might be beneficial to call this a ``translation recognizer'' - is the pair of strings (input/output) a valid translation from one to the other

2. generator - generates the language of corresponding strings in the input/output pair

   again, it might be beneficial to call this a ``translation generator'' - each transition can be considered as generating the output character from the input character (or vice versa).

3. translator - reads the input string and produces the output string

4. set relater - computes a relation between the set of input strings and the set of output strings

Test the following on the above example to see what the output string would be.

    baab
    baabb
    bab
    babaaba