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I'm considering turing machines with tape alphabet equal to $\{0,1\}$ and the blank symbol equal to $0$. Is there a name for this type of Turing machines? Isn't this type the one computers are based on?

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That would be a Turing machine with a binary tape alphabet. The term binary Turing machine also seems to be common enough (a Google search gives just under 1000 results for the term).

As David Richerby pointed out in the other answer, though, Turing machines are not what computers are based on. Rather, (modern) computers are based on register machines (Wikipedia). What we do have is an equivalence between the two computation models.

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Not sure about a name, but real computers aren't based on Turing machines at all.

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