Exact Join Detection for Convex Polyhedra and Other Numerical Abstractions

TL;DR

This paper investigates the problem of join detection across various numerical abstractions, providing necessary and sufficient conditions, improved algorithms for convex polyhedra, and new algorithms for other abstractions, all implemented and validated.

Contribution

It introduces a unified abstract framework for join detection in numerical abstractions, improves algorithms for convex polyhedra, and develops new algorithms for other abstractions, with implementations in the Parma Polyhedra Library.

Findings

New necessary and sufficient conditions for join detection in numerical abstractions.

An improved algorithm for convex polyhedra with better worst-case complexity.

New algorithms for other numerical abstractions with experimental validation.

Abstract

Deciding whether the union of two convex polyhedra is itself a convex polyhedron is a basic problem in polyhedral computations; having important applications in the field of constrained control and in the synthesis, analysis, verification and optimization of hardware and software systems. In such application fields though, general convex polyhedra are just one among many, so-called, numerical abstractions, which range from restricted families of (not necessarily closed) convex polyhedra to non-convex geometrical objects. We thus tackle the problem from an abstract point of view: for a wide range of numerical abstractions that can be modeled as bounded join-semilattices --that is, partial orders where any finite set of elements has a least upper bound--, we show necessary and sufficient conditions for the equivalence between the lattice-theoretic join and the set-theoretic union. For the case of closed convex polyhedra --which, as far as we know, is the only one already studied in the literature-- we improve upon the state-of-the-art by providing a new algorithm with a better worst-case complexity. The results and algorithms presented for the other numerical abstractions are new to this paper. All the algorithms have been implemented, experimentally validated, and made available in the Parma Polyhedra Library.