Adapting Real Quantifier Elimination Methods for Conflict Set Computation

The satisfiability problem in real closed fields is decidable. In the context of satisfiability modulo theories, the problem restricted to conjunctive sets of literals, that is, sets of polynomial constraints, is of particular importance. One of the central problems is the computation of good explanations of the unsatisfiability of such sets, i.e.\ obtaining a small subset of the input constraints whose conjunction is already unsatisfiable. We adapt two commonly used real quantifier elimination methods, cylindrical algebraic decomposition and virtual substitution, to provide such conflict sets and demonstrate the performance of our method in practice

Special Algorithm for Stability Analysis of Multistable Biological Regulatory Systems

We consider the problem of counting (stable) equilibriums of an important family of algebraic differential equations modeling multistable biological regulatory systems. The problem can be solved, in principle, using real quantifier elimination algorithms, in particular real root classification algorithms. However, it is well known that they can handle only very small cases due to the enormous computing time requirements. In this paper, we present a special algorithm which is much more efficient than the general methods. Its efficiency comes from the exploitation of certain interesting structures of the family of differential equations.Comment: 24 pages, 5 algorithms, 10 figure

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

Thirty Years of Virtual Substitution

International audienceIn 1988, Weispfenning published a seminal paper introducing a substitution technique for quantifier elimination in the linear theories of ordered and valued fields. The original focus was on complexity bounds including the important result that the decision problem for Tarski Algebra is bounded from below by a double exponential function. Soon after, Weispfenning's group began to implement substitution techniques in software in order to study their potential applicability to real world problems. Today virtual substitution has become an established computational tool, which greatly complements cylindrical algebraic decomposition. There are powerful implementations and applications with a current focus on satisfia-bility modulo theory solving and qualitative analysis of biological networks

Adapting Real Quantifier Elimination Methods for Conflict Set Computation

International audienceThe satisfiability problem in real closed fields is decidable. In the context of satisfiability modulo theories, the problem restricted to conjunctive sets of literals, that is, sets of polynomial constraints, is of particular importance. One of the central problems is the computation of good explanations of the unsatisfiability of such sets, i.e. obtaining a small subset of the input constraints whose conjunction is already unsatisfiable. We adapt two commonly used real quantifier elimination methods, cylindrical algebraic decomposition and virtual substitution, to provide such conflict sets and demonstrate the performance of our method in practice

Identifying the parametric occurrence of multiple steady states for some biological networks

We consider a problem from biological network analysis of determining regions in a parameter space over which there are multiple steady states for positive real values of variables and parameters. We describe multiple approaches to address the problem using tools from Symbolic Computation. We describe how progress was made to achieve semi-algebraic descriptions of the multistationarity regions of parameter space, and compare symbolic results to numerical methods. The biological networks studied are models of the mitogen-activated protein kinases (MAPK) network which has already consumed considerable effort using special insights into its structure of corresponding models. Our main example is a model with 11 equations in 11 variables and 19 parameters, 3 of which are of interest for symbolic treatment. The model also imposes positivity conditions on all variables and parameters. We apply combinations of symbolic computation methods designed for mixed equality/inequality systems, specifically virtual substitution, lazy real triangularization and cylindrical algebraic decomposition, as well as a simplification technique adapted from Gaussian elimination and graph theory. We are able to determine multistationarity of our main example over a 2-dimensional parameter space. We also study a second MAPK model and a symbolic grid sampling technique which can locate such regions in 3-dimensional parameter space.Comment: 60 pages - author preprint. Accepted in the Journal of Symbolic Computatio

A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications

International audienceEffective quantifier elimination procedures for first-order theories provide a powerful tool for genericallysolving a wide range of problems based on logical specifications. In contrast to general first-order provers, quantifierelimination procedures are based on a fixed set of admissible logical symbolswith an implicitly fixed semantics. Thisadmits the use of sub-algorithms from symbolic computation. We are going to focus on quantifier elimination forthe reals and its applications giving examples from geometry, verification, and the life sciences. Beyond quantifierelimination we are going to discuss recent results with a subtropical procedure for an existential fragment of thereals. This incomplete decision procedure has been successfully applied to the analysis of reaction systems inchemistry and in the life sciences

A computational approach to polynomial conservation laws

For polynomial ODE models, we introduce and discuss the concepts of exact and approximate conservation laws, which are the first integrals of the full and truncated sets of ODEs. For fast-slow systems, truncated ODEs describe the fast dynamics. We define compatibility classes as subsets of the state space, obtained by equating the conservation laws to constants. A set of conservation laws is complete when the corresponding compatibility classes contain a finite number of steady states. Complete sets of conservation laws can be used for model order reduction and for studying the multistationarity of the model. We provide algorithmic methods for computing linear, monomial, and polynomial conservation laws of polynomial ODE models and for testing their completeness. The resulting conservation laws and their completeness are either independent or dependent on the parameters. In the latter case, we provide parametric case distinctions. In particular, we propose a new method to compute polynomial conservation laws by comprehensive Gröbner systems and syzygies

