CTC

Speech recognition has an alignment problem: you have the audio and you have the transcript, but not which frames produced which letters, and hand-labelling that is hopeless. CTC removes the question. It adds a blank symbol meaning "nothing new here", then scores a transcript by summing over every frame-to-letter alignment that could have produced it. You never have to say where a letter starts.

Per-frame outputs over the alphabet plus a blank, trained by maximising the total probability of every alignment that collapses to the target — collapse being merge repeats, then drop blanks. Frames are conditionally independent given the audio, which is what makes the dynamic program cheap and also what makes CTC a poor language model: it cannot express that one output depends on the last. Restoring that dependency without giving up streaming is exactly what RNN-T adds.

p(y|x) = Σ sub _{a ∈ B super ^-1 (y)}  Π sub _t  p(a sub _t  | x), where B merges repeated labels and then removes blanks. The sum runs over a count of alignments exponential in T and is computed in O(T·|y|) by the same forward–backward recursion an HMM uses. The ordering inside B is load-bearing: merge-then-drop is why "hello" needs a blank between its two l's to survive the collapse.