A Turing machine is a mathematical model of computation describing an abstract machine^[1] that manipulates symbols on a strip of tape according to a table of rules.^[2] Despite the model’s simplicity, it is capable of implementing any computer algorithm.^[3]

The machine operates on an infinite^[4] memory tape divided into discrete cells,^[5] each of which can hold a single symbol drawn from a finite set of symbols called the alphabet of the machine. It has a “head” that, at any point in the machine’s operation, is positioned over one of these cells, and a “state” selected from a finite set of states. At each step of its operation, the head reads the symbol in its cell. Then, based on the symbol and the machine’s own present state, the machine writes a symbol into the same cell, and moves the head one step to the left or the right,^[6] or halts the computation. The choice of which replacement symbol to write, which direction to move the head, and whether to halt is based on a finite table that specifies what to do for each combination of the current state and the symbol that is read. As with a real computer program, it is possible for a Turing machine to go into an infinite loop which will never halt.