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Turing-complete

# Turing-complete

## 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.

## 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.