Formula for the Search Space as a Union of Complete Sequences

In a sequence generation process, a sequence is considered 'complete' if it ends with a special ⟨EOS⟩ token. At a given step i, the set of all candidate sequences is Yi = {('The', 'cat', 'sat'), ('The', 'cat', 'slept', '⟨EOS⟩'), ('The', 'dog'), ('The', 'dog', 'barked', '⟨EOS⟩')}. If Ψ(Yi) represents the subset of sequences in Yi that are complete, which of the following correctly identifies Ψ(Yi)?

Defining the Set of Complete Sequences

In a step-wise sequence generation process, the set denoted by Ψ(Yi) contains all candidate sequences from step i that will be expanded to create new, longer sequences in the subsequent step.

In a step-by-step sequence generation process, a set of candidate sequences is maintained and expanded at each step. Suppose at a given step, the current set of candidate sequences is:

["The cat sat", "The dog ran ⟨EOS⟩", "The cat slept on"]

Assuming ⟨EOS⟩ is a special token indicating the end of a sequence, which of these sequences will be used as a basis for generating longer sequences in the next step?

Applying a Stopping Condition in Sequence Expansion

Efficiency of Sequence Expansion