New concepts for real quantifier elimination by virtual substitution

Quantifier elimination methods for real closed fields are an intensively studied subject from both theoretical and practical points of view. This thesis studies quantifier elimination based on virtual substitution with a particular focus on practically applicable methods and techniques. We develop a novel, stand-alone, and modular quantifier elimination framework for virtual substitution that can in principle be extended to arbitrary but bounded degrees of quantified variables. The framework subsumes previous virtual substitution algorithms. Quantifier elimination algorithms are obtained via instantiation of our quantifier elimination algorithm scheme with three precisely specified subalgorithms. We give instantiations of our scheme up to degree three of a quantified variable, which yields a quantifier elimination algorithm by virtual substitution for the cubic case. Compared to previous virtual substitution-based approaches, we propose novel improvements like smaller elimination sets and clustering. Furthermore, we exploit the Boolean structure and develop a structural quantifier elimination algorithm scheme. This allows us to take advantage of subformulas containing equations or negated equations, simplify virtual substitution results, and develop flexible bound selection strategies. We also revisit the established technique of degree shifts and show how to generalize this within our structural quantifier elimination algorithm scheme. Restricting ourselves to existential problems, we address the established concept of extended quantified elimination, which in addition to quantifier-free equivalents yields answers for existentially quantified variables. We show how to realize this concept within our quantifier elimination algorithm scheme. Moreover, we generalize our post-processing method for eliminating nonstandard symbols from answers to the general case. Our implementation of most of the concepts developed in this thesis is the first implementation of a cubic virtual substitution method. Experimental results comparing our implementation with the established original implementation of the quadratic virtual substitution in the Redlog computer logic system demonstrate the relevance of our novel techniques: On more than two hundred quantifier elimination problems---considered in more than sixty scientific publications during the past twenty years---we never eliminate fewer quantifiers than the Redlog's original implementation. For a considerable number of problems we eliminate more quantifiers.Quantoreneliminationsverfahren für reelle abgeschlossene Körper sind sowohl von der theoretischen als auch von der praktischen Seite ein intensiv studiertes Thema. Diese Dissertation befasst sich mit Quantorenelimination basierend auf virtueller Substitution. Im Mittelpunkt stehen praktisch anwendbare Methoden und Techniken. Wir entwicklen ein neues, unabhängiges und modulares Quantoreneliminationsrahmenkonzept für virtuelle Substitution, das im Prinzip auf beliebige Grade von quantifizierten Variablen erweitert werden kann. Unser Rahmenkonzept subsumiert existierende auf virtueller Substitution beruhende Algorithmen. Konkrete Algorithmen enstehen als Instanzen unseres Quantoreneliminationsalgorithmusschemas mit drei genau spezifizierten Subalgorithmen. Wir präsentieren Instanzen bis zu Grad drei einer quantifizierten Variable. Die liefern einen Algorithmus beruhend auf virtueller Substitution für den kubischen Fall. Im Vergleich mit anderen Verfahren basierend auf virtueller Substitution präsentieren wir zahlreiche Verbesserungen wie etwa kleinere Eliminationsmengen oder Clustering. Außerdem nutzen wir die Boolsche Struktur aus und entwickeln ein strukturelles Quantoreneliminationsalgorithmusschema. Somit können wir Gleichungen oder negierte Gleichungen ausnutzen, Ergebnisse der virtuellen Substitution vereinfachen und flexible Schrankenauswahlstrategien entwickeln. Wir studieren auch die bekannte Technik des degree shifts, die in manchen Fällen den Grad der quantifizierten Variablen reduzieren kann. Wir zeigen wie man diese Technik in unserem Quantoreneliminationsalgorithmusschema realisiert und verallgemeinert. Für reelle existentielle Probleme diskutieren wir das Konzept der erweiterten Quantorenelimination, die zu quantorenfreien Äquivalenten auch Antworten für die quantifizierten Variablen liefert. Wir zeigen wie sich dieses Konzept in unserem Quantoreneliminationsalgorithmusschema realisieren lässt. Zusätzlich verallgemeinern wir unser Postprocessingverfahren zur Elimination von Nichtstandardsymbolen aus Antworten. Unsere Implementierung unterstützt die meisten in dieser Arbeit vorgestellte Konzepte und stellt damit die erste Implementierung einer kubischen Methode basierend auf virtueller Substitution dar. Praktische Rechenexperimente, in denen wir unsere Implementierung mit bekannten im Computerlogik-System Redlog implementierten Verfahren für quadratische virtuelle Substitution verglichen, zeigen die Relevanz unserer Techniken: Auf mehr als 200 in mehr als sechzig wissenschaftlichen Publikationen beschriebenen Quantoreneliminationsproblemen eliminiert unsere Implementierung niemals weniger Quantoren als die existierende Implementierung in Redlog. Für eine signifikante Anzahl von Problemen können wir sogar mehr Quantoren eliminieren

