Difference between revisions of "Turing completeness"

Revision as of 06:02, 16 February 2016

In computability theory, a system of data-manipulation rules (such as a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing complete or computationally universal if it can be used to simulate any single-taped Turing machine.

The concept is named after English mathematician Alan Turing.

A classic example is lambda calculus.

A closely related concept is that of Turing equivalence -- two computers P and Q are called equivalent if P can simulate Q and Q can simulate P.

According to the Church–Turing thesis, which conjectures that the Turing machines are the most powerful computing machines, for every real-world computer there exists a Turing machine that can simulate its computational aspects.

Universal Turing machines can simulate any Turing machine and by extension the computational aspects of any possible real-world computer.

Example

To show that something is Turing complete, it is enough to show that it can be used to simulate some Turing complete system.

For example, an imperative language is Turing complete if it has conditional branching (e.g., "if" and "goto" statements, or a "branch if zero" instruction.) and the ability to change an arbitrary amount of memory locations (e.g., the ability to maintain an arbitrary number of variables).

Since this is almost always the case, most (if not all) imperative languages are Turing complete if the limitations of finite memory are ignored.

See also

• Alan Turing
• Computability theory
• Functional completeness
• Mathematical logic
• Mathematics
• Turing machine

External links

• Turing completeness @ Wikipedia