Cylindrical Decomposition Research Articles

Lessons on Datasets and Paradigms in Machine Learning for Symbolic Computation: A Case Study on CAD

Symbolic Computation algorithms and their implementation in computer algebra systems often contain choices which do not affect the correctness of the output but can significantly impact the resources required: such choices can benefit from having them made separately for each problem via a machine learning model. This study reports lessons on such use of machine learning in symbolic computation, in particular on the importance of analysing datasets prior to machine learning and on the different machine learning paradigms that may be utilised. We present results for a particular case study, the selection of variable ordering for cylindrical algebraic decomposition, but expect that the lessons learned are applicable to other decisions in symbolic computation. We utilise an existing dataset of examples derived from applications which was found to be imbalanced with respect to the variable ordering decision. We introduce an augmentation technique for polynomial systems problems that allows us to balance and further augment the dataset, improving the machine learning results by 28% and 38% on average, respectively. We then demonstrate how the existing machine learning methodology used for the problem—classification—might be recast into the regression paradigm. While this does not have a radical change on the performance, it does widen the scope in which the methodology can be applied to make choices.

Dynamics of a Cournot game with bounded rational firms and various scale effects.

In this paper, we focus on a dynamic Cournot game in the market with a nonlinear (isoelastic) demand function. In our model, there are N competing firms featured by nonlinear cost functions, which enhances our model's resemblance to real-world scenarios. Firstly, we focus on the homogeneous case where firms' marginal costs change at equal rates. We establish analytical expressions of the market supply at equilibrium and perform comparative static analysis. In addition, we investigate the local stability under different economies of scale and show that there could be multiple stable equilibria if firms face economies of scale. The heterogeneous case where firms' marginal costs change at distinct rates is much more complex, thus we investigate the duopoly game with only two firms involved. Methods of symbolic computations such as triangular decomposition and partial cylindrical algebraic decomposition are employed in the analytical investigations of the equilibrium, which is nearly impossible by using the pencil-and-paper approach since the closed-form equilibrium is quite complicated. According to the computational results, we derive that two stable positive equilibria may coexist if both firms face economies of scale. Additionally, we conduct preliminary numerical simulations and find two different types of complex dynamics of the model considered in this paper: complex trajectories such as periodic and chaotic orbits may appear; the topological structure of the basins of attraction may be complex.

Levelwise construction of a single cylindrical algebraic cell

Satisfiability modulo theories (SMT) solvers check the satisfiability of quantifier-free first-order logic formulae over different theories. We consider the theory of non-linear real arithmetic where the formulae are logical combinations of polynomial constraints. Here a commonly used tool is the cylindrical algebraic decomposition (CAD) to decompose the real space into cells where the constraints are truth-invariant through the use of projection polynomials.A CAD encodes more information than necessary for checking satisfiability. One approach to address this is to repackage the CAD theory into a search-based algorithm: one that guesses sample points to satisfy the formula, and generalizes guesses that conflict constraints to cylindrical cells around samples which are avoided in the continuing search. This can lead to a satisfying assignment more quickly, or conclude unsatisfiability with far fewer cells. A notable example of this approach is Jovanović and de Moura's NLSAT algorithm. Since these cells are being produced locally to a sample there is scope to use fewer projection polynomials than the traditional CAD projection. The original NLSAT algorithm reduced the set a little; while Brown's single cell construction reduced it much further still. However, it refines a cell polynomial-by-polynomial, meaning the shape and size of the cell produced depends on the order in which the polynomials are considered.The present paper proposes a method to construct such cells levelwise, i.e. built level-by-level according to a variable ordering instead of polynomial-by-polynomial for all levels. We still use a reduced number of projection polynomials, but can now consider a variety of different reductions and use heuristics to select the projection polynomials in order to optimize the shape of the cell under construction. The new method can thus improve the performance of the NLSAT algorithm. We formulate all the necessary theory that underpins the algorithm as a proof system: while not a common presentation for work in this field, it is valuable in allowing an elegant decoupling of heuristic decisions from the main algorithm and its proof of correctness. We expect the symbolic computation community may find uses for it in other areas too. In particular, the proof system could be a step towards formal proofs for non-linear real arithmetic.This work has been implemented in the SMT-RAT solver and the benefits of the levelwise construction are validated experimentally on the SMT-LIB benchmark library. We also compare several heuristics for the construction and observe that each heuristic has strengths offering potential for further exploitation of the new approach.

