What is Turing Complete?
What does the expression "Turing Complete" mean?
Can you give a simple explanation, without going into too many theoretical details?
Solution 1:
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.
Solution 2:
Here is the simplest explanation
Alan Turing created a machine that can take a program, run that program, and show some result. But then he had to create different machines for different programs. So he created "Universal Turing Machine" that can take ANY program and run it.
Programming languages are similar to those machines (although virtual). They take programs and run them. Now, a programing language is called "Turing complete", if it can run any program (irrespective of the language) that a Turing machine can run given enough time and memory.
For example: Let's say there is a program that takes 10 numbers and adds them. A Turing machine can easily run this program. But now imagine that for some reason your programming language can't perform the same addition. This would make it "Turing incomplete" (so to speak). On the other hand, if it can run any program that the universal Turing machine can run, then it's Turing complete.
Most modern programming languages (e.g. Java, JavaScript, Perl, etc.) are all Turing complete because they each implement all the features required to run programs like addition, multiplication, if-else condition, return statements, ways to store/retrieve/erase data and so on.
Update: You can learn more on my blog post: "JavaScript Is Turing Complete" — Explained
Solution 3:
Informal Definition
A Turing complete language is one that can perform any computation. The Church-Turing Thesis states that any performable computation can be done by a Turing machine. A Turing machine is a machine with infinite random access memory and a finite 'program' that dictates when it should read, write, and move across that memory, when it should terminate with a certain result, and what it should do next. The input to a Turing machine is put in its memory before it starts.
Things that can make a language NOT Turing complete
A Turing machine can make decisions based on what it sees in memory - The 'language' that only supports +, -, *, and / on integers is not Turing complete because it can't make a choice based on its input, but a Turing machine can.
A Turing machine can run forever - If we took Java, Javascript, or Python and removed the ability to do any sort of loop, GOTO, or function call, it wouldn't be Turing complete because it can't perform an arbitrary computation that never finishes. Coq is a theorem prover that can't express programs that don't terminate, so it's not Turing complete.
A Turing machine can use infinite memory - A language that was exactly like Java but would terminate once it used more than 4 Gigabytes of memory wouldn't be Turing complete, because a Turing machine can use infinite memory. This is why we can't actually build a Turing machine, but Java is still a Turing complete language because the Java language has no restriction preventing it from using infinite memory. This is one reason regular expressions aren't Turing complete.
A Turing machine has random access memory - A language that only lets you work with memory through push and pop operations to a stack wouldn't be Turing complete. If I have a 'language' that reads a string once and can only use memory by pushing and popping from a stack, it can tell me whether every ( in the string has its own ) later on by pushing when it sees ( and popping when it sees ). However, it can't tell me if every ( has its own ) later on and every [ has its own ] later on (note that ([)] meets this criteria but ([]] does not). A Turing machine can use its random access memory to track ()'s and []'s separately, but this language with only a stack cannot.
A Turing machine can simulate any other Turing machine - A Turing machine, when given an appropriate 'program', can take another Turing machine's 'program' and simulate it on arbitrary input. If you had a language that was forbidden from implementing a Python interpreter, it wouldn't be Turing complete.
Examples of Turing complete languages
If your language has infinite random access memory, conditional execution, and some form of repeated execution, it's probably Turing complete. There are more exotic systems that can still achieve everything a Turing machine can, which makes them Turing complete too:
- Untyped lambda calculus
- Conway's game of life
- C++ Templates
- Prolog
Solution 4:
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'".
Solution 5:
Fundamentally, Turing-completeness is one concise requirement, unbounded recursion.
Not even bounded by memory.
I thought of this independently, but here is some discussion of the assertion. My definition of LSP provides more context.
The other answers here don't directly define the fundamental essence of Turing-completeness.