Turing Machine


In theoretical computer science, a Turing machine is a mathematical model of computation that provides a foundational abstraction of how computers function. Invented by Alan Turing in 1936, the Turing machine is a remarkably simple concept, yet it is theoretically capable of implementing any computer algorithm that can be devised.

Key Components

  • Tape: An infinitely long strip of tape divided into discrete squares or "cells." Each cell can hold a single symbol from a finite alphabet.
  • Head: A device that can read and write symbols on the tape, and can move one cell left or right at a time.
  • State Register: Stores the current state of the Turing machine, drawn from a finite set of possible states.
  • Table of Rules (Transition Function): A set of instructions that determine the Turing machine's actions based on its current state and the symbol it is reading on the tape. A rule will typically say something like: "If in state 2 and you see the symbol '0', write the symbol '1', move the head to the right, and change to state 7."


The Turing machine begins in a designated start state, with its head positioned on a particular cell of the tape. It then operates as follows:

  1. Read: The machine reads the symbol currently under the head.
  2. Apply Rule: Based on the current state and the symbol read, the machine looks up the relevant rule in its table of rules.
  3. Write: The machine overwrites the symbol in the current cell with a new symbol (which might be the same symbol).
  4. Move: The machine moves its head one cell to the left or right.
  5. Change State: The machine changes its internal state according to the rule.
  6. Repeat: It repeats these steps until it reaches a designated halt state.


  • Foundation of Computability: Turing machines capture the fundamental logic of computation. Any problem solvable by a modern computer can, in principle, be solved by a Turing machine (assuming enough time and memory).
  • Church-Turing Thesis: This widely accepted principle states that any real-world calculation or process that can be defined effectively as an algorithm can be carried out by a Turing machine.
  • Limits of Computation: Although theoretically very powerful, Turing machines helped illustrate the inherent limits of computation. Specifically, the 'halting problem' proves the existence of problems that no Turing machine (and hence no computer algorithm) can definitively solve.

Additional Notes

  • There are many variants of Turing machines, some with multiple tapes, more complex movement capabilities, or non-deterministic transitions.
  • While a theoretical concept, Turing machines influenced the design of early computers and continue to be an essential tool for investigating computation, algorithms, and complexity theory.