Termination of linear programs with nonlinear constraints

Tiwari (2004), proved that the termination problem of a class of linear programs (loops with linear loop conditions and updates) over the reals is decidable through Jordan forms and eigenvector computation. Braverman (2006) proved that it is also decidable over the integers. Following their work, we consider the termination problems of three more general classes of programs which are loops with linear updates and three kinds of polynomial loop conditions, i.e., strict constraints, non-strict constraints and both strict and non-strict constraints, respectively. First, we prove that the termination problems of such loops over the integers are all undecidable. Then, for each class we provide an algorithm to decide the termination of such programs over the reals. The algorithms are complete for those programs satisfying a property, Non-Zero Minimum.