Hilbert’s tenth problem: Diophantine equations: positive aspects of a negative solution

Davis, Martin; Matijasevič, Yuri; Robinson, Julia

doi:10.1090/pspum/028.2/0432534

Cited by 224 publications

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“…Matijasevich. (See [3] and [4].) Since the time when this result was obtained, similar questions have been raised for other fields and rings.…”

Section: Introductionmentioning

confidence: 59%

On Diophantine definability and decidability in some infinite totally real extensions of ℚ

Shlapentokh

2003

Trans. Amer. Math. Soc.

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Abstract. Let M be a number field, and W M a set of its non-Archimedean. . , pr} be a finite set of prime numbers. Let F inf be the field generated by all the p j i -th roots of unity as j → ∞ and i = 1, . . . , r. Let K inf be the largest totally real subfield of F inf . Then for any ε > 0, there exist a number field M ⊂ K inf , and a set W M of non-Archimedean primes of M such that W M has density greater than 1 − ε, and Z has a Diophantine definition over the integral closure of O M,W M in K inf .

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“…Matijasevich. (See [3] and [4].) Since the time when this result was obtained, similar questions have been raised for other fields and rings.…”

Section: Introductionmentioning

confidence: 59%

On Diophantine definability and decidability in some infinite totally real extensions of ℚ

Shlapentokh

2003

Trans. Amer. Math. Soc.

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“…QFPAbit) by in- Note that those results only hold for fixed-size bit-vector logics. For example, allowing multiplication (in combination with addition) makes non-fixed-size bit-vector logics undecidable [22]. We are not aware of any complexity results concerning non-fixed-size bit-vector logics with slicing or shift by constant.…”

Section: Discussionmentioning

confidence: 99%

More on the Complexity of Quantifier-Free Fixed-Size Bit-Vector Logics with Binary Encoding

Fröhlich

Kovásznai

Biere

2013

Computer Science – Theory and Applications

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Abstract. Bit-precise reasoning is important for many practical applications of Satisfiability Modulo Theories (SMT). In recent years, efficient approaches for solving fixed-size bit-vector formulas have been developed. From the theoretical point of view, only few results on the complexity of fixed-size bit-vector logics have been published. Most of these results only hold if unary encoding on the bit-width of bit-vectors is used. In previous work [1], we showed that binary encoding adds more expressiveness to bit-vector logics, e.g. it makes fixed-size bit-vector logic without uninterpreted functions nor quantification NExpTime-complete. In this paper, we look at the quantifier-free case again and propose two new results. While it is enough to consider logics with bitwise operations, equality, and shift by constant to derive NExpTime-completeness, we show that the logic becomes PSpace-complete if, instead of shift by constant, only shift by 1 is permitted, and even NP-complete if no shifts are allowed at all.

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“…It is only in this case that Peerose is justified in somewhat loosely defining the Entscheidungsproblem as referring to "all the problems of mathematics" (p. 34). Penrose's conflation of the Entscheidungsproblem with Hilbert's 10 th problem of 1900, which merely asked for an algorithm for the solvability of Diophantine equations, is likewise justified after the fact: It is a corollary of the methods used to give a negative solution to Hilbert's tenth problem that the question of whether any given Turing machine will eventually halt, and hence the Entscheidungsproblem, can be encoded as a Diophantine problem (Davis et al 1976 What minds can do, Penrose claims, is to see or judge that certain mathematical propositions are true by "Insight" rather than mechanical proof-Penrose then argues that there could be no algorithm, or at any rate no practical algorithm, for Insight. This Ignores an Independently plausible possibility: The algorithms that minds use for judging mathematical truth are not algorithms "for" Insight -but they nevertheless work very well.…”

Section: Is Mathematical Insight Algorithmic? Martin Davismentioning

confidence: 99%

Physics of brain-mind interaction

Eccles

1990

Behav Brain Sci

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Hilbert’s tenth problem: Diophantine equations: positive aspects of a negative solution

Cited by 224 publications

References 0 publications

On Diophantine definability and decidability in some infinite totally real extensions of ℚ

On Diophantine definability and decidability in some infinite totally real extensions of ℚ

More on the Complexity of Quantifier-Free Fixed-Size Bit-Vector Logics with Binary Encoding

Physics of brain-mind interaction

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