Some ways to reduce the space dimension in polyhedra computations

Halbwachs, Nicolas; Merchat, David; Gonnord, Laure

doi:10.1007/s10703-006-0013-2

Cited by 40 publications

(30 citation statements)

References 20 publications

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“…Despite their expressiveness and 40 years of research, polyhedra are little used in verification because operations on polyhedra are still costly and do not scale to large programs [13]. Usually, they are restricted to a small subset of program variables such as loop indices [14] -including more variables would mean skyrocketing costs.…”

Section: The Challenge Of Verification Using Polyhedramentioning

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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Section: The Challenge Of Verification Using Polyhedramentioning

confidence: 99%

“…The explosive nature of the generator representation motivated approaches that detect when a polyhedron is a Cartesian product of polyhedra and compute generator representations separately for each element of the product, thereby avoiding exponential blowup in the case of the hypercube [13,31].…”

Section: Related Workmentioning

confidence: 99%

Scalable Minimizing-Operators on Polyhedra via Parametric Linear Programming

2017

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“…Noticeable eorts have been put both to reduce the loss of precision due to the upper bound operation, and to accelerate the convergence of the Kleene iterative algorithm [15,22,21,4], but they do not track disjunctive information.…”

Section: Related Workmentioning

confidence: 99%

The Domain of Parametric Hypercubes for Static Analysis of Computer Games Software

Costantini

Ferrara

Maggiore

et al. 2013

Lecture Notes in Computer Science

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Abstract. Computer Games Software deeply relies on physics simulations, which are particularly demanding to analyze because they manipulate a large amount of interleaving oating point variables. Therefore, this application domain is an interesting workbench to stress the trade-o between accuracy and eciency of abstract domains for static analysis. In this paper, we introduce Parametric Hypercubes, a novel disjunctive non-relational abstract domain. Its main features are: (i) it combines the low computational cost of operations on (selected) multidimensional intervals with the accuracy provided by lifting to a power-set disjunctive domain, (ii) the compact representation of its elements allows to limit the space complexity of the analysis, and (iii) the parametric nature of the domain provides a way to tune the accuracy/eciency of the analysis by just setting the widths of the hypercubes sides. The rst experimental results on a representative Computer Games case study outline both the eciency and the precision of the proposal.

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“…Calculating a new set of Cutting Planes will refine the rational vertex k, 1 2 to k − 1, 1 2 . Thus k − 1 more steps are necessary to obtain a Z-polyhedron [31] where linear independence is detected between sets of constraints and exploited by applying meet and join on these smaller sets. The Octagon domain was generalised into the Octahedron domain [17], allowing more than two variables with zero or unary coefficients whilst maintaining a hull operation that is polynomial in the number of variables.…”

Section: Related Workmentioning

confidence: 99%

The two variable per inequality abstract domain

Simon

King

2010

Higher-Order Symb Comput

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Abstract. This article presents the Two Variable Per Inequality abstract domain (TVPI domain for short). This so-called weakly-relational domain is able to express systems of linear inequalities where each inequality has at most two variables. The domain represents a sweet-point in the performance-cost tradeoff between the faster Octagon domain and the more expressive domain of general convex polyhedra. In particular, we detail techniques to closely approximate integral TVPI systems, thereby finessing the problem of excessively growing coefficients, yielding -to our knowledge -the only relational domain that combines linear relations with arbitrary coefficients and strongly polynomial performance. Keywords: polyhedral analysis, integer programming, abstract interpretationStatic analysis methods have evolved from inferring prerequisite invariants for compiler optimisations to tools in their own right that are able to prove the absence of run-time errors of software. The abstract interpretation framework [21] provides a way of constructing and justifying program analyses. The key idea is to simulate each operation in a program with an abstract analog that operates on a description of the program state, rather than the state itself. The descriptions are chosen to trace a property of interest, for instance, an interval that encloses all possible values of a variable would be of value when reasoning about array bounds. These descriptions constitute what is known as an abstract domain. An abstract domain is usually presented as a lattice D, , , and the analysis itself is formulated as a set of recursive equations over D. The recursive equations are solved iteratively, until a fixpoint is reached which is detected using the ordering predicate . Each equation expresses how a program statement transforms the state at one program point to the state at another. Equations may be recursive because of loops in the program. The meet and join † This paper is a revised extract of the first author's PhD thesis [65], which, in turn, extends [72].

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Some ways to reduce the space dimension in polyhedra computations

Cited by 40 publications

References 20 publications

Scalable Minimizing-Operators on Polyhedra via Parametric Linear Programming

Scalable Minimizing-Operators on Polyhedra via Parametric Linear Programming

The Domain of Parametric Hypercubes for Static Analysis of Computer Games Software

The two variable per inequality abstract domain

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