Explainable AI Insights for Symbolic Computation: A case study on selecting the variable ordering for cylindrical algebraic decomposition

In recent years there has been increased use of machine learning (ML) techniques within mathematics, including symbolic computation where it may be applied safely to optimise or select algorithms. This paper explores whether using explainable AI (XAI) techniques on such ML models can offer new insight for symbolic computation, inspiring new implementations within computer algebra systems that do not directly call upon AI tools. We present a case study on the use of ML to select the variable ordering for cylindrical algebraic decomposition. It has already been demonstrated that ML can make the choice well, but here we show how the SHAP tool for explainability can be used to inform new heuristics of a size and complexity similar to those human-designed heuristics currently commonly used in symbolic computation.

Clustering in the Lazard method for Cylindrical Algebraic Decomposition

Cylindrical Algebraic Decomposition (CAD) is an algorithm originally proposed by Collins in [4]. Given a set of multivariate polynomials, CAD decomposes the multidimensional real space into connected subsets called cells, within which those polynomials are sign-invariant.

Faithful Real-Time Animation of Parametrized (Semi-) Algebraic Expressions via Cylindrical Algebraic Decomposition

We present some current improvements, implemented in the software package GeoGebra Discovery, that combine symbolic computation and graphics algorithms to faithfully visualize (semi-)algebraic expressions. Our implementation allows fluid animation of set of (semi-)algebraic sets of dimension 1 in a desktop application or a web browser. We use the Tarski library to create a cylindrical algebraic decomposition of the input, and its plot2d command, which is processed further in GeoGebra Discovery, to provide the user with a familiar look and feel.

Choosing better variable orderings for cylindrical algebraic decomposition via exploiting chordal structure

As is well-known, the choice of variable ordering while computing cylindrical algebraic decomposition (CAD) has a great effect on the time and memory use of the computation as well as the number of sample points computed. In this paper, we indicate that typical CAD algorithms, if executed with respect to a special kind of variable orderings (called “the perfect elimination orderings”, PEO), naturally preserve chordality, which is well compatible with an important (variable) sparsity pattern called “the correlative sparsity”. If the associated graph of the polynomial system in question is chordal (resp., is nearly chordal), then a PEO of the associated graph (resp., of a minimal chordal completion of the associated graph) can be a better variable ordering for the CAD computation than other naive variable orderings in the sense that it results in a much smaller full set of projection polynomials and thus more efficient computation. A new (m,d)-property of the full set of CAD projection polynomials obtained via a PEO is given, which indicates that when the corresponding perfect elimination tree has a lower height, the full set of projection polynomials also tends to have a smaller “size”. Furthermore, combining the lower-tree-height rule with Brown's heuristics, a new procedure is proposed to choose better PEOs for CAD computation.

Polynomial superlevel set representation of the multistationarity region of chemical reaction networks

In this paper we introduce a new representation for the multistationarity region of a reaction network, using polynomial superlevel sets. The advantages of using this polynomial superlevel set representation over the already existing representations (cylindrical algebraic decompositions, numeric sampling, rectangular divisions) is discussed, and algorithms to compute this new representation are provided. The results are given for the general mathematical formalism of a parametric system of equations and so may be applied to other application domains.

A General Framework for Secrecy Performance Analysis via Quantifier Elimination

The problems of finding accurate limits of integration in symbolic form are non-trivial when analyzing the secrecy outage probability (SOP) over complicated communication schemes. To identify these limits, we formulate an existential quantifier elimination (QE) problem based on the description of the secrecy outage events. After that, the resulting QE problem can be solved using QE algorithms such as cylindrical algebraic decomposition (CAD) via existing solvers. Since CAD is an exact algorithm, our approach does not require Monte-Carlo simulations to verify the correctness of the SOP integration limits. Finally, two complex SOP analysis scenarios for cognitive radio systems are presented to demonstrate the generality of the proposed method.

