We’ll work with the sequence Z = [ ϵ , y 1 , ϵ , y 2 , … , ϵ , y U , ϵ ] Z \enspace =\enspace [\epsilon, ~y_1, ~\epsilon, ~y_2,~ \ldots, ~\epsilon, ~y_U, ~\epsilon] Z=[ϵ, y1​, ϵ, y2​, …, ϵ, yU​, ϵ] which is Y Y Y with an ϵ \epsilon ϵ at the beginning, end, and between every character. α s , t = \alpha_{s, t} \; = α s , t ​ = ( α s − 2 , t − 1 + α s − 1 , t − 1 + α s , t − 1 ) ⋅ (\alpha_{s-2, t-1} + \alpha_{s-1, t-1} + \alpha_{s, t-1}) \quad\quad \cdot ( α s − 2 , t − 1 ​ + α s − 1 , t − 1 ​ + α s , t − 1 ​ ) ⋅ The CTC probability of the three valid subsequences after t − 1 t-1 t − 1 input steps. The probability of all alignments which end in ϵ \epsilon ϵ and the probability of all alignments which don’t end in ϵ .