Termination of initialized two variable homogeneous linear loops

In this paper, we consider the termination problems about uninitialized and initialized two variable homogeneous linear loops. Through analysing the properties of the boundaries of the set NT composed of all points on which a given uninitialized two variable homogeneous linear loop does not terminate, we reconstruct a group of simpler and complete quantified formulas to check whether a given uninitialized loop terminates. Furthermore, we present another group of quantified formulas to check the termination of initialized two variable homogeneous linear loops.