Question

The halting problem cannot be solved for Turing complete languages and it can be solved trivially for some non-TC languages like regexes where it always halts.

I was wondering if there are any languages that has both the ability to halt and not halt but admits an algorithm that can determine whether it halts.

Answer 1

Yes. One important class of this kind are primitive recursive functions. This class includes all of the basic things you expect to be able to do with numbers (addition, multiplication, etc.), as well as some complex classes like @adrian has mentioned (regular expressions/finite automata, context-free grammars/pushdown automata). There do, however, exist functions that are not primitive recursive, such as the Ackermann function.

It’s actually pretty easy to understand primitive recursive functions. They’re the functions that you could get in a programming language that had no true recursion (so a function f cannot call itself, whether directly or by calling another function g that then calls f, etc.) and has no while-loops, instead having bounded for-loops. A bounded for-loop is one like ‘for i from 1 to r’ where r is a variable that has already been computed earlier in the program; also, i cannot be modified within the for-loop. The point of such a programming language is that every program halts.

Most programs we write are actually primitive recursive (I mean, can be translated into such a language).