Hierarchical Analysis of Loops With Relaxed Abstract Transformers

Numerical computation is often involved in software of embedded control systems, cyber-physical systems, artificial neural network systems, big data processing systems, etc. Automatically discovering numerical loop invariants is fundamental for checking the safety of such software. Abstract interpretation provides a framework to automatically discover sound invariants but which may be not precise enough due to over-approximations. One major source of precision loss is due to the limited linear expressiveness of most widely used numerical abstract domains and the widening operation. This becomes more serious when analyzing all variables simultaneously as a whole for programs that involve nonlinear behaviors. Based on the observation that the dependency among variables in a loop can be hierarchical, in this article, we propose a hierarchical static analysis to analyze a loop by utilizing relaxed abstract transformers. The main idea is to first partition all variables involved in a loop into different hierarchical layers, then compute invariants over the variables layer by layer in a bottom-up manner. During the iterative process, the computed invariants over lower layer variables are then used to relax transfer functions when analyzing the higher layer variables. One benefit of our method lies in that it can generate linear invariants to soundly enclose nonlinear behaviors in a loop. Finally, we present encouraging experimental results on benchmark programs involving nonlinear behaviors.