So reading through this paper:
http://www.cs.nyu.edu/~mohri/pub/fla.pdf
I see that a weighted finite state transducer (WFST) is a semiring, and many operations on WFST can be expressed in terms of "sum" and "product" over the semiring. For example, composition of Transducers one and two is:
(T1 ◦ T2)(x, y) = ⊕ z∈∆∗ T1(x, z)⊗T2(z, y)
But I can't seem to find an explanation on how do pure sum and product of WFST, and am having trouble backing out the operation from the composition example above.
A demonstration over this example would be much appreciated:
format: state1 state2, input alphabet : output alphabet, transition prob
T1
0 1 a : b, 0.1
0 2 b : b, 0.2
2 3 b : b, 0.3
0 0 a : a, 0.4
1 3 b : a, 0.5
T2
0 1 b : a, 0.1
1 2 b : a, 0.2
1 1 a : d, 0.3
1 2 a : c, 0.4
Example taken from: How to perform FST (Finite State Transducer) composition
--------------- update ------------
Found the answer in this document: http://www.cs.nyu.edu/~mohri/pub/hwa.pdf
page 12
