The Verified Polyhedron Library: an Overview

Abstract

The Verified Polyhedra Library operates upon a constraint-only representation of convex polyhedra and provides all common operations (image, pre-image, projection, convex hull, widening, inclusion and equality tests.. .). Optionally, the soundness of the results is checked by a layer certified in Coq.

Verified code generation for the polyhedral model

“…We also need a number of operations on polyhedra: emptiness check, intersection, canonicalization, projection on the first dimensions, etc. To this end, we reused the Verified Polyhedron Library (VPL) [Boulmé et al 2018], a mixed Coq-OCaml library that uses a posteriori certification: operations over polyhedra are implemented efficiently in OCaml and produce Farkas-style certificates [Schrijver 1998, corollary 10.1a]; the results and their certificates are then checked by Coq functions that have been proved (once and for all) to be sound (if the checker succeeds, the result is correct) [Fouilhé and Boulmé 2014]. Since certificate checking may fail, most operations over polyhedra live in a monad that supports errors and also the possibility that the OCaml implementation is not pure (e.g.…”

“…As shown in Figure 1, the code generator comprises four passes: schedule elimination, replacing the input schedule by the identity schedule (section 4); decomposition into abstract loops expressed in the intermediate language PolyLoop (section 5); elimination of redundant constraints produced by the previous pass (section 6); and generation of concrete loops expressed in a simple sequential imperative language, Loop (section 7). We formalized the code generator, its source, intermediate for (int i = 0; i < n; i++ ) { for (int j = 0; j < n; j++ ) { C [i][j] = 0; for (int k = 0; k < n; k++ ) { C and target languages, and its correctness proofs using the Coq proof assistant and the VPL [Boulmé et al 2018] verified library of polyhedral operations (section 8). The Coq development is available at https://github.com/Ekdohibs/PolyGen.…”

Parallel Parametric Linear Programming Solving, and Application to Polyhedral Computations

“…Jones and Maciejowski [21] applied reverse search to parametric linear programming; they however warn that while they have better asymptotic complexity than other approaches, the constant hidden in the big-O notation is huge and they warn that their approach is likely to be interesting only on larger examples. In contrast, we base ourselves on an approach already used in a sequential library that is competitive with double description approaches even on problems in moderate dimension [5].…”

“…Why this curse of dimensionality? There exist multiple libraries for computing over polyhedra (NewPolka, 3 Parma Polyhedra Library, 4 PolyLib, 5 CDD. .…”

“…We instead turned to parametric linear programming. Image, projection, convex hull can all be formulated as solutions to linear programs where parameters occur linearly within the objective [25,5]: given a system of equalities and inequalities, defining a convex polyhedron P in higher dimension, and a parametric bilinear objective function f (x, λ) where x is the point and λ a vector of parameters, give for each λ a vertex x * of P such that f (x * , λ) is minimal. A solution to such a program is a quasi-partition of the space of parameters λ into convex polyhedra, with one optimum associated to each polyhedron.…”