The halting problem forbids the existence of oracles which can decide with certainty in some finite time that an arbitrary program will halt. Loosen any of thes

The halting problem forbids the existence of oracles which can decide with certainty in some finite time that an arbitrary program will halt. Loosen any of these qualifiers and the problem is not necessarily undecidable. Loosening the "arbitrary program" restriction, the problem of analyzing the termination of a subset of programs which doesn't encode looping or recursion constructs ( i.e. simply typed lambda calculus ) is trivially decidable. A totality checker like in Agda can decide whether some programs halt given a set of restrictions on the program.