What are practical guidelines for evaluating a language's "Turing Completeness"?

Question

I've read "what-is-turing-complete" and the wikipedia page, but I'm less interested in a formal proof than in the practical implications of being Turing Complete.

What I'm actually trying to decide is if the toy language I've just designed could be used as a general-purpose language. I know I can prove it is if I can write a Turing machine with it. But I don't want to go through that exercise until I'm fairly certain of success.

Is there a minimum set of features without which Turing Completeness is impossible? Is there a set of features which virtually guarantees completeness?

(My guess is that conditional branching and a readable/writeable memory store will get me most of the way there)

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EDIT:

I think I've gone off on a tangent by saying "Turing Complete". I'm trying to guess with reasonable confidence that a newly invented language with a certain feature set (or alternately, a VM with a certain instruction set) would be able to compute anything worth computing. I know proving you can build a Turing machine with it is one way, but not the only way.

What I was hoping for was a set of guidelines like: "if it can do X,Y,and Z, it can probably do anything".

Answer 1

You need some form of dynamic allocation construct (malloc ornew or cons will do) and either recursive functions or some other way of writing an infinite loop. If you have those and can do anything at all interesting, you're almost certainly Turing-complete.

The lambda calculus is equivalent in power to a Turing machine, and if you implement lambda calculus it is actually pretty fun writing lambda calculus programs. Way more fun than writing program for a Turing machine!

The only practical implication of Turing-completeness I'm aware of is that you can write programs that don't terminate. I've used a couple of special-purpose languages that guarantee termination and therefore are not Turing-complete. Sometimes it is useful to give up the extra expressive power in exchange for guaranteed termination.

Answer 2

'Turing Completeness' describes the property of being able to express any arbitrary algorithmic computation, which was the point of Turing's Machine in the first place. A language or logical system can be described as 'Turing Complete' if it has this property. From a practical perspective all general purpose programming languages - and a surprisingly large number of special purpose ones - can do this for a suitably loose definition (see below).

However, a strict definition of Turing Completeness implies infinite storage capacity, which is of course not physically possible. Given this, no physical machine can possibly be Turing Complete, but this constraint is usually relaxed (at least informally) when ascribing Turing Completeness to a programming language. One trivial test of Turing Completeness for a language is whether the language can be used to implement a Turing Machine simulator.

An example of a widespread system that is not Turing Complete is Relational Algebra, the theoretical basis behind SQL as described in Codd's paper A relational model for large shared data banks. Relational Algebra has the property of Godel Completeness, which means that it can express any computation that can be defined in terms of first-order predicate calculus (i.e. ordinary logical expressions). However, it is not Turing-Complete as it cannot express an arbitrary algorithmic computation.

Note that most if not all all practical SQL dialects extend the pure relational model with procedural constructs to the extent that they are Turing Complete by the definition as normally applied to programming languages. However, an individual SQL query by and large is not.

Some more egregious examples of Turing Complete domain-specific languages are TeX and sendmail.cf,. In the latter case there is actually a famous-ish example of someone using sendmail.cf to implement a universal Turing Machine simulator.