Reachability-based Acyclicity Analysis by Abstract Interpretation

TL;DR

This paper introduces a formal static analysis method based on abstract interpretation to determine whether data structures are acyclic in programming languages like Java, aiding in termination and resource usage analysis.

Contribution

It formalizes a novel combined reachability and cyclicity analysis using abstract interpretation, providing a sound and optimal method for inferring acyclicity.

Findings

Analysis is sound and optimal within the abstract domain

Enables bounding loop iterations based on data structure depth

Improves termination and resource usage analysis in dynamic memory languages

Abstract

In programming languages with dynamic use of memory, such as Java, knowing that a reference variable x points to an acyclic data structure is valuable for the analysis of termination and resource usage (e.g., execution time or memory consumption). For instance, this information guarantees that the depth of the data structure to which x points is greater than the depth of the data structure pointed to by x.f for any field f of x. This, in turn, allows bounding the number of iterations of a loop which traverses the structure by its depth, which is essential in order to prove the termination or infer the resource usage of the loop. The present paper provides an Abstract-Interpretation-based formalization of a static analysis for inferring acyclicity, which works on the reduced product of two abstract domains: reachability, which models the property that the location pointed to by a variable w can be reached by dereferencing another variable v (in this case, v is said to reach w); and cyclicity, modeling the property that v can point to a cyclic data structure. The analysis is proven to be sound and optimal with respect to the chosen abstraction.