Efficiently and Effectively Recognizing Toricity of Steady State Varieties

We consider the problem of testing whether the points in a complex or real variety with non-zero coordinates form a multiplicative group or, more generally, a coset of a multiplicative group. For the coset case, we study the notion of shifted toric varieties which generalizes the notion of toric varieties. This requires a geometric view on the varieties rather than an algebraic view on the ideals. We present algorithms and computations on 129 models from the BioModels repository testing for group and coset structures over both the complex numbers and the real numbers. Our methods over the complex numbers are based on Gr\"obner basis techniques and binomiality tests. Over the real numbers we use first-order characterizations and employ real quantifier elimination. In combination with suitable prime decompositions and restrictions to subspaces it turns out that almost all models show coset structure. Beyond our practical computations, we give upper bounds on the asymptotic worst-case complexity of the corresponding problems by proposing single exponential algorithms that test complex or real varieties for toricity or shifted toricity. In the positive case, these algorithms produce generating binomials. In addition, we propose an asymptotically fast algorithm for testing membership in a binomial variety over the algebraic closure of the rational numbers

First-Order Tests for Toricity

Motivated by problems arising with the symbolic analysis of steady state ideals in Chemical Reaction Network Theory, we consider the problem of testing whether the points in a complex or real variety with non-zero coordinates form a coset of a multiplicative group. That property corresponds to Shifted Toricity, a recent generalization of toricity of the corresponding polynomial ideal. The key idea is to take a geometric view on varieties rather than an algebraic view on ideals. Recently, corresponding coset tests have been proposed for complex and for real varieties. The former combine numerous techniques from commutative algorithmic algebra with Gr\"obner bases as the central algorithmic tool. The latter are based on interpreted first-order logic in real closed fields with real quantifier elimination techniques on the algorithmic side. Here we take a new logic approach to both theories, complex and real, and beyond. Besides alternative algorithms, our approach provides a unified view on theories of fields and helps to understand the relevance and interconnection of the rich existing literature in the area, which has been focusing on complex numbers, while from a scientific point of view the (positive) real numbers are clearly the relevant domain in chemical reaction network theory. We apply prototypical implementations of our new approach to a set of 129 models from the BioModels repository

First-Order Tests for Toricity

Motivated by problems arising with the symbolic analysis of steady state ideals in Chemical Reaction Network Theory, we consider the problem of testing whether the points in a complex or real variety with non-zero coordinates form a coset of a multiplicative group. That property corresponds to Shifted Toricity, a recent generalization of toricity of the corresponding polynomial ideal. The key idea is to take a geometric view on varieties rather than an algebraic view on ideals. Recently, corresponding coset tests have been proposed for complex and for real varieties. The former combine numerous techniques from commutative algorithmic algebra with Gr\"obner bases as the central algorithmic tool. The latter are based on interpreted first-order logic in real closed fields with real quantifier elimination techniques on the algorithmic side. Here we take a new logic approach to both theories, complex and real, and beyond. Besides alternative algorithms, our approach provides a unified view on theories of fields and helps to understand the relevance and interconnection of the rich existing literature in the area, which has been focusing on complex numbers, while from a scientific point of view the (positive) real numbers are clearly the relevant domain in chemical reaction network theory. We apply prototypical implementations of our new approach to a set of 129 models from the BioModels repository

On the Complexity of Solving Zero-Dimensional Polynomial Systems via Projection

Given a zero-dimensional polynomial system consisting of n integer polynomials in n variables, we propose a certified and complete method to compute all complex solutions of the system as well as a corresponding separating linear form l with coefficients of small bit size. For computing l, we need to project the solutions into one dimension along O(n) distinct directions but no further algebraic manipulations. The solutions are then directly reconstructed from the considered projections. The first step is deterministic, whereas the second step uses randomization, thus being Las-Vegas. The theoretical analysis of our approach shows that the overall cost for the two problems considered above is dominated by the cost of carrying out the projections. We also give bounds on the bit complexity of our algorithms that are exclusively stated in terms of the number of variables, the total degree and the bitsize of the input polynomials

