First-Order Tests for Toricity

Abstract

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 Gröbner 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.

Testing Binomiality of Chemical Reaction Networks Using Comprehensive Gröbner Systems

“…We have given efficient algorithms for testing toricity in [27]. We also have considered the binomiality from a first-order logic point of view and gave efficient computational results and studied biomodels systematically via quantifier elimination [44,27]. Other than those, we have considered the concept of unconditional binomiality, which considers rate constants as variables, and gave polynomial time linear algebra and graph theoretical approaches for detecting binomiality [42,43].…”

“…In [27], the two main tools from computer algebra,, quantifier elimination [12,26,50] and Gröbner bases [5,6,18,19] are used. Also recently, the authors introduced a first order logic test for toricity [44]. An efficient linear algebra method for testing unconditional binomiality has been presented in [43] and a graph-theoretical equivalent of the method is given in [42].…”

“…However, the fact that specialisation of the rate constants may affect the computations has not beed considered. The authors' previous work on toricity [27,44] considers the toricity problem when the parameters have already been assigned real positive values. Other articles of the authors have considered unconditional binomiality, that is, when the rate constants are considered variables [43,42].…”

A Graph Theoretical Approach for Testing Binomiality of Reversible Chemical Reaction Networks

“…The latter has been implemented in Maple and Macaulay II in [19,18]. A geometric view towards toricity has been considered by Grigoriev et al in [16], introducing shifted toricity and presenting algorithms using quantifier elimination [8,15,30] and Gröbner bases [3,4] A first order logic test for toricity has been given in [26]. In [25] unconditional binomiality has been introduced and for reversible reactions a polynomial time algorithm, based on a linear algebra approach, has been presented.…”

A Linear Algebra Approach for Detecting Binomiality of Steady State Ideals of Reversible Chemical Reaction Networks

“…Introducing shifted toricity, they presented algorithms, complexity bounds as well as experimental results for testing toricity using two important tools from symbolic computation, quantifier elimination [8,20,35] and Gröbner bases [4,5,11,12]. Recently, first order logic test for toricity have been introduced [32].…”