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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
Complexity of computations with Pfaffian and Noetherian functions
 Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, volume 137 of NATO Science Series II
, 2004
"... This paper is a survey of the upper bounds on the complexity of basic algebraic and geometric operations with Pfaffian and Noetherian functions, and with sets definable by these functions. Among other results, we consider bounds on Betti numbers of subPfaffian sets, multiplicities of Pfaffian inter ..."
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Cited by 23 (6 self)
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This paper is a survey of the upper bounds on the complexity of basic algebraic and geometric operations with Pfaffian and Noetherian functions, and with sets definable by these functions. Among other results, we consider bounds on Betti numbers of subPfaffian sets, multiplicities of Pfaffian intersections, effective �Lojasiewicz inequality for Pfaffian functions, computing frontier and closure of restricted semiPfaffian sets, constructing smooth stratifications and cylindrical cell decompositions (including an effective version of the complement theorem for restricted subPfaffian sets), relative closures of nonrestricted semiPfaffian sets and bounds on the number of their connected components, bounds on multiplicities of isolated solutions of systems of Noetherian equations. 1
Complexity of cylindrical decompositions of subPfaffian sets
 J. Pure Appl. Algebra
, 2001
"... Abstract. We construct an algorithm for a cylindrical cell decomposition of aclosedcubeI n ⊂ R n compatible with a “restricted ” subPfaffian subset Y ⊂ I n, provided an oracle deciding consistency of a system of Pfaffian equations and inequalities is given. In particular, the algorithm produces the ..."
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Cited by 17 (8 self)
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Abstract. We construct an algorithm for a cylindrical cell decomposition of aclosedcubeI n ⊂ R n compatible with a “restricted ” subPfaffian subset Y ⊂ I n, provided an oracle deciding consistency of a system of Pfaffian equations and inequalities is given. In particular, the algorithm produces the complement ˜Y = I n \ Y. The complexity bound of the algorithm, the number and formats of cells are doubly exponential in n 3.
Complexity Lower Bounds for Computation Trees with Elementary Transcendental Function Gates (Extended Abstract)
 Theoretical Computer Science
, 1996
"... We consider computation trees which admit as gate functions along with the usual arithmetic operations also algebraic or transcendental functions like exp; log; sin; square root (defined in the relevant domains) or much more general, Pfaffian functions. A new method for proving lower bounds on the d ..."
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Cited by 8 (4 self)
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We consider computation trees which admit as gate functions along with the usual arithmetic operations also algebraic or transcendental functions like exp; log; sin; square root (defined in the relevant domains) or much more general, Pfaffian functions. A new method for proving lower bounds on the depth of these trees is developed which allows to prove a lower bound \Omega\Gamma p log N ) for testing membership to a convex polyhedron with N facets of all dimensions, provided that N is large enough. This method differs essentially from the approaches adopted for algebraic computation trees ([1], [4], [26], [13]). 1 Pfaffian computation trees We consider the following computation model, a generalization of the algebraic computation trees (see, e.g., [1], [26]). Definition 1. Pfaffian computation tree T is a tree at every node v of which a Pfaffian function f v in variables X 1 ; : : : ; Xn is attached, which satisfies the following properties. Let f v0 ; : : : ; f v l ; f v l+1 = ...
Deciding LinearTranscendental Problems
, 2000
"... We present a decision procedure for lineartranscendental problems formalized in a suitable firstorder language. The problems are formalized by formulas with arbitrary quantified linear variables and a block of quantifiers with respect to mixed lineartranscendental variables. Variables may range b ..."
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Cited by 2 (0 self)
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We present a decision procedure for lineartranscendental problems formalized in a suitable firstorder language. The problems are formalized by formulas with arbitrary quantified linear variables and a block of quantifiers with respect to mixed lineartranscendental variables. Variables may range both over the reals and over the integers. The transcendental functions admitted are characterized axiomatically; they include the exponential function applied to a polynomial, hyperbolic functions and their inverses, and the arcustangent. The decision procedure is explicit and implementable; it is based on mixed realinteger linear elimination, the symbolic test point method, elementary analysis, and Lindemann's theorem. As a byproduct we obtain sample solutions for existential formulas and a qualitative description of the connected components of the satisfaction set wrt. a mixed lineartranscendental variable.