Identifying Minimal Changes in the Zone Abstract Domain
Kenny Ballou, Elena Sherman
TL;DR
This work tackles the scalability of relational numerical analyses by introducing semantic, Zone-specific minimization techniques to identify the minimal set of inequalities that can change after a program state update. It proposes a family of algorithms—RemoveSpurious, ConnectedComponents, NodeNeighbors, and MinimalNeighbors—to compute the smallest affected subset of Zone constraints while mitigating spurious edges through a Z0-based reduction and accommodating widening/merges in the DFA framework. Empirically, the approach yields large state-size reductions (over 70% fewer variables and ~30% fewer inequalities) and substantial speedups in domain comparisons, with notable improvements in cross-domain precision (e.g., Zones vs Predicates incomparabilities dropping dramatically and equalities with Intervals increasing). These results clarify how fine-grained, semantics-aware changes enhance both accuracy and efficiency, and point to future extensions to other relational domains and adaptive analysis strategies.
Abstract
Verification techniques express program states as logical formulas over program variables. For example, symbolic execution and abstract interpretation encode program states as a set of integer inequalities. However, for real-world programs these formulas tend to become large, which affects scalability of analyses. To address this problem, researchers developed complementary approaches which either remove redundant inequalities or extract a subset of inequalities sufficient for specific reasoning. For arbitrary integer inequalities, such reduction approaches either have high complexities or over-approximate. However, efficiency and precision of these approaches can be improved for a restricted type of logical formulas used in relational numerical abstract domains. While previous work investigated custom efficient redundant inequality elimination for Zones states, our work examines custom semantic slicing algorithms that identify a minimal set of changed inequalities in Zones states. The client application of the minimal changes in Zones is an empirical study on comparison between invariants computed by data-flow analysis using Zones, Intervals and Predicates numerical domains. In particular, evaluations compare how our proposed algorithms affect the precision of comparing Zones vs. Intervals and Zones vs. Predicates abstract domains. The results show our techniques reduce the number of variables by more than 70% and the number of inequalities by 30%, compared to full states. The approach refines the granularity of comparison between domains, reducing incomparable invariants between Zones and Predicates from 52% to 4%, and increases equality of Intervals and Zones, invariants from 27% to 71%. The techniques improve the comparison efficiency by reducing total runtime for all subject comparisons for Zones and Predicates from over 4 minutes to a few seconds.