Bounds for Polynomials on Algebraic Numbers and Application to Curve Topology

Let $$P \in {\mathbb {Z}}[X, Y]$$ be a given square-free polynomial of total degree d with integer coefficients of bitsize less than $$\tau $$ , and let $$V_{{\mathbb {R}}} (P) := \{ (x,y) \in {\mathbb {R}}^2\mid P (x,y) = 0 \}$$ be the real planar algebraic curve implicitly defined as the vanishing set of P. We give a deterministic algorithm to compute the topology of $$V_{{\mathbb {R}}} (P)$$ in terms of a simple straight-line planar graph $${\mathcal {G}}$$ that is isotopic to $$V_{{\mathbb {R}}} (P)$$ . The upper bound on the bit complexity of our algorithm is in $${\tilde{O}}(d^5 \tau + d^6)$$ (The expression “the complexity is in $${\tilde{O}}(f(d,\tau ))$$ ” with f a polynomial in $$d,\tau $$ is an abbreviation for the expression “there exists a positive integer c such that the complexity is in $$O(({\log d }\log \tau )^c f(d,\tau ))$$ ”); which matches the current record bound for the problem of computing the topology of a planar algebraic curve. However, compared to existing algorithms with comparable complexity, our method does not consider any change of coordinates, and more importantly the returned simple planar graph $${\mathcal {G}}$$ yields the cylindrical algebraic decomposition information of the curve in the original coordinates. Our result is based on two main ingredients: First, we derive amortized quantitative bounds on the roots of polynomials with algebraic coefficients as well as adaptive methods for computing the roots of bivariate polynomial systems that actually exploit this amortization. The results we obtain are more general that the previous literature. Our second ingredient is a novel approach for the computation of the local topology of the curve in a neighborhood of all singular points.

Design space description through adaptive sampling and symbolic computation

AbstractDesign space definition is one of the key parts in pharmaceutical research and development. In this article, we propose a novel solution strategy to explicitly describe the design space without recourse decisions. First, to smooth the boundary, the Kreisselmeier–Steinhauser (KS) function is applied to aggregate all inequality constraints. Next, for creating a surrogate polynomial model of the KS function, we focus on finding sampling points on the boundary of KS space. After performing Latin hypercube sampling (LHS), two methods are presented to efficiently expand the boundary points, that is, line projection to the boundary through any two feasible LHS points and perturbation around the adaptive sampling points. Finally, a symbolic computation method, cylindrical algebraic decomposition, is applied to transform the surrogate model into a series of explicit and triangular subsystems, which can be converted to describe the KS space. Two case studies show the efficiency of the proposed algorithm.

A parameter space method for analyzing Hopf bifurcation of fractional-order nonlinear systems with multiple-parameter

Hopf bifurcation analysis of fractional-order nonlinear systems with multiple-parameter is discussed in this paper. The regions and boundaries corresponding to Hopf bifurcation conditions are described in parameter space. Based on cylindrical algebraic decomposition, the parameter space is decomposed into finite number of connected regions by some boundaries. Then parameter space method is proposed to determine stable parameter region and Hopf bifurcation parameter hypersurface. One example illustrates the effectiveness of the method.

Exotic Bifurcations in Three Connected Populations with Allee Effect

We consider three connected populations with strong Allee effect, and give a complete classification of the steady state structure of the system with respect to the Allee threshold and the dispersal rate, describing the bifurcations at each critical point where the number of steady states change. One may expect that by increasing the dispersal rate between the patches, the system would become more well-mixed, hence simpler. However, we show that it is not always the case, and the number of steady states may (temporarily) go up by increasing the dispersal rate. Besides sequences of pitchfork and saddle-node bifurcations, we find triple-transcritical bifurcations and also a sun-ray shaped bifurcation where twelve steady states meet at a single point then disappear. The major tool of our investigations is a novel algorithm that decomposes the parameter space with respect to the number of steady states and finds the bifurcation values using cylindrical algebraic decomposition with respect to the discriminant variety of the polynomial system.

Revisiting reachability in Polynomial Interrupt Timed Automata

Polynomial Interrupt Timed Automata (PolITA) are finite automata with clocks organized along hierarchical levels. These clocks are equipped with an interruption mechanism, well suited to the modeling of real-time operating systems. Moreover, transitions between states contain polynomial guards and updates. The reachability problem in this class is known to be in 2EXPTIME with a decision procedure based on the cylindrical algebraic decomposition. We improve this complexity to EXPSPACE mainly using a combinatorial argument and we include a reduction leading to a PSPACE lower bound.

The DEWCAD project

This abstract seeks to introduce the ISSAC community to the DEWCAD project, which is based at Coventry University and the University of Bath, in the United Kingdom. The project seeks to push back the Doubly Exponential Wall of Cylindrical Algebraic Decomposition, through the integration of SAT/SMT technology, the extension of Lazard projection theory, and the development of new algorithms based on CAD technology but without producing CADs themselves. The project also seeks to develop applications of CAD and will focus on applications in the domains of economics and bio-network analysis.

