What are the practical limitations of a non-turing complete language like Coq?

Question

As there are non-Turing complete languages out there, and given I didn\'t study Comp Sci at university, could someone explain something that a Turing-incomplete language (like C

Answer 1

The most direct answer is: a machine/language that is not Turing complete cannot be used to implement/simulate Turing machines. This comes from the basic definition of Turing completeness: a machine/language is turing complete if it can implement/simulate Turing machines.

So, what are the practical implications? Well, there is a proof that anything that can be shown to be turing complete can solve all computable problems. Which by definition means that anything that is not turing complete has the handicap that there are some computable problems that it can't solve. What those problems are depends on what features are missing that makes the system non-Turing complete.

For example if a language doesn't support looping or recursion or implicitly loops cannot be Turing complete because Turing machines can be programmed to run forever. Consequently that language cannot solve problems requiring loops.

Another example is if a language doesn't support lists or arrays (or allow you to emulate them for example using the filesystem) then it cannot implement a Turing machine because Turing machines require arbitrary random access to memory. Consequently that language cannot solve problems requiring arbitrary random access to memory.

So, the feature that is missing that classifies a language to be non-Turing complete is the very thing that practically limits the usefulness of the language. So the answer is, it depends: what makes the language non-Turing complete?