A Certifying Frontend for (Sub)polyhedral Abstract Domains

Abstract. Coq provides linear arithmetic tactics like omega or lia. Currently, these tactics either fully prove the goal in progress, or fail. We propose to improve this behavior: when the goal is not provable in linear arithmetic, we inject in hypotheses new equalities discovered from the linear inequalities. These equalities may help other Coq tactics to discharge the goal. In other words, we apply -in interactive proofsone of the seminal idea of SMT-solving: combining tactics by exchanging equalities. The paper describes how we have implemented equality learning in a new Coq tactic, dealing with linear arithmetic over rationals. It also illustrates how this tactic interacts with other Coq tactics.

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“…The tactic uses the linear constraints defined in the VPL [11], that we recall here. Type var is the type of variables in polyhedra.…”

Section: Certified Farkas Operations On Linear Constraintsmentioning

confidence: 99%

“…The previous Coq frontend of the VPL [11] would also allow to perform such proofs by reflection. Here, we believe than the HOAS approach followed in Section 4.3 is much simpler and more efficient than this previous implementation (where substitutions were very inefficiently encoded with lists of constraints).…”

Section: Conclusion and Related Workmentioning

confidence: 99%

A Coq Tactic for Equality Learning in Linear Arithmetic

Boulmé

Maréchal

2018

Interactive Theorem Proving

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“…This explains the reputation of polyhedra as unwieldy except in very low dimension, and motivated the design of the Verimag Verified Polyhedra Library (VPL) that operates on constraints-only representations [8,10]. An advantage of that approach is that it is easy to log enough information to independently check that the computed polyhedron includes the exact polyhedron that should be computed, which suffices for proving that static analysis is sound [9,10]; the certificate checker was implemented and proved correct in COQ. 4 The consequence is that many operations of the VPL, such as assignment, convex hull or Minkowski sum, were encoded as projection, finally performed by Fourier-Motzkin elimination [2].…”

Section: The Challenge Of Verification Using Polyhedramentioning

confidence: 99%

“…We took a step further and developed a generic PLP-solver exploiting insights by [18,17]. Our solver, implemented in OCAML, works over rationals and generates COQ-certificates of correctness of its computations, similar to those in VPL [8,9,10].…”

Section: Related Workmentioning

confidence: 99%

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Scalable Minimizing-Operators on Polyhedra via Parametric Linear Programming

2017

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Convex polyhedra capture linear relations between variables. They are used in static analysis and optimizing compilation. Their high expressiveness is however barely used in verification because of their cost, often prohibitive as the number of variables involved increases. Our goal in this article is to lower this cost. Whatever the chosen representation of polyhedra -as constraints, as generators or as bothexpensive operations are unavoidable. That cost is mostly due to four operations: conversion between representations, based on Chernikova's algorithm, for libraries in double description; convex hull, projection and minimization, in the constraintsonly representation of polyhedra. Libraries operating over generators incur exponential costs on cases common in program analysis. In the Verimag Polyhedra Library this cost was avoided by a constraints-only representation and reducing all operations to variable projection, classically done by Fourier-Motzkin elimination. Since Fourier-Motzkin generates many redundant constraints, minimization was however very expensive. In this article, we avoid this pitfall by expressing projection as a parametric linear programming problem. This dramatically improves efficiency, mainly because it avoids the post-processing minimization. We show how our new approach can be up to orders of magnitude faster than the previous approach implemented in the Verimag Polyhedra Library that uses only constraints and Fourier-Motzkin elimination, and on par with the conventional double description approach, as implemented in well-known libraries.

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

Coti

Monniaux

2020

Concurrency and Computation

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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...

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A Certifying Frontend for (Sub)polyhedral Abstract Domains

Cited by 18 publications

References 16 publications

A Coq Tactic for Equality Learning in Linear Arithmetic

A Coq Tactic for Equality Learning in Linear Arithmetic

Scalable Minimizing-Operators on Polyhedra via Parametric Linear Programming

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

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