Computational Models and Complexities of Tarski’s Fixed Points∗

We consider two models of computation for Tarski’s order preserving function f related to fixed points in a complete lattice: the oracle function model and the polynomial function model. We develop a complete understanding under the oracle function model for finding a Tarski’s fixed point as well as determining the uniqueness of Tarski’s fixed point in both the lexicographic ordering and the componentwise ordering lattices. Moreover, we present a polynomial-time reduction of an integer program to an order preserving mapping f from a lattice L into itself. As a result of this reduction, we prove that, when f is given as a polynomial function, determining whether or not f has a unique fixed point is Co-NP hard.