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38
Generating Polynomial Orderings for Termination Proofs
 In Proc. 6th RTA, LNCS 914
, 1995
"... Most systems for the automation of termination proofs using polynomial orderings are only semiautomatic, i.e. the "right" polynomial ordering has to be given by the user. We show that a variation of Lankford's partial derivative technique leads to an easier and slightly more powerful method than mo ..."
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Cited by 46 (22 self)
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Most systems for the automation of termination proofs using polynomial orderings are only semiautomatic, i.e. the "right" polynomial ordering has to be given by the user. We show that a variation of Lankford's partial derivative technique leads to an easier and slightly more powerful method than most other semiautomatic approaches. Based on this technique we develop a method for the automated synthesis of a suited polynomial ordering.
Testing Stability by Quantifier Elimination
, 1997
"... this paper, we show how to write all common stability problems as quantifierelimination ..."
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Cited by 28 (5 self)
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this paper, we show how to write all common stability problems as quantifierelimination
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
On the passage from local to global in number theory
 Bull. Amer. Math. Soc. (N.S
, 1993
"... Would a reader be able to predict the branch of mathematics that is the subject of this article if its title had not included the phrase “in Number Theory”? The distinction “local ” versus “global”, with various connotations, has found a home in almost every part of mathematics, local problems being ..."
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Cited by 21 (0 self)
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Would a reader be able to predict the branch of mathematics that is the subject of this article if its title had not included the phrase “in Number Theory”? The distinction “local ” versus “global”, with various connotations, has found a home in almost every part of mathematics, local problems being often a steppingstone to
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
Comparison of Several Decision Algorithms for the Existential Theory of the Reals
, 1991
"... In this paper we compare the complexities of the following three decision algorithms on existential sentences over the reals: Collins (1975), Grigor'ev and Vorobjov (1988), and Renegar (1989). Let n be the number of variables, m the number of polynomials, d the total degree, and L the coefficient bi ..."
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Cited by 18 (1 self)
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In this paper we compare the complexities of the following three decision algorithms on existential sentences over the reals: Collins (1975), Grigor'ev and Vorobjov (1988), and Renegar (1989). Let n be the number of variables, m the number of polynomials, d the total degree, and L the coefficient bit length. The table below shows their(already known) theoretical complexities, along with their estimated running time for small inputs (n = m = d = L = 2) on currently available machines: Algorithm Theoretical n = m = d = L = 2 Collins L 3 (md) 2 O(n) 1 second Grigor 0 ev=Vorobjov L(md) n 2 AE 1 million years Renegar L(log L)(log log L)(md) O(n) AE 1 million years Thus it suggests that Collins' algorithm is the fastest among them for inputs which can be decided in a reasonable amount of time.
Analytic cell decomposition and analytic motivic integration
 ANN. SCI. ÉCOLE NORM. SUP
, 2006
"... ..."
Combined Decision Techniques for the Existential Theory of the Reals
 CALCULEMUS
, 2009
"... Methods for deciding quantifierfree nonlinear arithmetical conjectures over *** are crucial in the formal verification of many realworld systems and in formalised mathematics. While nonlinear (rational function) arithmetic over *** is decidable, it is fundamentally infeasible: any general decisi ..."
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Cited by 10 (5 self)
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Methods for deciding quantifierfree nonlinear arithmetical conjectures over *** are crucial in the formal verification of many realworld systems and in formalised mathematics. While nonlinear (rational function) arithmetic over *** is decidable, it is fundamentally infeasible: any general decision method for this problem is worstcase 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 highdimensional 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 "sweetspots." We discuss highlevel mathematical and design aspects of RAHD and illustrate its use on a number of examples.
Tilings with congruent tiles
 Bull. Amer. Maths. Soc
, 1980
"... Introduction. The purpose of this paper is to survey recent results related to the second part of Hubert's eighteenth problem (see Hilbert [1900]). This problem, which is concerned with tilings of Euclidean space by congruent polyhedra, will be stated below after the necessary terminology has been i ..."
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Cited by 9 (0 self)
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Introduction. The purpose of this paper is to survey recent results related to the second part of Hubert's eighteenth problem (see Hilbert [1900]). This problem, which is concerned with tilings of Euclidean space by congruent polyhedra, will be stated below after the necessary terminology has been introduced. Although Hubert's original question was answered by one of his
Parallelization of Quantifier Elimination on Workstation Network
 In Proceedings of AAECC 10 (Puerto Rico), Springer Lecture Notes in Computer Science 673
, 1991
"... This paper reports our effort to parallelize on a network of workstations the quantifier elimination algorithm over the reals which was devised by Collins and improved by the author. The preliminary experiments show both sublinear and superlinear speedups due to speculative parallelism. On the pro ..."
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Cited by 8 (3 self)
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This paper reports our effort to parallelize on a network of workstations the quantifier elimination algorithm over the reals which was devised by Collins and improved by the author. The preliminary experiments show both sublinear and superlinear speedups due to speculative parallelism. On the problems we have tested so far, the efficiencies range between 70% and 320%. 1 Introduction This paper reports our effort to parallelize on a network of workstations the quantifier elimination algorithm over the reals which was devised by Collins [9] and improved by the author [20, 11]. The preliminary experiments show both sublinear and superlinear speedups due to speculative parallelism. On the problems we have tested so far, the efficiencies range between 70% and 320%. Quantifier elimination in a given theory is a procedure that constructs from a quantified formula an equivalent quantifierfree formula. The decision problem is, then, a special case where there is no free variable in the...