Cylindrical algebraic decompositions for boolean combinations

There has been recent interest in the use of machine learning (ML) approaches within mathematical software to make choices that impact on the computing performance without affecting the mathematical correctness of the result. We address the problem of selecting the variable ordering for cylindrical algebraic decomposition (CAD), an important algorithm in Symbolic Computation. Prior work to apply ML on this problem implemented a Support Vector Machine (SVM) to select between three existing human-made heuristics, which did better than anyone heuristic alone. Here we extend this result by training ML models to select the variable ordering directly, and by trying out a wider variety of ML techniques. We experimented with the NLSAT dataset and the Regular Chains Library CAD function for Maple 2018. For each problem, the variable ordering leading to the shortest computing time was selected as the target class for ML. Features were generated from the polynomial input and used to train the following ML models: k-nearest neighbours (KNN) classifier, multi-layer perceptron (MLP), decision tree (DT) and SVM, as implemented in the Python scikit-learn package. We also compared these with the two leading human-made heuristics for the problem: the Brown heuristic and sotd. On this dataset all of the ML approaches outperformed the human-made heuristics, some by a large margin.

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“…7 Proportion of x1 occuring in polynomials. 8 Proportion of x2 occuring in polynomials. 9 Proportion of x0 occuring in monomials.…”

Section: Featuresmentioning

confidence: 99%

Comparing Machine Learning Models to Choose the Variable Ordering for Cylindrical Algebraic Decomposition

England

Florescu

2019

Lecture Notes in Computer Science

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“…A key improvement to CAD has been in the projection operator to reduce the number of projection polynomials computed (Hong, 1990;McCallum, 1998McCallum, , 1999bMcCallum, , 2001Brown, 2001;Bradford et al, 2013a;Han et al, 2014;Bradford et al, 2016).…”

Section: Projection Operatorsmentioning

confidence: 99%

“…We build on recent work by Bradford et al (2016) to measure the dominant term in bounds on the number of CAD cells produced. Numerous studies have shown this to be closely correlated to the computation time (Dolzmann et al, 2004;Bradford et al, 2013aBradford et al, , 2014. We assume CAD input with m polynomials of maximum degree d in any one of n variables.…”

Section: Complexity Analysis Of Cad With Ecmentioning

confidence: 99%

Cylindrical algebraic decomposition with equational constraints

England

Bradford

Davenport

2020

Journal of Symbolic Computation

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Cylindrical Algebraic Decomposition (CAD) has long been one of the most important algorithms within Symbolic Computation, as a tool to perform quantifier elimination in first order logic over the reals. More recently it is finding prominence in the Satisfiability Checking community as a tool to identify satisfying solutions of problems in nonlinear real arithmetic.The original algorithm produces decompositions according to the signs of polynomials, when what is usually required is a decomposition according to the truth of a formula containing those polynomials. One approach to achieve that coarser (but hopefully cheaper) decomposition is to reduce the polynomials identified in the CAD to reflect a logical structure which reduces the solution space dimension: the presence of Equational Constraints (ECs). This paper may act as a tutorial for the use of CAD with ECs: we describe all necessary background and the current state of the art. In particular, we present recent work on how McCallum's theory of reduced projection may be leveraged to make further savings in the lifting phase: both to the polynomials we lift with and the cells lifted over. We give a new complexity analysis to demonstrate that the double exponent in the worst case complexity bound for CAD reduces in line with the number of ECs. We show that the reduction can apply to both the number of polynomials produced and their degree.

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“…NuCAD and the model-based construction also provide a natural way to take strong advantage of the particulars of the input formula from which the decomposition is constructed. Much of what has been done in the way of CAD research has been based on trying to do exactly this: partial CAD, equational constraints [7,11], divide & conquer [14], and truth table invariant CAD [1], for example. This paper's contributions are the introduction of the Open NuCAD, a model-based algorithm for efficiently constructing NuCADs along with a proof of correctness, and empirical data demonstrating how much more efficiently NuCAD represents semi-algebraic sets than CAD.…”

Section: Illustrates Thismentioning

confidence: 99%

Open Non-uniform Cylindrical Algebraic Decompositions

Brown

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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This paper introduces the notion of an Open Non-uniform Cylindrical Algebraic Decomposition (NuCAD), and presents an efficient model-based algorithm for constructing an Open NuCAD from an input formula. Using a limited experimental implementation of the algorithm, we demonstrate the effectiveness of the approach. NuCAD generalizes Cylindrical Algebraic Decomposition (CAD) as defined by Collins in his seminal work from the early 1970s, and extended in concepts like Hong's partial CAD. A NuCAD, like a CAD, is a decomposition of R n into cylindrical cells. But unlike a CAD, the cells in a NuCAD need not be arranged cylindrically. It is in this sense that NuCADs are not uniformly cylindrical. However, NuCADs -like CADs -carry a treelike structure that relates different cells. It is a very different tree but, as with the CAD tree structure, it allows some operations to be performed efficiently, for example locating the containing cell for an arbitrary input point.

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Cylindrical algebraic decompositions for boolean combinations

Cited by 35 publications

References 24 publications

Comparing Machine Learning Models to Choose the Variable Ordering for Cylindrical Algebraic Decomposition

Comparing Machine Learning Models to Choose the Variable Ordering for Cylindrical Algebraic Decomposition

Cylindrical algebraic decomposition with equational constraints

Open Non-uniform Cylindrical Algebraic Decompositions

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