Section: New Results

Termination

We have explored the analysis of program termination and the inference of sufficient conditions to ensure the definite termination of programs using abstract interpretation techniques. Following [40] , we employ a backward analysis over an abstract domain of ranking functions.

We present in [24] and [23] a parameterized abstract domain that infers sufficient conditions for program termination by automatically synthesizing piecewise-defined ranking functions over natural numbers. The analysis uses over-approximations but we prove its soundness, meaning that all program executions respecting these sufficient conditions are indeed terminating. The abstract domain is parameterized by a numerical abstract domain for environments and a numerical abstract domain for functions. This parameterization allows to easily tune the trade-off between precision and cost of the analysis. We describe an instantiation of this generic domain with intervals and affine functions. We define all abstract operators, including widening to ensure convergence. To experiment with this domain, we have implemented a research prototype static analyzer FuncTion ( 5.6 ) that yielded interesting preliminary results.

We observed that, in some important cases (such as programs with unbounded non-determinism), there does not exist any ranking function over natural numbers. In [22] and [29] we propose a new abstract domain to automatically infer ranking functions over ordinals. We extended the domain of piecewise-defined natural-valued ranking functions introduced in the previous section to polynomials in $\omega$, where the polynomial coefficients are natural-valued functions of the program variables. The abstract domain is parametric in the choice of the maximum degree of the polynomial, and the types of functions used as polynomial coefficients. We have enriched the FuncTion prototype analyzer ( 5.6 ) with an instantiation of our domain using affine functions as polynomial coefficients. We successfully analyzed small but intricate examples that are out of the reach of existing methods. To our knowledge this is the first abstract domain able to reason about ordinals. Handling ordinals leads to a powerful approach for proving termination of imperative programs, which in particular subsumes existing techniques based on lexicographic ranking functions.