What is Turing Complete?

Question

What does the expression "Turing Complete" mean?

Can you give a simple explanation, without going into too many theoretical details?

Answer 1

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).

Answer 2

As Waylon Flinn said:

Turing Complete means that it is at least as powerful as a Turing Machine.

I believe this is incorrect, a system is Turing complete if it's exactly as powerful as the Turing Machine, i.e. every computation done by the machine can be done by the system, but also every computation done by the system can be done by the Turing machine.

Answer 3

Can a relational database input latitudes and longitudes of places and roads, and compute the shortest path between them - no. This is one problem that shows SQL is not Turing complete.

But C++ can do it, and can do any problem. Thus it is.

Answer 4

Here is the simplest explanation

Alan Turing created a machine that can take a program and 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 that 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. Turing machine can easily run this program. But now imagine for some reason your programming language can't do the same addition then it's Turing machine incomplete. On the other hand, if it can run any program like the universal turing machine can run, then it's Turing complete.

Most modern programming languages like Java, JavaScript, Perl etc are all turing complete because they all 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

Answer 5

What i understand in simple words:

Turing Complete : A programming language / program that can do computation, is Turing complete.

For example :

1. Can you add two numbers using Just HTML. (Ans is 'No', you have to use javascript to perform addition.), Hence HTML is not Turing Complete.

2. Languages like Java , C++, Python, Javascript, Solidity for Ethereum etc are Turing Complete because you can do computation like adding two numbers using this languages.

Hope this helps.

Answer 6

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