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50
QEPCAD B: A program for computing with semialgebraic sets using CADs
 SIGSAM BULLETIN
, 2003
"... 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 199 ..."
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Cited by 56 (1 self)
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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.
Safety verification of hybrid systems by constraint propagation based abstraction refinement
, 2005
"... This paper deals with the problem of safety verification of nonlinear hybrid systems. We start from a classical method that uses interval arithmetic to check whether trajectories can move over the boundaries in a rectangular grid. We put this method into an abstraction refinement framework and impr ..."
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Cited by 44 (10 self)
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This paper deals with the problem of safety verification of nonlinear hybrid systems. We start from a classical method that uses interval arithmetic to check whether trajectories can move over the boundaries in a rectangular grid. We put this method into an abstraction refinement framework and improve it by developing an additional refinement step that employs interval constraint propagation to add information to the abstraction without introducing new grid elements. Moreover, the resulting method allows switching conditions, initial states and unsafe states to be described by complex constraints instead of sets that correspond to grid elements. Nevertheless, the method can be easily implemented since it is based on a welldefined set of constraints, on which one can run any constraint propagation based solver. Tests of such an implementation are promising.
Logical Representations for Automated Reasoning about Spatial Relationships
, 1997
"... This thesis investigates logical representations for describing and reasoning about spatial situations. Previously proposed theories of spatial regions are investigated in some detail  especially the 1storder theory of Randell, Cui and Cohn (1992). The difficulty of achieving effective automated ..."
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Cited by 33 (5 self)
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This thesis investigates logical representations for describing and reasoning about spatial situations. Previously proposed theories of spatial regions are investigated in some detail  especially the 1storder theory of Randell, Cui and Cohn (1992). The difficulty of achieving effective automated reasoning with these systems is observed. A new approach is presented, based on encoding spatial relations in formulae of 0order (`propositional ') logics. It is proved that entailment, which is valid according to the standard semantics for these logics, is also valid with respect to the spatial interpretation. Consequently, wellknown mechanisms for propositional reasoning can be applied to spatial reasoning. Specific encodings of topological relations into both the modal logic S4 and the intuitionistic propositional calculus are given. The complexity of reasoning using the intuitionistic representation is examined and a procedure is presented which is shown to be of O(n 3 ) complexity ...
Improved Projection for Cylindrical Algebraic Decomposition
 Journal of Symbolic Computation
, 2001
"... This technical report is a preliminary version of a paper on improved projection for Cylindrical Algebraic Decomposition. It is being made available for ISSAC 2000 because of its bearing on [Bro00]. McCallum's projection operator for Cylindrical Algebraic Decomposition (CAD) [McC98, McC88, McC84 ..."
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Cited by 30 (3 self)
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This technical report is a preliminary version of a paper on improved projection for Cylindrical Algebraic Decomposition. It is being made available for ISSAC 2000 because of its bearing on [Bro00]. McCallum's projection operator for Cylindrical Algebraic Decomposition (CAD) [McC98, McC88, McC84] represented a huge step forward for the practical utility of the CAD algorithm. This report presents a simple theorem showing that the mathematics in McCallum's paper actually point to a better projection operator than he proposes  a reduced McCallum projection. As with McCallum's projection, the reduced projection does not simply speed up CAD computation for problems that are currently solvable in practice, but actually increases the scope of problems that can realistically be attacked via CAD's. Additionally, the same methods are used to show that McCallum's projection can be reduced still further when CAD is applied to certain types of commonly occurring quantifier eliminat...
Computational Real Algebraic Geometry
 HANDBOOK OF DISCRETE AND COMPUTATIONAL GEOMETRY, CHAPTER 29
, 1997
"... Computational real algebraic geometry studies various algorithmic questions dealing with the real solutions of a system of equalities, inequalities, and inequations of polynomials over the real numbers. This emerging field is largely motivated by the power and elegance with which it solves a broad ..."
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Cited by 25 (5 self)
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Computational real algebraic geometry studies various algorithmic questions dealing with the real solutions of a system of equalities, inequalities, and inequations of polynomials over the real numbers. This emerging field is largely motivated by the power and elegance with which it solves a broad and general class of problems arising in robotics, vision, computer aided design, geometric theorem proving, etc. The algorithmic problems that arise in this context are formulated as decision problems for the firstorder theory of reals and the related problems of quantifier elimination (Section 1). The associated geometric structures are then examined via an exploration of the semialgebraic sets (Section 2). Algorithmic problems for semialgebraic sets are considered next. In particular, there is a discussion of real algebraic numbers and their representation which relies on such classical theorems as Stu
Efficient solving of quantified inequality constraints over the real numbers
 ACM Transactions on Computational Logic
"... Let a quantified inequality constraint over the reals be a formula in the firstorder 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 ..."
