Based on the principle of how the search space expands at each step of generation, which model described below would face a significantly greater computational challenge when performing an exhaustive search to generate a sequence of 10 tokens? Justify your answer by explaining how the size of the vocabulary influences the total number of sequences that must be evaluated.

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In sequence generation, the set of sequences generated at step $$i$$, denoted $$Y_i$$, can be obtained by expanding each sequence in the previous set $$Y_{i-1}$$ with all possible next tokens from the vocabulary $$V$$. This is expressed recursively as the Cartesian product: $$Y_i = Y_{i-1} \times V$$. This operation denotes that each sequence in $$Y_{i-1}$$ is concatenated with each token in $$V$$. However, if a sequence in $$Y_{i-1}$$ is already complete (for instance, ending with an $$\langle \mathrm{EOS} \rangle$$ token), it will not be expanded any further.

Formula for the Expansion of the Search Space at Each Step

A language model with a vocabulary of 5,000 unique tokens is generating text. If, at a certain stage, the model is considering 20 different unique sequences of tokens, each 4 tokens long, how many total unique sequences of 5 tokens can possibly be generated in the very next step?

A language model with a vocabulary of 50,000 tokens is generating a sequence. Explain why generating the 10th token in a sequence is computationally a much larger problem than generating the 2nd token, even though both involve selecting just one token. Base your explanation on the growth of the total number of possible sequences.

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