### math

#### How do you define summary and extension of weighted finite state transducers?

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

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