Discovering non-terminating inputs for multi-path polynomial programs

We investigate the termination problems of multi-path polynomial programs (MPPs) with equational loop guards. To establish sufficient conditions for termination and nontermination simultaneously, we first propose the notion of strong/weak non-termination which under/over-approximates non-termination. Based on polynomial ideal theory, we show that the set of all strong non-terminating inputs (SNTI) and weak non-terminating inputs (WNTI) both correspond to the real varieties of certain polynomial ideals. Furthermore, we prove that the variety of SNTI is computable, and under some sufficient conditions the variety of WNTI is also computable. Then by checking the computed SNTI and WNTI varieties in parallel, termination properties of a considered MPP can be asserted. As a consequence, we establish a new framework for termination analysis of MPPs.