A theory of function-induced-orders to study recursion termination

Termination property of functions is an important issue in computability theory. In this paper, we show that repeated iterations of a function can induce an order amongst the elements of its domain set. Hasse diagram of the poset, thus obtained, is shown to look like a forest of trees, with a possible base set and a generator set (defined in the paper). Isomorphic forests may arise for different functions and equivalences classes are, thus, formed. Based on this analysis, a study of the class of deterministically terminating functions is presented, in which the existence of a Self-Ranking Program, which can prove its own termination, and a Universal Terminating Function, from which every other terminating function can be derived, is conjectured.
Comment: Not relevant anymore