Decomposing Non-Redundant Sharing by Complementation

TL;DR

This paper investigates the use of complementation for domain decomposition in abstract interpretation, revealing that removing redundancies prior to complementation yields more minimal and meaningful decompositions.

Contribution

It introduces a schema for subdomains of set-sharing, clarifies the role of non-redundant domains, and emphasizes the importance of redundancy removal before domain decomposition.

Findings

Redundant information in set-sharing domains affects decomposition results.

Replacing SH with its non-redundant version PSD leads to proper decompositions.

Redundancy removal before complementation is essential for minimal domain decompositions.

Abstract

Complementation, the inverse of the reduced product operation, is a technique for systematically finding minimal decompositions of abstract domains. File' and Ranzato advanced the state of the art by introducing a simple method for computing a complement. As an application, they considered the extraction by complementation of the pair-sharing domain PS from the Jacobs and Langen's set-sharing domain SH. However, since the result of this operation was still SH, they concluded that PS was too abstract for this. Here, we show that the source of this result lies not with PS but with SH and, more precisely, with the redundant information contained in SH with respect to ground-dependencies and pair-sharing. In fact, a proper decomposition is obtained if the non-redundant version of SH, PSD, is substituted for SH. To establish the results for PSD, we define a general schema for subdomains of SH that includes PSD and Def as special cases. This sheds new light on the structure of PSD and exposes a natural though unexpected connection between Def and PSD. Moreover, we substantiate the claim that complementation alone is not sufficient to obtain truly minimal decompositions of domains. The right solution to this problem is to first remove redundancies by computing the quotient of the domain with respect to the observable behavior, and only then decompose it by complementation.