Formal Definition of Turing Machines

A Turing machine M is a 7-tuple (Q,

, q₀, #, F) where

Q is a set of states,
is a finite set of symbols, the input alphabet,
is a finite set of symbols, the tape alphabet,
is the partial transition function,
# is a symbol called blank,
q₀ Q is the initial state,
F Q is a set of final states.

Because the Turing machine has to be able to find its input, and to know when it has processed all of that input, we require:

The tape is initially blank (every symbol is #) except possibly for a finite, contiguous sequence of symbols.
If there are initially nonblank symbols on the tape, the tape head is initially positioned on one of them.

Most other textbooks make no distinction between (the input alphabet) and (the tape alphabet). Our textbook does this to emphasize that the "input" (the nonblank symbols on the tape) does not contain #. Also, there may be more symbols in than are present in the input.