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40
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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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
A HOL theory of Euclidean space
 In Hurd and Melham [7
, 2005
"... Abstract. We describe a formalization of the elementary algebra, topology and analysis of finitedimensional Euclidean space in the HOL Light theorem prover. (Euclidean space is R N with the usual notion of distance.) A notable feature is that the HOL type system is used to encode the dimension N in ..."
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Abstract. We describe a formalization of the elementary algebra, topology and analysis of finitedimensional Euclidean space in the HOL Light theorem prover. (Euclidean space is R N with the usual notion of distance.) A notable feature is that the HOL type system is used to encode the dimension N in a simple and useful way, even though HOL does not permit dependent types. In the resulting theory the HOL type system, far from getting in the way, naturally imposes the correct dimensional constraints, e.g. checking compatibility in matrix multiplication. Among the interesting later developments of the theory are a partial decision procedure for the theory of vector spaces (based on a more general algorithm due to Solovay) and a formal proof of various classic theorems of topology and analysis for arbitrary Ndimensional Euclidean space, e.g. Brouwer’s fixpoint theorem and the differentiability of inverse functions. 1 1 The problem with R N
Real World Verification
"... Abstract. Scalable handling of real arithmetic is a crucial part of the verification of hybrid systems, mathematical algorithms, and mixed analog/digital circuits. Despite substantial advances in verification technology, complexity issues with classical decision procedures are still a major obstacle ..."
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Cited by 13 (4 self)
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Abstract. Scalable handling of real arithmetic is a crucial part of the verification of hybrid systems, mathematical algorithms, and mixed analog/digital circuits. Despite substantial advances in verification technology, complexity issues with classical decision procedures are still a major obstacle for formal verification of realworld applications, e.g., in automotive and avionic industries. To identify strengths and weaknesses, we examine state of the art symbolic techniques and implementations for the universal fragment of realclosed fields: approaches based on quantifier elimination, Gröbner Bases, and semidefinite programming for the Positivstellensatz. Within a uniform context of the verification tool KeYmaera, we compare these approaches qualitatively and quantitatively on verification benchmarks from hybrid systems, textbook algorithms, and on geometric problems. Finally, we introduce a new decision procedure combining Gröbner Bases and semidefinite programming for the real Nullstellensatz that outperforms the individual approaches on an interesting set of problems.
Linear Approximation of Planar Spatial Databases Using TransitiveClosure Logic
 In Proceedings 19th ACM Symposium on Principles of Database Systems
, 2000
"... We consider spatial databases in the plane that can be defined by polynomial constraint formulas. Motivated by applications in geographic information systems, we investigate linear approximations of spatial databases and study in which language they can be expressed effectively. Specifically, we sho ..."
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Cited by 10 (5 self)
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We consider spatial databases in the plane that can be defined by polynomial constraint formulas. Motivated by applications in geographic information systems, we investigate linear approximations of spatial databases and study in which language they can be expressed effectively. Specifically, we show that they cannot be expressed in the standard firstorder query language for polynomial constraint databases but that an extension of this firstorder language with transitive closure suces to express the approximation query in an effective manner. Furthermore, we introduce an extension of transitiveclosure logic and show that this logic is complete for the computable queries on linear spatial databases. This result together with our first result implies that this extension of transitiveclosure logic can express all computable topological queries on arbitrary spatial databases in the plane.
THE PROBABILITY THAT A SLIGHTLY PERTURBED NUMERICAL ANALYSIS PROBLEM IS DIFFICULT
, 2008
"... We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of illposed inputs. Several applications to linear and polynomial equation solving ..."
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Cited by 8 (8 self)
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We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of illposed inputs. Several applications to linear and polynomial equation solving show that the estimates obtained in this way are easy to derive and quite accurate. The main theorem is based on a volume estimate of εtubular neighborhoods around a real algebraic subvariety of a sphere, intersected with a spherical disk of radius σ. Besides ε and σ, this bound depends only on the dimension of the sphere and on the degree of the defining equations.
The Structure of Differential Invariants and Differential Cut Elimination
, 2011
"... not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution or government. Keywords: Proof theory, differential equations, differential cut elimination, logics of programs, The biggest challenge in hybrid systems verification is the handling o ..."
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Cited by 7 (7 self)
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not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution or government. Keywords: Proof theory, differential equations, differential cut elimination, logics of programs, The biggest challenge in hybrid systems verification is the handling of differential equations. Because computable closedform solutions only exist for very simple differential equations, proof certificates have been proposed for more scalable verification. Search procedures for these proof certificates are still rather adhoc, though, because the problem structure is only understood poorly. We investigate differential invariants, which can be checked for invariance along a differential equation just by using their differential structure and without having to solve the differential equation. We study the structural properties of differential invariants. To analyze tradeoffs for proof search complexity, we identify more than a dozen relations between several classes of differential invariants and compare their deductive power. As our main results, we analyze the deductive power of differential cuts and the deductive power of differential invariants with auxiliary differential variables. We refute the differential cut elimination hypothesis and show that differential cuts are fundamental proof principles that strictly increase the deductive power. We also prove that
A MorseSard theorem for the distance function on Riemannian manifolds
 Manuscr. Math
"... Abstract. We prove that the set of critical values of the distance function from a submanifold of a complete Riemannian manifold is of Lebesgue measure zero. In this way, we extend a result of Itoh and Tanaka. 1. ..."
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Abstract. We prove that the set of critical values of the distance function from a submanifold of a complete Riemannian manifold is of Lebesgue measure zero. In this way, we extend a result of Itoh and Tanaka. 1.
On the complexity of deciding connectedness and computing Betti numbers of a complex algebraic variety
 J. Complexity
"... We extend the lower bounds on the complexity of computing Betti numbers proved in [6] to complex algebraic varieties. More precisely, we first prove that the problem of deciding connectedness of a complex affine or projective variety given as the zero set of integer polynomials is PSPACEhard. Then ..."
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We extend the lower bounds on the complexity of computing Betti numbers proved in [6] to complex algebraic varieties. More precisely, we first prove that the problem of deciding connectedness of a complex affine or projective variety given as the zero set of integer polynomials is PSPACEhard. Then we prove PSPACEhardness for the more general problem of deciding whether the Betti number of fixed order of a complex affine or projective variety is at most some given integer. Key words: connected components, Betti numbers, PSPACE, lower bounds 1
Pseudozeros Of Multivariate Polynomials
 Math. Comp
, 2000
"... . The pseudozero set of a system f of polynomials in n complex variables is the subset of C n which is the union of the zero  sets of all polynomial systems g that are near to f in a suitable sense. This concept is made precise and general properties of pseudozero sets are established. In partic ..."
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. The pseudozero set of a system f of polynomials in n complex variables is the subset of C n which is the union of the zero  sets of all polynomial systems g that are near to f in a suitable sense. This concept is made precise and general properties of pseudozero sets are established. In particular it is shown that under wide circumstances, the pseudozero set is a semialgebraic set. Also, estimates are given for the size of the projections of pseudozero sets into coordinate directions. Several examples are presented illustrating some of the general theory developed here. Finally, algorithmic ideas are proposed for solving multivariate polynomials. 1. Introduction 1.1. Summary. The pseudozero set of a general polynomial in a single variable was investigated in [18]. Our purpose here is to extend some ideas from that work to systems of polynomials in several variables, with special attention to the case in which the zero set of the system consists of finitely many points. Recall th...