A Poly-algorithmic Approach to Quantifier Elimination

Cylindrical Algebraic Decomposition (CAD) was the first practical means for doing real quantifier elimination (QE), and is still a major method, with many improvements since Collins' original method. Nevertheless, its complexity is inherently doubly exponential in the number of variables. Where applicable, virtual term substitution (VTS) is more effective, turning a QE problem in $n$ variables to one in $n-1$ variables in one application, and so on. Hence there is scope for hybrid methods: doing VTS where possible then using CAD. This paper describes such a poly-algorithmic implementation, based on the second author's Ph.D. thesis. The version of CAD used is based on a new implementation of Lazard's recently-justified method, with some improvements to handle equational constraints

Algorithmic Reduction of Biological Networks With Multiple Time Scales

We present a symbolic algorithmic approach that allows to compute invariant manifolds and corresponding reduced systems for differential equations modeling biological networks which comprise chemical reaction networks for cellular biochemistry, and compartmental models for pharmacology, epidemiology and ecology. Multiple time scales of a given network are obtained by scaling, based on tropical geometry. Our reduction is mathematically justified within a singular perturbation setting using a recent result by Cardin and Teixeira. The existence of invariant manifolds is subject to hyperbolicity conditions, which we test algorithmically using Hurwitz criteria. We finally obtain a sequence of nested invariant manifolds and respective reduced systems on those manifolds. Our theoretical results are generally accompanied by rigorous algorithmic descriptions suitable for direct implementation based on existing off-the-shelf software systems, specifically symbolic computation libraries and Satisfiability Modulo Theories solvers. We present computational examples taken from the well-known BioModels database using our own prototypical implementations

Parametric Toricity of Steady State Varieties of Reaction Networks

We study real steady state varieties of the dynamics of chemical reaction networks. The dynamics are derived using mass action kinetics with parametric reaction rates. The models studied are not inherently parametric in nature. Rather, our interest in parameters is motivated by parameter uncertainty, as reaction rates are typically either measured with limited precision or estimated. We aim at detecting toricity and shifted toricity, using a framework that has been recently introduced and studied for the non-parametric case over both the real and the complex numbers. While toricity requires that the variety specifies a subgroup of the direct power of the multiplicative group of the underlying field, shifted toricity requires only a coset. In the non-parametric case these requirements establish real decision problems. In the presence of parameters we must go further and derive necessary and sufficient conditions in the parameters for toricity or shifted toricity to hold. Technically, we use real quantifier elimination methods. Our computations on biological networks here once more confirm shifted toricity as a relevant concept, while toricity holds only for degenerate parameter choices.Comment: Computations available as ancillary file

A CDCL-style calculus for solving non-linear constraints

In this paper we propose a novel approach for checking satisfiability of non-linear constraints over the reals, called ksmt. The procedure is based on conflict resolution in CDCL style calculus, using a composition of symbolical and numerical methods. To deal with the non-linear components in case of conflicts we use numerically constructed restricted linearisations. This approach covers a large number of computable non-linear real functions such as polynomials, rational or trigonometrical functions and beyond. A prototypical implementation has been evaluated on several non-linear SMT-LIB examples and the results have been compared with state-of-the-art SMT solvers.Comment: 17 pages, 3 figures; accepted at FroCoS 2019; software available at <http://informatik.uni-trier.de/~brausse/ksmt/

An introduction to interval-based constraint processing.

Constraint programming is often associated with solving problems over finite domains. Many applications in engineering, CAD and design, however, require solving problems over continuous (real-valued) domains. While simple constraint solvers can solve linear constraints with the inaccuracy of floating-point arithmetic, methods based on interval arithmetic allow exact (interval) solutions over a much wider range of problems. Applications of interval-based programming extend the range of solvable problems from non-linear polynomials up to those involving ordinary differential equations. In this text, we give an introduction to current approaches, methods and implementations of interval-based constraint programming and solving. Special care is taken to provide a uniform and consistent notation, since the literature in this field employs many seemingly different, but yet conceptually related, notations and terminology