The Verified Polyhedron Library: an Overview

Boulmé, Sylvain; Marechaly, Alexandre; Monniaux, David; Périn, Michaël; Yu, Hang

doi:10.1109/synasc.2018.00014

Cited by 16 publications

(16 citation statements)

References 21 publications

(30 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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.…”

Section: Operations On Polyhedramentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Verified code generation for the polyhedral model

Courant

Leroy

2021

Proc. ACM Program. Lang.

View full text Add to dashboard Cite

The polyhedral model is a high-level intermediate representation for loop nests that supports elegantly a great many loop optimizations. In a compiler, after polyhedral loop optimizations have been performed, it is necessary and difficult to regenerate sequential or parallel loop nests before continuing compilation. This paper reports on the formalization and proof of semantic preservation of such a code generator that produces sequential code from a polyhedral representation. The formalization and proofs are mechanized using the Coq proof assistant.

show abstract

Section: Operations On Polyhedramentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Verified code generation for the polyhedral model

Courant

Leroy

2021

Proc. ACM Program. Lang.

View full text Add to dashboard Cite

show abstract

“…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].…”

Section: Related Workmentioning

confidence: 99%

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

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Parallel Parametric Linear Programming Solving, and Application to Polyhedral Computations

Coti

Monniaux

2019

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

show abstract

A task‐based approach to parallel parametric linear programming solving, and application to polyhedral computations

Coti

Monniaux

2020

Concurrency and Computation

Self Cite

View full text Add to dashboard Cite

Parametric linear programming is a central operation for polyhedral computations, as well as in certain control applications. Here, we propose a task-based scheme for parallelizing it, with quasi-linear speedup over large problems. This type of parallel applications is challenging, because several tasks might be computing the same region. In this article, we are presenting the algorithm itself with a parallel redundancy elimination algorithm, and conducting a thorough performance analysis. K E Y W O R D S parallel parametric linear programming, polyhedra 1 INTRODUCTION A convex polyhedron in dimension n is the solution set over Q n (or, equivalently, 1 R n) of a system of inequalities (with integer or rational coefficients). Since all the polyhedra we consider here are convex, we shall talk of polyhedra for short. There also exist integer polyhedra, defined over Z n , but they are very different in many respects; we shall not consider them here. 2 Computations over polyhedra arise in low dimension (n = 2 and n = 3) for modeling physical objects, but here we are interested in higher dimensions. Polyhedra in higher dimension are typically used to enclose the set of reachable states of systems whose state can be expressed, at least partially, as a vector of reals or rationals. For instance, one can study the flow of ordinary differential equations, or more generally the trajectories of hybrid systems, by enclosing the trajectories into a succession of convex polyhedra. If the internal state of a control system is expressed by a vector of n numeric variables, one may prove that this system never encounters a bad condition by exhibiting a polyhedron P such that the initial state belongs to P, no bad condition belongs to P, and all possible time steps of the control system leave P stable (i.e., it is not possible to execute a step starting in P and ending outside of P). One generally proves the correctness of programs by exhibiting inductive invariants-an inductive invariant for a loop is a set containing the precondition of the loop, stable by moving to the next loop iteration, and implying the desired postcondition. Since providing inductive invariants by hand is tedious, it is desirable to automate that process. One approach for doing so is abstract interpretation, where an ascending sequence of sets of states is computed until reaching a fixed point. One may, for instance, search such sets as products of intervals, an approach known as interval analysis, but the lack of relationships between the dimensions tends to severely limit the kind of properties that can be proved (e.g., one cannot have an invariant i < n where n is a parameter: this is neither an interval on i nor on n nor a combination thereof). Cousot and Halbwachs proposed searching for polyhedral inductive invariants. 1,2 The operations needed there are projection, more generally image by an affine transformation, convex hull, and inclusion (or equality) test, together with an extrapolation operator known as widening. 1 Whether emptiness tests, inclusio...

show abstract

The Verified Polyhedron Library: an Overview

Cited by 16 publications

References 21 publications

Verified code generation for the polyhedral model

Verified code generation for the polyhedral model

Parallel Parametric Linear Programming Solving, and Application to Polyhedral Computations

A task‐based approach to parallel parametric linear programming solving, and application to polyhedral computations

Contact Info

Product

Resources

About