This report introduces QEPCAD B, a program for computing with real algebraic sets using cylindrical algebraic decomposition (CAD). QEPCAD B both extends and improves upon the QEPCAD system for quantifier elimination by partial cylindrical algebraic decomposition written by Hoon Hong in the early 1990s. This paper briefly discusses some of the improvements in the implementation of CAD and quantifier elimination via CAD, and provides somewhat more detail on extensions to the system that go beyond quantifier elimination. The author is responsible for most of the extended features of QEPCAD B, but improvements to the basic CAD implementation and to the SACLIB library on which QEPCAD is based are the results of many people’s work, including: George E.

. We present a new, "elementary" quantifier elimination method for various special cases of the general quantifier elimination problem for the first--order theory of real numbers. These include the elimination of one existential quantifier 9x in front of quantifier--free formulas restricted by a non-trivial quadratic equation in x (the case considered also in [7]), and more generally in front of arbitrary quantifier--free formulas involving only polynomials that are quadratic in x. The method generalizes the linear quantifier elimination method by virtual substitution of test terms in [9]. It yields a quantifier elimination method for an arbitrary number of quantifiers in certain formulas involving only linear and quadratic occurences of the quantified variables. Moreover, for existential formulas ' of this kind it yields sample answers to the query represented by '. The method is implemented in reduce as part of the redlog package (see [4, 5]). Experiments show that the method is appl...

this paper, we show how to write all common stability problems as quantifier-elimination

Many problems in mathematics, logic, computer science, and engineering can be reduced to the problem of testing positiveness of polynomials. Although the problem is decidable (Tarski 1930), the general decision methods are not always practically applicable due to their high computational time requirements. Thus several partial methods were proposed in the field of term rewriting systems. In this paper, we exactly determine how much partial these methods are, and propose simpler and/or more efficient methods with the same power.

Introduction Many interesting control system design and analysis problems can be recast as systems of inequalities for multivariate polynomials in real variables. In particular, for linear time-invariant systems, important control issues such as robust stability and robust performance can be reduced to such systems. Typically, the variables in the (multivariate) polynomials come from plant (controlled system) and compensator (controller) parameters. In this chapter, we describe a method for solving such systems of inequalities. By solving we mean that we end up with a collection of axis-parallel boxes in the parameter space whose union provides an inner approximation of the solution set, i.e., the polynomial inequalities are fulfilled for each parameter vector taken from such a box. This method is based on the expansion of a multivariate polynomial into Bernstein polynomials. It provides an alternative to symbolic methods like quantifier elimination whose application to control

Let a quantified inequality constraint over the reals be a formula in the first-order predicate language over the structure of the real numbers, where the allowed predicate symbols are ≤ and <. Solving such constraints is an undecidable problem when allowing function symbols such sin or cos. In the paper we give an algorithm that terminates with a solution for all, except for very special, pathological inputs. We ensure the practical efficiency of this algorithm by employing constraint programming techniques. 1

Cylindrical algebraic decomposition is one of the most important tools for computing with semi-algebraic sets, while triangular decomposition is among the most important approaches for manipulating constructible sets. In this paper, for an arbitrary finite set F ⊂ R[y1,..., yn] we apply comprehensive triangular decomposition in order to obtain an F-invariant cylindrical decomposition of the n-dimensional complex space, from which we extract an F-invariant cylindrical algebraic decomposition of the n-dimensional real space. We report on an implementation of this new approach for constructing cylindrical algebraic decompositions.

This paper applies interval methods to a classical problem in computer algebra. Let a quantified constraint be a first-order formula over the real numbers. As shown by A. Tarski in the 1930's, such constraints, when restricted to the predicate symbols , are in general solvable. However, the problem becomes undecidable, when we add function symbols like sin. Furthermore, all exact algorithms known up to now are too slow for big examples, do not provide partial information before computing the total result, cannot satisfactorily deal with interval constants in the input, and often generate huge output. As a remedy we propose an approximation method based on interval arithmetic. It uses a generalization of the notion of cylindrical decomposition -- as introduced by G. Collins. We describe an implementation of the method and demonstrate that, for quantied constraints without equalities, it can efficiently give approximate information on problems that are too hard for current exact methods.

Methods for deciding quantifier-free non-linear arithmetical conjectures over *** are crucial in the formal verification of many real-world systems and in formalised mathematics. While non-linear (rational function) arithmetic over *** is decidable, it is fundamentally infeasible: any general decision method for this problem is worst-case exponential in the dimension (number of variables) of the formula being analysed. This is unfortunate, as many practical applications of real algebraic decision methods require reasoning about high-dimensional conjectures. Despite their inherent infeasibility, a number of different decision methods have been developed, most of which have "sweet spots" --- e.g., types of problems for which they perform much better than they do in general. Such "sweet spots" can in many cases be heuristically combined to solve problems that are out of reach of the individual decision methods when used in isolation. RAHD ("Real Algebra in High Dimensions") is a theorem prover that works to combine a collection of real algebraic decision methods in ways that exploit their respective "sweet-spots." We discuss high-level mathematical and design aspects of RAHD and illustrate its use on a number of examples.

We introduce an efficient algorithm for determining a suitable projection order for performing cylindrical algebraic decomposition. Our algorithm is motivated by a statistical analysis of comprehensive test set computations. This