Proving termination with transition invariants of height omega
  The Termination Theorem by Podelski and Rybalchenko states that the reduction relations which are terminating from any initial state are exactly the reduction relations whose transitive closure, restricted to the accessible states, is included in some finite union of well-founded relations. An alternative statement of the theorem is that terminating reduction relations are precisely those having a "disjunctively well-founded transition invariant". From this result the same authors and Byron Cook designed an algorithm checking a sufficient condition for termination for a while-if program. The algorithm looks for a disjunctively well-founded transition invariant, made of well-founded relations of height omega, and if it finds it, it deduces the termination for the while-if program using the Termination Theorem.   This raises an interesting question: What is the status of reduction relations having a disjunctively well-founded transition invariant where each relation has height omega? An answer to this question can lead to a characterization of the set of while-if programs which the termination algorithm can prove to be terminating. The goal of this work is to prove that they are exactly the set of reduction relations having height omega^n for some n < omega. Besides, if all the relations in the transition invariant are primitive recursive and the reduction relation is the graph of the restriction to some primitive recursive set of a primitive recursive map, then a final state is computable by some primitive recursive map in the initial state.   As a corollary we derive that the set of functions, having at least one implementation in Podelski Rybalchenko while-if language with a well-founded disjunctively transition invariant where each relation has height omega, is exactly the set of primitive recursive functions. 
Stefano Berardi, Paulo Oliva, Silvia Steila
cs.LO