There is growing interest in termination reasoning for non-linear programs and, meanwhile, recent dynamic strategies have shown they are able to infer invariants for such challenging programs. These advances led us to hypothesize that perhaps such dynamic strategies for non-linear invariants could be adapted to learn recurrent sets (for non-termination) and/or ranking functions (for termination).

In this paper, we exploit dynamic analysis and draw termination and non-termination as well as static and dynamic strategies closer together in order to tackle non-linear programs. For termination, our algorithm infers ranking functions from concrete transitive closures, and, for non-termination, the algorithm iteratively collects executions and dynamically learns conditions to refine recurrent sets. Finally, we describe an integrated algorithm that allows these algorithms to mutually inform each other, taking counterexamples from a failed validation in one endeavor and crossing both the static/dynamic and term./non-term.~ lines, to create new execution samples for the other one.

We have implemented these algorithms in a new tool called DynamiTe. For non-linear programs, there are currently no SV-COMP termination benchmarks so we created new sets of 37 terminating and 37 non-terminating programs. Our empirical evaluation shows that we can effectively guess (and sometimes even validate) ranking functions and recurrent sets for programs with non-linear behaviors. Furthermore, we show that counterexamples from one failed validation can be used to generate executions for a dynamic analysis of the opposite property. Although we are focused on non-linear programs, as a point of comparison, we compare DynamiTe’s performance on linear programs with that of the state-of-the-art tool, Ultimate. Although DynamiTe is an order of magnitude slower it is nonetheless somewhat competitive and sometimes finds ranking functions where Ultimate was unable to. Ultimate cannot, however, handle the non-linear programs in our new benchmark suite.