New concepts for real quantifier elimination by virtual substitution

Quantifier elimination methods for real closed fields are an intensively studied subject from both theoretical and practical points of view. This thesis studies quantifier elimination based on virtual substitution with a particular focus on practically applicable methods and techniques. We develop a novel, stand-alone, and modular quantifier elimination framework for virtual substitution that can in principle be extended to arbitrary but bounded degrees of quantified variables. The framework subsumes previous virtual substitution algorithms. Quantifier elimination algorithms are obtained via instantiation of our quantifier elimination algorithm scheme with three precisely specified subalgorithms. We give instantiations of our scheme up to degree three of a quantified variable, which yields a quantifier elimination algorithm by virtual substitution for the cubic case. Compared to previous virtual substitution-based approaches, we propose novel improvements like smaller elimination sets and clustering. Furthermore, we exploit the Boolean structure and develop a structural quantifier elimination algorithm scheme. This allows us to take advantage of subformulas containing equations or negated equations, simplify virtual substitution results, and develop flexible bound selection strategies. We also revisit the established technique of degree shifts and show how to generalize this within our structural quantifier elimination algorithm scheme. Restricting ourselves to existential problems, we address the established concept of extended quantified elimination, which in addition to quantifier-free equivalents yields answers for existentially quantified variables. We show how to realize this concept within our quantifier elimination algorithm scheme. Moreover, we generalize our post-processing method for eliminating nonstandard symbols from answers to the general case. Our implementation of most of the concepts developed in this thesis is the first implementation of a cubic virtual substitution method. Experimental results comparing our implementation with the established original implementation of the quadratic virtual substitution in the Redlog computer logic system demonstrate the relevance of our novel techniques: On more than two hundred quantifier elimination problems---considered in more than sixty scientific publications during the past twenty years---we never eliminate fewer quantifiers than the Redlog\u27s original implementation. For a considerable number of problems we eliminate more quantifiers.Quantoreneliminationsverfahren für reelle abgeschlossene Körper sind sowohl von der theoretischen als auch von der praktischen Seite ein intensiv studiertes Thema. Diese Dissertation befasst sich mit Quantorenelimination basierend auf virtueller Substitution. Im Mittelpunkt stehen praktisch anwendbare Methoden und Techniken. Wir entwicklen ein neues, unabhängiges und modulares Quantoreneliminationsrahmenkonzept für virtuelle Substitution, das im Prinzip auf beliebige Grade von quantifizierten Variablen erweitert werden kann. Unser Rahmenkonzept subsumiert existierende auf virtueller Substitution beruhende Algorithmen. Konkrete Algorithmen enstehen als Instanzen unseres Quantoreneliminationsalgorithmusschemas mit drei genau spezifizierten Subalgorithmen. Wir präsentieren Instanzen bis zu Grad drei einer quantifizierten Variable. Die liefern einen Algorithmus beruhend auf virtueller Substitution für den kubischen Fall. Im Vergleich mit anderen Verfahren basierend auf virtueller Substitution präsentieren wir zahlreiche Verbesserungen wie etwa kleinere Eliminationsmengen oder Clustering. Außerdem nutzen wir die Boolsche Struktur aus und entwickeln ein strukturelles Quantoreneliminationsalgorithmusschema. Somit können wir Gleichungen oder negierte Gleichungen ausnutzen, Ergebnisse der virtuellen Substitution vereinfachen und flexible Schrankenauswahlstrategien entwickeln. Wir studieren auch die bekannte Technik des degree shifts, die in manchen Fällen den Grad der quantifizierten Variablen reduzieren kann. Wir zeigen wie man diese Technik in unserem Quantoreneliminationsalgorithmusschema realisiert und verallgemeinert. Für reelle existentielle Probleme diskutieren wir das Konzept der erweiterten Quantorenelimination, die zu quantorenfreien Äquivalenten auch Antworten für die quantifizierten Variablen liefert. Wir zeigen wie sich dieses Konzept in unserem Quantoreneliminationsalgorithmusschema realisieren lässt. Zusätzlich verallgemeinern wir unser Postprocessingverfahren zur Elimination von Nichtstandardsymbolen aus Antworten. Unsere Implementierung unterstützt die meisten in dieser Arbeit vorgestellte Konzepte und stellt damit die erste Implementierung einer kubischen Methode basierend auf virtueller Substitution dar. Praktische Rechenexperimente, in denen wir unsere Implementierung mit bekannten im Computerlogik-System Redlog implementierten Verfahren für quadratische virtuelle Substitution verglichen, zeigen die Relevanz unserer Techniken: Auf mehr als 200 in mehr als sechzig wissenschaftlichen Publikationen beschriebenen Quantoreneliminationsproblemen eliminiert unsere Implementierung niemals weniger Quantoren als die existierende Implementierung in Redlog. Für eine signifikante Anzahl von Problemen können wir sogar mehr Quantoren eliminieren