Reasoning Method between Polynomial Error Assertions

Error coefficients are ubiquitous in systems. In particular, errors in reasoning verification must be considered regarding safety-critical systems. We present a reasoning method that can be applied to systems described by the polynomial error assertion (PEA). The implication relationship between PEAs can be converted to an inclusion relationship between zero sets of PEAs; the PEAs are then transformed into first-order polynomial logic. Combined with the quantifier elimination method, based on cylindrical algebraic decomposition, the judgment of the inclusion relationship between zero sets of PEAs is transformed into judgment error parameters and specific error coefficient constraints, which can be obtained by the quantifier elimination method. The proposed reasoning method is validated by proving the related theorems. An example of intercepting target objects is provided, and the correctness of our method is tested through large-scale random cases. Compared with reasoning methods without error semantics, our reasoning method has the advantage of being able to deal with error parameters.

Projection-based robust optimization with symbolic computation

Robust counterpart reformulation is a common technique used to deal with data uncertainty in robust optimization (RO) problems. The derivation of the robust counterpart formulation using the duality theory is nontrivial, especially for complex uncertainty sets. To reduce the dependence on robust counterparts, a novel method is proposed in this article for RO problems. Based on the feasible space projection, the proposed method can locate robust solutions without formulating the robust counterparts. RO model can be reformulated as a semi-algebraic system and a modified cylindrical algebraic decomposition method is applied to project the high-dimensional feasible space on the low-dimensional space of the objective function and uncertainty parameters. By solving the maximization problem and using the max-max decision criterion, the final robust solution can be selected. The case studies, involving problems of nonlinear programming (NLP) and robust design optimization problems, show that the proposed method can obtain the robust solution effectively.

Analytical solution of volumetric flexibility through symbolic computation

Volumetric flexibility index is an indicator for describing the flexibility of a process design. However, there is no prior literature on how the theoretical value of the volumetric flexibility index can be obtained. In this work, we propose the first systematic method to accurately derive the analytical solution of the volumetric flexibility index through symbolic computation. First, based on the cylindrical algebraic decomposition method, the feasible space can be decomposed into finite valid cells with analytical expressions. Then, the integral operation is implemented to calculate the hypervolume for each cell. Finally, hypervolumes of all feasible cells are summed to determine the volumetric flexibility index. Case studies show that the proposed method can obtain the theoretical value of the volumetric flexibility index for nonconvex systems. In addition, the explicit information between the hypervolume and design variables can be derived, which can be further applied to hypervolume optimization problems under different designs.

On exact Reznick, Hilbert-Artin and Putinar's representations

We consider the problem of computing exact sums of squares (SOS) decompositions for certain classes of non-negative multivariate polynomials, relying on semidefinite programming (SDP) solvers.We provide a hybrid numeric-symbolic algorithm computing exact rational SOS decompositions with rational coefficients for polynomials lying in the interior of the SOS cone. The first step of this algorithm computes an approximate SOS decomposition for a perturbation of the input polynomial with an arbitrary-precision SDP solver. Next, an exact SOS decomposition is obtained thanks to the perturbation terms and a compensation phenomenon. We prove that bit complexity estimates on output size and runtime are both polynomial in the degree of the input polynomial and singly exponential in the number of variables. Next, we apply this algorithm to compute exact Reznick, Hilbert-Artin's representation and Putinar's representations respectively for positive definite forms and positive polynomials over basic compact semi-algebraic sets. We also report on practical experiments done with the implementation of these algorithms and existing alternatives such as the critical point method and cylindrical algebraic decomposition.

Deciding the consistency of non-linear real arithmetic constraints with a conflict driven search using cylindrical algebraic coverings

We present a new algorithm for determining the satisfiability of conjunctions of non-linear polynomial constraints over the reals, which can be used as a theory solver for satisfiability modulo theory (SMT) solving for non-linear real arithmetic. The algorithm is a variant of Cylindrical Algebraic Decomposition (CAD) adapted for satisfiability, where solution candidates (sample points) are constructed incrementally, either until a satisfying sample is found or sufficient samples have been sampled to conclude unsatisfiability. The choice of samples is guided by the input constraints and previous conflicts.The key idea behind our new approach is to start with a partial sample; demonstrate that it cannot be extended to a full sample; and from the reasons for that rule out a larger space around the partial sample, which build up incrementally into a cylindrical algebraic covering of the space. There are similarities with the incremental variant of CAD, the NLSAT method of Jovanović and de Moura, and the NuCAD algorithm of Brown; but we present worked examples and experimental results on a preliminary implementation to demonstrate the differences to these, and the benefits of the new approach.