Formulating problems for real algebraic geometry

We discuss issues of problem formulation for algorithms in real algebraic geometry, focussing on quantifier elimination by cylindrical algebraic decomposition. We recall how the variable ordering used can have a profound effect on both performance and output and summarise what may be done to assist with this choice. We then survey other questions of problem formulation and algorithm optimisation that have become pertinent following advances in CAD theory, including both work that is already published and work that is currently underway. With implementations now in reach of real world applications and new theory meaning algorithms are far more sensitive to the input, our thesis is that intelligently formulating problems for algorithms, and indeed choosing the correct algorithm variant for a problem, is key to improving the practical use of both quantifier elimination and symbolic real algebraic geometry in general.Comment: To be presented at The "Encuentros de \'Algebra Computacional y Aplicaciones, EACA 2014" (Meetings on Computer Algebra and Applications) in Barcelon

Розробка алгоритму з покращеною релевантністю локалізації координат вектора для інтелектуальних сенсорів

There are sensors of vector quantities whose field characteristics are described by the equations of quadrics. These sensors have improved sensitivity and smaller dimensions, the "payment" for which is, in fact, the non-linearity of field characteristics. In order to use such sensors, one has to solve the system of three quadric equations. Given the labor-intensity of this process, the sensors are designed intelligent – a finished device includes microcontrollers or other units that are able to process results of measurements. These devices are characterized by "curtailed" software and hardware capacities that necessitate the development of algorithms and their implementations with regard to these constraints.Classic algorithms for solving the systems of polynomial equations are not appropriate because of their violating the requirements to minimal resource consumption. The search for solutions of the systems of quadric equations is carried out in two stages – first, numerical fields that potentially contain intersections are localized, and then in these regions the search is conducted for accurate solutions by numerical methods. The success of using numerical methods depends on the quality of the conducted localization of solutions. The means of localization are not sufficiently worked out. Earlier, an algorithm was developed using interval arithmetic, the implementation of which by a microcontroller of the ARM Cortex-M4 architecture proved its capacity to find all, without exception, regions with the sought solutions of a system of equations, however, in addition to the "useful" regions, this algorithm finds as well the regions that do not actually contain the solutions. This fact led to unnecessary waste of time trying to find exact solutions in the regions where they do not exist.Thus, there was a need to search for alternative ways to the localization of solutions for the systems of quadrics equations. It is natural to search for these methods, based on the properties of quadrics in particular and continuous differentiable functions in general. A foundation of the proposed algorithm is the fact that a function in a closed region acquires its maximum or minimum value either at the borders or in critical points. If we accept, as a closed region, a rectangular parallelepiped, then its boundaries are its six sides, the boundaries of sides are its edges, and the boundaries of edges are the tops of parallelepiped. On the sides of the parallelepiped, function of three variables is reduced to function of two variables, on the edges – to functions of one variable. In the case of quadrics, finding the critical points of function on the edges comes down to solving a linear equation, and the critical points of function on the sides – to solving a system of two linear equations with two unknowns. Therefore, it is sufficient to check the signs of function at the tops of rectangular parallelepiped and those critical points of function that belong to the examined region. If in all these points the signs of function are the same, then there is no any point inside where function takes the 0 value. Thus, checking the signs of all functions that represent the left parts of the quadrics equation allows us to "reject" the regions, where there possible may not be any points of intersection. Instead of remembering the values of functions (valid numbers), it is sufficient to keep the signs of functions (one bit), which provides for the less consumption of operative memory. The tests proved that the proposed new algorithm is applicable for the implementation in the micro programming software, thus providing for a higher relevance of the found regions in comparison with the algorithm-analogue. An increase in relevancy is explained by the fact that interval arithmetic always implies overstated evaluations because, as the lower boundaries of intervals, the minimum permissible values are accepted, and as the upper limits – maximum permissible values. Checking the signs of functions in the selected points is free from the revaluation of results. The new algorithm somewhat deteriorated performance of permanent memory and execution time, however, these costs are compensated for by the further search for the solutions for a smaller number of irrelevant regions.Исследована проблема недостаточного развития алгоритмов, применимых для реализации в микропрограммном обеспечении для поиска точек пересечения квадрик. Разработан, реализован и изучен алгоритм локализации точек пересечения квадрик на основе свойств непрерывных дифференцируемых функций в замкнутой области. Новым алгоритмом достигается релевантность результатов выше, чем у его единственного аналога. Результаты предназначены для интеллектуальных сенсоровДосліджено проблему недостатнього розвитку алгоритмів, реалізованих у мікропрограмному забезпеченні для пошуку точок перетину квадрик. Розроблено, реалізовано та досліджено алгоритм локалізації точок перетину квадрик на основі властивостей неперервних диференційовних функцій у замкнутій області. Новим алгоритмом досягається вища релевантність результатів, ніж його єдиним аналогом. Результати призначені для інтелектуальних сенсорів векторних величи

Truth table invariant cylindrical algebraic decomposition

When using cylindrical algebraic decomposition (CAD) to solve a problem with respect to a set of polynomials, it is likely not the signs of those polynomials that are of paramount importance but rather the truth values of certain quantifier free formulae involving them. This observation motivates our article and definition of a Truth Table Invariant CAD (TTICAD). In ISSAC 2013 the current authors presented an algorithm that can efficiently and directly construct a TTICAD for a list of formulae in which each has an equational constraint. This was achieved by generalising McCallum's theory of reduced projection operators. In this paper we present an extended version of our theory which can be applied to an arbitrary list of formulae, achieving savings if at least one has an equational constraint. We also explain how the theory of reduced projection operators can allow for further improvements to the lifting phase of CAD algorithms, even in the context of a single equational constraint. The algorithm is implemented fully in Maple and we present both promising results from experimentation and a complexity analysis showing the benefits of our contributions.Comment: 40 page

Using Machine Learning to Improve Cylindrical Algebraic Decomposition

Cylindrical Algebraic Decomposition (CAD) is a key tool in computational algebraic geometry, best known as a procedure to enable Quantifier Elimination over real-closed fields. However, it has a worst case complexity doubly exponential in the size of the input, which is often encountered in practice. It has been observed that for many problems a change in algorithm settings or problem formulation can cause huge differences in runtime costs, changing problem instances from intractable to easy. A number of heuristics have been developed to help with such choices, but the complicated nature of the geometric relationships involved means these are imperfect and can sometimes make poor choices. We investigate the use of machine learning (specifically support vector machines) to make such choices instead. Machine learning is the process of fitting a computer model to a complex function based on properties learned from measured data. In this paper we apply it in two case studies: the first to select between heuristics for choosing a CAD variable ordering; the second to identify when a CAD problem instance would benefit from Groebner Basis preconditioning. These appear to be the first such applications of machine learning to Symbolic Computation. We demonstrate in both cases that the machine learned choice outperforms human developed heuristics.This work was supported by EPSRC grant EP/J003247/1; the European Union’s Horizon 2020 research and innovation programme under grant agreement No 712689 (SC2); and the China Scholarship Council (CSC)