A Turing machine has a tape, a head that can move left or right, a finite set of states, and a rule for each pair of state and symbol. That is all. No memory beyond the tape, no arithmetic, no jumps. And in 1936 Alan Turing showed that anything a person can compute, this thing can compute — which is the reason the machine is a machine and not a metaphor.
Click a table cell to cycle its write symbol and direction. Use the dropdown to set the next state.
The image is doing more work than it looks. A Turing machine's tape is unbounded — it extends forever to the right, which is the whole point, because that is where the machine writes its output and keeps its working memory. The tape is blank except where the head has been, and blank is a symbol like any other.
Choosing that picture was not a metaphor but a claim. Turing argued that a person calculating is doing exactly this: reading a symbol, deciding what to write next, and moving one place along. Everything else — the paper, the pencil, the desk — is scaffolding for a process that is already purely symbolic.
The proof has two halves. First, that a machine like this can be built out of switches and wires — that it is not a magical object. Second, and much harder, that any procedure a person could carry out could be written down as a set of rules of this kind, so that a machine following the rules would do the same thing.
It is a mechanical argument, made without reference to consciousness, and it is the reason a modern computer is called a stored-program machine: the program is written on the same tape the machine works on.