Incrementally Closing Octagons

TL;DR

This paper introduces new quadratic incremental algorithms for closure operations in the octagon abstract domain, improving efficiency and correctness proofs for relational numeric constraints.

Contribution

It presents novel quadratic incremental algorithms for closure, strong closure, and integer closure in the octagon domain, with correctness proofs and performance evaluation.

Findings

Algorithms are quadratic and incremental, enhancing efficiency.

Correctness of the algorithms is formally proven.

Performance improvements are demonstrated through measurements.

Abstract

The octagon abstract domain is a widely used numeric abstract domain expressing relational information between variables whilst being both computationally efficient and simple to implement. Each element of the domain is a system of constraints where each constraint takes the restricted form $\pm x_i \pm x_j \leq d$. A key family of operations for the octagon domain are closure algorithms, which check satisfiability and provide a normal form for octagonal constraint systems. We present new quadratic incremental algorithms for closure, strong closure and integer closure and proofs of their correctness. We highlight the benefits and measure the performance of these new algorithms.