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Cited by 25 (7 self)
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Let a quantified inequality constraint over the reals be a formula in the firstorder 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
A proofproducing decision procedure for real arithmetic
 Automated deduction – CADE20. 20th international conference on automated deduction
, 2005
"... Abstract. We present a fully proofproducing implementation of a quantifierelimination procedure for real closed fields. To our knowledge, this is the first generally useful proofproducing implementation of such an algorithm. Whilemany problems within the domain are intractable, we demonstrate conv ..."
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Cited by 24 (3 self)
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Abstract. We present a fully proofproducing implementation of a quantifierelimination procedure for real closed fields. To our knowledge, this is the first generally useful proofproducing implementation of such an algorithm. Whilemany problems within the domain are intractable, we demonstrate convincing examples of its value in interactive theorem proving. 1 Overview and related work Arguably the first automated theorem prover ever written was for a theory of lineararithmetic [8]. Nowadays many theorem proving systems, even those normally classified as `interactive ' rather than `automatic', contain procedures to automate routinearithmetical reasoning over some of the supported number systems like N, Z, Q, R and C. Experience shows that such automated support is invaluable in relieving users ofwhat would otherwise be tedious lowlevel proofs. We can identify several very common limitations of such procedures: Often they are restricted to proving purely universal formulas rather than dealingwith arbitrary quantifier structure and performing general quantifier elimination. Often they are not complete even for the supported class of formulas; in particular procedures for the integers often fail on problems that depend inherently on divisibility properties (e.g. 8x y 2 Z. 2x + 1 6 = 2y) They seldom handle nontrivial nonlinear reasoning, even in such simple cases as 8x y 2 R. x> 0 ^ y> 0) xy> 0, and those that do [18] tend to use heuristicsrather than systematic complete methods. Many of the procedures are standalone decision algorithms that produce no certificate of correctness and do not produce a `proof ' in the usual sense. The earliest serious exception is described in [4]. Many of these restrictions are not so important in practice, since subproblems arising in interactive proof can still often be handled effectively. Indeed, sometimes the restrictions are unavoidable: Tarski's theorem on the undefinability of truth implies thatthere cannot even be a complete semidecision procedure for nonlinear reasoning over
Continuous FirstOrder Constraint Satisfaction
 ARTIFICIAL INTELLIGENCE, AUTOMATED REASONING, AND SYMBOLIC COMPUTATION, NUMBER 2385 IN LNCS
, 2002
"... This paper shows how to use constraint programming techniques for solving firstorder constraints over the reals (i.e., formulas in the firstorder predicate language over the structure of the real numbers). More specifically, based on a narrowing operator that implements an arbitrary notion of con ..."
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Cited by 22 (12 self)
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This paper shows how to use constraint programming techniques for solving firstorder constraints over the reals (i.e., formulas in the firstorder predicate language over the structure of the real numbers). More specifically, based on a narrowing operator that implements an arbitrary notion of consistency for atomic constraints over the reals (e.g., boxconsistency), the paper provides a narrowing operator for firstorder constraints that implements a corresponding notion of firstorder consistency, and a solver based on such a narrowing operator. As a consequence, this solver can take over various favorable properties from the field of constraint programming.
Verifying nonlinear real formulas via sums of squares
 Theorem Proving in Higher Order Logics, TPHOLs 2007, volume 4732 of Lect. Notes in Comp. Sci
, 2007
"... Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable ‘sum of squares ’ certificates ..."
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Cited by 19 (2 self)
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Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable ‘sum of squares ’ certificates can be found by sophisticated numerical methods such as semidefinite programming, yet the actual verification of the resulting proof is straightforward even in a highly foundational theorem prover. We will describe our experience with an implementation in HOL Light, noting some successes as well as difficulties. We also describe a new approach to the univariate case that can handle some otherwise difficult examples. 1 Verifying nonlinear formulas over the reals Over the real numbers, there are algorithms that can in principle perform quantifier elimination from arbitrary firstorder formulas built up using addition, multiplication and the usual equality and inequality predicates. A classic example of such a quantifier elimination equivalence is the criterion for a quadratic equation to have a real root: ∀a b c. (∃x. ax 2 + bx + c = 0) ⇔ a = 0 ∧ (b = 0 ⇒ c = 0) ∨ a � = 0 ∧ b 2 ≥ 4ac