Better Answers to Real Questions

We consider existential problems over the reals. Extended quantifier elimination generalizes the concept of regular quantifier elimination by providing in addition answers, which are descriptions of possible assignments for the quantified variables. Implementations of extended quantifier elimination via virtual substitution have been successfully applied to various problems in science and engineering. So far, the answers produced by these implementations included infinitesimal and infinite numbers, which are hard to interpret in practice. We introduce here a post-processing procedure to convert, for fixed parameters, all answers into standard real numbers. The relevance of our procedure is demonstrated by application of our implementation to various examples from the literature, where it significantly improves the quality of the results

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

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

Algorithmic strategies for applicable real quantifier elimination

One of the most important algorithms for real quantifier elimination is the quantifier elimination by virtual substitution introduced by Weispfenning in 1988. In this thesis we present numerous algorithmic approaches for optimizing this quantifier elimination algorithm. Optimization goals are the actual running time of the implementation of the algorithm and the size of the output formula. Strategies for obtaining these goals include simplification of first-order formulas,reduction of the size of the computed elimination set, and condensing a new replacement for the virtual substitution. Local quantifier elimination computes formulas that are equivalent to the input formula only nearby a given point. We can make use of this restriction for further optimizing the quantifier elimination by virtual substitution. Finally we discuss how to solve a large class of scheduling problems by real quantifier elimination. To optimize our algorithm for solving scheduling problems we make use of the special form of the input formula and of additional information given by the description of the scheduling problemEines der bedeutendsten Verfahren zur reellen Quantorenelimination ist die Quantorenelimination mittels virtueller Substitution, die von Weispfenning 1988 eingeführt wurde. In der vorliegenden Arbeit werden zahlreiche algorithmische Strategien zur Optimierung dieses Verfahrens präsentiert. Optimierungsziele der Arbeit waren dabei die tatsächliche Laufzeit der Implementierung des Algorithmus sowie die Größe der Ausgabeformel. Zur Optimierung werden dabei die Simplifikation vonFormeln erster Stufe, die Reduktion der Größe der Eliminationsmenge sowie das Condensing, ein Ersatz für die virtuelle Substitution,untersucht. Lokale Quantorenelimination berechnet Formeln, die nur inder Nähe eines gegebenen Punktes äquivalent zur Eingabeformel ist. Diese Einschränkung erlaubt es, das Verfahren weiter zu verbessern.Als Anwendung des Eliminationsverfahren diskutieren wir abschließend, wie man eine große Klasse von Schedulingproblemen mittels reeller Quantorenelimination lösen kann. In diesem Fall benutzen wir die spezielle Struktur der Eingabeformel und zusätzliche Informationen über das Schedulingproblem, um die Quantorenelimination mittels virtueller Substitution problemspezifisch zu optimieren