A Decision Method for Elementary Algebra and Geometry

Tarski, Alfred; McKinsey, J. C. C.

doi:10.1525/9780520348097

Cited by 1,066 publications

(445 citation statements)

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“…Indeed, when is infinite (later on we shall consider the case ⊂ ℂ n ) and for generic functions f , g i , the nonnegativity decision problem is undecidable. In this paper, we consider the case where gambles are (complex) multivariate polynomials of degree at most d. In this case, by Tarski-Seidenberg's quantifier elimination theory [59,60], the problem (4) becomes decidable but still intractable, being in general NP-hard. From this perspective, the classical theory is therefore not suitable for constituting a realistic model of rationality.…”

Section: Algorithmic Desirabilitymentioning

confidence: 99%

The Weirdness Theorem and the Origin of Quantum Paradoxes

2021

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We argue that there is a simple, unique, reason for all quantum paradoxes, and that such a reason is not uniquely related to quantum theory. It is rather a mathematical question that arises at the intersection of logic, probability, and computation. We give our ‘weirdness theorem’ that characterises the conditions under which the weirdness will show up. It shows that whenever logic has bounds due to the algorithmic nature of its tasks, then weirdness arises in the special form of negative probabilities or non-classical evaluation functionals. Weirdness is not logical inconsistency, however. It is only the expression of the clash between an unbounded and a bounded view of computation in logic. We discuss the implication of these results for quantum mechanics, arguing in particular that its interpretation should ultimately be computational rather than exclusively physical. We develop in addition a probabilistic theory in the real numbers that exhibits the phenomenon of entanglement, thus concretely showing that the latter is not specific to quantum mechanics.

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Section: Algorithmic Desirabilitymentioning

confidence: 99%

The Weirdness Theorem and the Origin of Quantum Paradoxes

2021

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“…1. R (Tarski, [31], also [11,5]) The theory of real-closed fields admits elimination of quantifiers in the language of ordered . fields, or in the langaage of field theory augmented with the predicate P 2 • 2.…”

Section: Model Theorymentioning

confidence: 99%

Quantifier Elimination in p-adic Fields

Dubhashi

1993

The Computer Journal

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We present a tutorial survey of quantifier-elimination and decision procedures in p-adic :fieIds. The p-adic :fieIds are studied in the (so-called) P,,-formalism of Angus Macintyre, for whieh motivation is provided through a rieh body of analogies with real-closed :fields. Quantifier-elimination and decision procedures are described proceeding via a Cylindrical Algebraic Decomposition of affine p-adic space. EfFective eomplexi.ty analyses are also provided.

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“…In theory and practice, most of the symbolic computations are done by numerical linear algebra running in the backyards. In several problems, however, the complexities of symbolic computations are enormous; they often scale exponentially or double-exponentially (or worse) with the numbers of the involved symbols [ACM84a,ACM84b,Tar98]. In these circumstances, if we could anticipate the exponential speed-up in symbolic computations by quantum computers, it is progress.…”

Section: Introductionmentioning

confidence: 99%

Quantum computation of Groebner basis

Kikuchi¹,

Kikuchi²

2021

Preprint

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In this essay, we examine the feasibility of quantum computation of Groebner basis which is a fundamental tool of algebraic geometry. The classical method for computing Groebner basis is based on Buchberger's algorithm, and our question is how to adopt quantum algorithm there. A Quantum algorithm for finding the maximum is usable for detecting head terms of polynomials, which are required for the computation of S-polynomials. The reduction of S-polynomials with respect to a Groebner basis could be done by the quantum version of Gauss-Jordan elimination of echelon which represents polynomials. However, the frequent occurrence of zero-reductions of polynomials is an obstacle to the effective application of quantum algorithms. This is because zero-reductions of polynomials occur in non-full-rank echelons, for which quantum linear systems algorithms (through the inversion of matrices) are inadequate, as ever-known quantum linear solvers (such as Harrow-Hassidim-Lloyd) require the clandestine computations of the inverses of eigenvalues. Hence, for the quantum computation of the Groebner basis, the schemes to suppress the zero-reductions are necessary. To this end, the F5 algorithm or its variant (F5C) would be the most promising, as these algorithms have countermeasures against the occurrence of zero-reductions and can construct full-rank echelons whenever the inputs are regular sequences. Between these two algorithms, the F5C is the better match for algorithms involving the inversion of matrices.

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A Decision Method for Elementary Algebra and Geometry

Cited by 1,066 publications

References 0 publications

The Weirdness Theorem and the Origin of Quantum Paradoxes

The Weirdness Theorem and the Origin of Quantum Paradoxes

Quantifier Elimination in p-adic Fields

Quantum computation of Groebner basis

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