Most systems for the automation of termination proofs using polynomial orderings are only semi-automatic, 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 semi-automatic approaches. Based on this technique we develop a method for the automated synthesis of a suited polynomial ordering.

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

Abstract. We present a fully proof-producing implementation of a quantifierelimination procedure for real closed fields. To our knowledge, this is the first generally useful proof-producing 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 low-level 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 partic-ular 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 non-trivial 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 certifi-cate 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 aris-ing 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

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 stepping-stone to

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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. 1 Introduction Since Tarski [33] gave the first decision algorithm for the first order theory of the reals, many other algorithms with better theoretical complexities have been proposed [32...

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 first-order 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

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.

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 sub-linear and super-linear 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 sub-linear and super-linear 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 quantifier--free formula. The decision problem is, then, a special case where there is no free variable in the...

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