A Few Graph-Based Relational Numerical Abstract Domains

TL;DR

This paper introduces a systematic method to design weakly relational numerical abstract domains from non-relational ones, enhancing static analysis capabilities with modular and extendable domain constructions.

Contribution

It presents a novel, modular approach to constructing weakly relational numerical abstract domains using potential graphs and shortest-path algorithms, enabling quick development of new domains.

Findings

Retrieves well-known and new domains using the proposed method.

Enhances static analysis through modular domain construction.

Demonstrates the applicability of the approach in Abstract Interpretation.

Abstract

This article presents the systematic design of a class of relational numerical abstract domains from non-relational ones. Constructed domains represent sets of invariants of the form (vj - vi in C), where vj and vi are two variables, and C lives in an abstraction of P(Z), P(Q), or P(R). We will call this family of domains weakly relational domains. The underlying concept allowing this construction is an extension of potential graphs and shortest-path closure algorithms in exotic-like algebras. Example constructions are given in order to retrieve well-known domains as well as new ones. Such domains can then be used in the Abstract Interpretation framework in order to design various static analyses. Amajor benfit of this construction is its modularity, allowing to quickly implement new abstract domains from existing ones.