What is Turing Complete?

Here's the briefest explanation:

A Turing Complete system means a system in which a program can be written that will find an answer (although with no guarantees regarding runtime or memory).

So, if somebody says "my new thing is Turing Complete" that means in principle (although often not in practice) it could be used to solve any computation problem.

Sometimes it's a joke... a guy wrote a Turing Machine simulator in vi, so it's possible to say that vi is the only computational engine ever needed in the world.

From wikipedia:

Turing completeness, named after Alan Turing, is significant in that every plausible design for a computing device so far advanced can be emulated by a universal Turing machine — an observation that has become known as the Church-Turing thesis. Thus, a machine that can act as a universal Turing machine can, in principle, perform any calculation that any other programmable computer is capable of. However, this has nothing to do with the effort required to write a program for the machine, the time it may take for the machine to perform the calculation, or any abilities the machine may possess that are unrelated to computation.

While truly Turing-complete machines are very likely physically impossible, as they require unlimited storage, Turing completeness is often loosely attributed to physical machines or programming languages that would be universal if they had unlimited storage. All modern computers are Turing-complete in this sense.

I don't know how you can be more non-technical than that except by saying "turing complete means 'able to answer computable problem given enough time and space'".

Turing Complete means that it is at least as powerful as a Turing Machine. This means anything that can be computed by a Turing Machine can be computed by a Turing Complete system.

No one has yet found a system more powerful than a Turing Machine. So, for the time being, saying a system is Turing Complete is the same as saying the system is as powerful as any known computing system (see Church-Turing Thesis).

I think the importance of the concept "Turing Complete" is in the the ability to identify a computing machine (not necessarily a mechanical/electrical "computer") that can have its processes be deconstructed into "simple" instructions, composed of simpler and simpler instructions, that a Universal machine could interpret and then execute.

I highly recommend The Annotated Turing

@Mark i think what you are explaining is a mix between the description of the Universal Turing Machine and Turing Complete.

Something that is Turing Complete, in a practical sense, would be a machine/process/computation able to be written and represented as a program, to be executed by a Universal Machine (a desktop computer). Though it doesn't take consideration for time or storage, as mentioned by others.

Super-brief summary from what Professor Brasilford explains in this video.

Turing Complete ≅ do anything that a Turing Machine can do.

1. It has conditional branching (i.e. "if statement"). Also, implies "go to" and thus permitting loop.

2. It gets arbitrary amount of memory (e.g. long enough tape) that the program needs.

In practical language terms familiar to most programmers, the usual way to detect Turing completeness is if the language allows or allows the simulation of nested unbounded while statements (as opposed to Pascal-style for statements, with fixed upper bounds).

A Turing Machine requires that any program can perform condition testing. That is fundamental.

Consider a player piano roll. The player piano can play a highly complicated piece of music, but there is never any conditional logic in the music. It is not Turing Complete.

Conditional logic is both the power and the danger of a machine that is Turing Complete.

The piano roll is guaranteed to halt every time. There is no such guarantee for a TM. This is called the “halting problem.”

We call a language Turing-complete if and only if (1) it is decidable by a Turing machine but (2) not by anything less capable than a Turing machine. For instance, the language of palindromes over the alphabet {a, b} is decidable by Turing machines, but also by pushdown automata; so, this language is not Turing-complete. Truly Turing-complete languages - ones that require the full computing power of Turing machines - are pretty rare. Perhaps the language of strings x.y.z where x is a number, y is a Turing-machine and z is an initial tape configuration, and y halts on z in fewer than x! steps - perhaps that qualifies (though it would need to be shown!)

A common imprecise usage confuses Turing-completeness with Turing-equivalence. Turing-equivalence refers to the property of a computational system which can simulate, and which can be simulated by, Turing machines. We might say Java is a Turing-equivalent programming language, for instance, because you can write a Turing-machine simulator in Java, and because you could define a Turing machine that simulates execution of Java programs. According to the Church-Turing thesis, Turing machines can perform any effective computation, so Turing-equivalence means a system is as capable as possible (if the Church-Turing thesis is true!)

Turing equivalence is a much more mainstream concern that true Turing completeness; this and the fact that "complete" is shorter than "equivalent" may explain why "Turing-complete" is so often misused to mean Turing-equivalent, but I digress.