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Turing-complete definition

A Turing-complete system (such as a computer or a programming language) is one that can simulate a Turing machine. This means the system can perform any calculation that any other programmable computer can, given enough time and memory. Turing completeness is a core concept in computer science, as it defines the capabilities of computing systems to solve problems.

See also: computer system

History of Turing completeness

  • Alan Turing's 1936 paper. The concept originates from Alan Turing's 1936 paper, “On Computable Numbers, with an Application to the Entscheidungsproblem.” Turing introduced the notion of a universal machine capable of computing anything computable. Now this concept is known as a Turing machine.
  • Church-Turing thesis. Alongside Turing's work, Alonzo Church developed lambda calculus, a formal system of mathematical logic. The Church-Turing thesis emerged, proposing that anything computationally expressible can be computed by a Turing machine.
  • Post-war developments. After World War II, the creation of actual computers brought these theoretical ideas into the practical realm. The concept of Turing completeness became crucial for early computer design.

Examples of Turing-complete systems

  • Programming languages. Most modern programming languages like Python, Java, and C++ are Turing-complete. They have conditional branching (if-else statements), loops (for, while), and the ability to use as much memory as needed. Given enough time and memory, this allows them to do any calculation that a Turing machine can.
  • Cellular automata. Conway's Game of Life, a simple cellular automaton, is Turing-complete. Despite its basic rules, it can create complex patterns acting like a computer's memory and logic, similar to a Turing machine.
  • Some esoteric programming languages. Languages like Brainfuck and Befunge, designed more for fun or challenge than practical use, are also Turing-complete. Despite their challenging syntax, they can compute anything just like standard programming languages.
  • Certain computational models. Lambda calculus uses function definition and application to represent any computable function. Post machines, with simple instructions and unbounded memory tape, can also simulate any computation a Turing machine can perform.
  • Blockchain smart contracts. Some blockchain platforms, like Ethereum, support Turing-complete smart contracts. They can be programmed to perform any algorithmic task, limited only by the computational resources of the network.