Loop invariants: analysis, classification, and examples

TL;DR

This paper systematically analyzes, classifies, and validates loop invariants across fundamental algorithms, providing insights into their patterns, derivation, and verification to improve program understanding and software verification.

Contribution

It introduces a taxonomy of loop invariants based on their derivation patterns, validated through mechanical verification, and offers insights for invariant inference and high-level specifications.

Findings

Identified common invariant patterns across diverse algorithms

Proposed a taxonomy based on invariant derivation from postconditions

Validated invariants using automated theorem proving

Abstract

Software verification has emerged as a key concern for ensuring the continued progress of information technology. Full verification generally requires, as a crucial step, equipping each loop with a "loop invariant". Beyond their role in verification, loop invariants help program understanding by providing fundamental insights into the nature of algorithms. In practice, finding sound and useful invariants remains a challenge. Fortunately, many invariants seem intuitively to exhibit a common flavor. Understanding these fundamental invariant patterns could therefore provide help for understanding and verifying a large variety of programs. We performed a systematic identification, validation, and classification of loop invariants over a range of fundamental algorithms from diverse areas of computer science. This article analyzes the patterns, as uncovered in this study, governing how invariants are derived from postconditions; it proposes a taxonomy of invariants according to these patterns, and presents its application to the algorithms reviewed. The discussion also shows the need for high-level specifications based on "domain theory". It describes how the invariants and the corresponding algorithms have been mechanically verified using an automated program prover; the proof source files are available. The contributions also include suggestions for invariant inference and for model-based specification.