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What you describe is essentially Turing machines with advice, the advice for length $i$ being simply the description of $T_i$. It is a classic result that the two models are equivalent in the case of poly-time TMs and poly-sized circuits, that is, both produce the same class $\mathsf{P}/\mathrm{poly}$. If the description length of $T_i$ is allowed to be ...


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