Turing completeness

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

• Abstract machine
• Alan Turing
• Church–Turing thesis
• Computability theory
• Computer science
• Functional completeness
• Lambda calculus
• Mathematical logic
• Mathematics
• Non-deterministic Turing machine
• Turing machine
• Universal Turing machine

External links

• Turing completeness @ Wikipedia
• PowerPoint is Turing complete, among other things @ BoingBoing - a Turing-complete punchcard-driven universal machine that is embodied entirely in Powerpoint Animations and can execute any arbitrary code (albeit very slowly) and presented it at CMU's SIGBOVIK 2017 conference to great hilarity.