Extending the Polyhedron Model to Inequality Systems with Non-linear Parameters using Quantifier Elimination

نویسندگان

  • Armin Größlinger
  • Martin Griebl
  • Christian Lengauer
  • Thomas Wondrak
چکیده

The polyhedron model has proved to be a useful tool in studying methods for the automatic parallelization of loop nests. Most of the mathematical tools developed for the polyhedron model require the coefficients of variables to be constants. This restriction has turned out to be a severe limitation for several recent developments in the polyhedron model. We show how the polyhedron model can be generalized to allow non-linear parameters, i.e., parameters appearing in the coefficients of variables. The mathematical method we use for this generalization is quantifier elimination in the real numbers. We demonstrate how existing algorithms can be generalized by the use of decision methods and give examples of new algorithms which use quantifier elimination directly to solve common problems in the polyhedron model.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

The challenges of non-linear parameters and variables in automatic loop parallelisation

With the rise of manycore processors, parallelism is becoming a mainstream necessity. Unfortunately, parallel programming is inherently more difficult than sequential programming; therefore, techniques for automatic parallelisation will become indispensable. We aim at extending the well-known polyhedron model, which promises this automation, beyond some of its current restrictions. Up to now, l...

متن کامل

On Algorithmic and Heuristic Approaches to Integral Problems in the Polyhedron Model with Non-linear Parameters

The polyhedron model provides one possible approach to model-based program analysis and transformation. Over the years, it has undergone many improvements in order to handle a still growing class of programs. A recent approach enabled the use of non-linear parameters, which was in that generality not possible before. While this new framework uses real quantifier elimination, the present work ex...

متن کامل

Introducing Non-linear Parameters to the Polyhedron Model

A mathematical model based on polyhedra (the so-called “polyhedron model”) serves as a foundation for model based loop program transformation like automatic parallelization. One of the restrictions present in the current polyhedron model is the requirement that the coefficients of variables must be numeric constants. This has been hindering some recent developments which require parametric coef...

متن کامل

A lower bound for the complexity of linear optimization from a quantifier-elimination point of view

We analyze the arithmetic complexity of the feasibility problem in linear optimization theory as a quantifier-elimination problem. For the case of polyhedra defined by 2n halfspaces in R we prove that, if dense representation is used to code polynomials, any quantifier-free formula expressing the set of parameters describing nonempty polyhedra has size Ω(4).

متن کامل

Extending Quantifier Elimination to Linear Inequalities on Bit-Vectors

We present an algorithm for existentially quantifying variables from conjunctions of linear modular equalities (LMEs), disequalities (LMDs) and inequalities (LMIs). We use sound but relatively less complete and cheaper heuristics first, and expensive but more complete techniques are used only when required. Our experiments demonstrate that our algorithm outperforms alternative quantifier elimin...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2009