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Noncomputable functions in the Blum-Shub-Smale model

Wesley Calvert, Ken Kramer, Russell Miller, Xizhong Zheng
2011 Logical Methods in Computer Science  
Working in the Blum-Shub-Smale model of computation on the real numbers, we answer several questions of Meer and Ziegler.  ...  First, we show that, for each natural number d, an oracle for the set of algebraic real numbers of degree at most d is insufficient to allow an oracle BSS-machine to decide membership in the set of algebraic  ...  Introduction Blum, Shub, and Smale introduced in [2] a notion of computation with full-precision real arithmetic, in which the ordered field operations are axiomatically computable, and the computable  ... 
doi:10.2168/lmcs-7(2:15)2011 fatcat:27plqhu75rbrfjlasygksjthwy

Recursively enumerable subsets of Rq in two computing models Blum-Shub-Smale machine and Turing machine

Ning Zhong
1998 Theoretical Computer Science  
In this paper we compare recursively enumerable subsets of R" in two computing models over real numbers: the Blum-Shub-Smale machine and the oracle Turing machine.  ...  On the other hand, Blum et al. [2] developed a model for computation over ordered rings in 1989. This model provides an interplay between algebra, analysis, scientific computation, and topology.  ...  Both the Blum-ShubSmale model (BSS model) and the Turing model bring the theory of computation into the domain of topology and make it possible to investigate the effectiveness of const~~tions in topology  ... 
doi:10.1016/s0304-3975(97)00008-x fatcat:v2vkfo77ere5hptrtcryohrzoq

Complexities Approach to Two Problems In Number Theory [article]

Yang Bai, Xiuli Wang
2016 arXiv   pre-print
problems in the non-logical discipline.Futhermore,we show that resource-bounded Kolmogorov Complexity and computational complexity can at least provide tips or principles to mathematical problems in the  ...  not belong to P (period number) in their paper,another is the problem about existence of bounded coefficients of continued fraction expansion of transcendental number.Thus we show a new approach to mathematical  ...  , see [8] , [1] for reference.here,the computation model and the definition is from [8] .We adopt some result from [1] , since the two models are not equivalent,and the model by Blum,Cucker,Shub,Smale  ... 
arXiv:1610.04026v3 fatcat:5udl5eurnzh63c6ysh6b7kta6y

Inverse Problems Are Solvable on Real Number Signal Processing Hardware [article]

Holger Boche and Adalbert Fono and Gitta Kutyniok
2022 arXiv   pre-print
This paper investigates the general computation framework of Blum-Shub-Smale (BSS) machines which allows the processing and storage of arbitrary real values.  ...  Our results emphasize the influence of the considered computing model in questions of algorithmic solvability of inverse problems.  ...  Kutyniok was supported in part by the ONE Munich Strategy Forum (LMU Munich, TU Munich, and the Bavarian Ministery for Science and Art) as well as  ... 
arXiv:2204.02066v2 fatcat:efvvpd2z7bbkhdjdyme6eusiaa

Analog computation via neural networks

Hava T Siegelmann, Eduardo D Sontag
1994 Theoretical Computer Science  
In Section 7, we briefly compare our model to the Blum, Shub, and Smale model of computation over the reals.  ...  Complexity over the reals Blum, Shub, and Smale introduced in [4] a powerful model of computation over the real numbers.  ... 
doi:10.1016/0304-3975(94)90178-3 fatcat:nd5xeibb2jec5c7stfcjwd4j5m

Counting problems over the reals

Klaus Meer
2000 Theoretical Computer Science  
The main emphasis of our results is model-theoretic.  ...  We follow the approaches of Blum et al. (1998) for computability over R and of Gr adel and Meer (1996) for descriptive complexity theory in this setting and give a complete characterization of such problems  ...  My particular thanks go to an anonymous referee for his very careful reading of the manuscript. His many valuable and precise comments helped to improve the presentation a lot.  ... 
doi:10.1016/s0304-3975(98)00190-x fatcat:go3liffwpbfqnlxtzkmuu5azoy

Effective completeness for real computation [article]

Wesley Calvert
2009 arXiv   pre-print
The main result of this paper, as previously presented to arxiv, was incorrect. See the full text for details and for reference to the remaining results.  ...  To address both of these concerns, Blum, Cucker, Shub, and Smale introduced the model of real computation [3, 2] (actually, computation over an arbitrary ring; in this paper, we use only the case of  ...  In this model, which will be defined at more length in the following section, addition, multiplication, and comparison in the standard order of real numbers are all axiomatically "computable," and "noncomputability  ... 
arXiv:0807.3558v2 fatcat:h7dezctafbebzjoayqxg2mqjzy

Computing over the Reals: Foundations for Scientific Computing [article]

Mark Braverman, Stephen Cook
2005 arXiv   pre-print
In the introduction we also discuss the alternative Blum-Shub-Smale model. In the final section we discuss the issue of whether physical systems could defeat the Church-Turing Thesis.  ...  We give a detailed treatment of the "bit-model" of computability and complexity of real functions and subsets of R^n, and argue that this is a good way to formalize many problems of scientific computation  ...  For scientific computing the most influential model in the algebraic setting is due to Blum, Shub and Smale (BSS) [BSS89] .  ... 
arXiv:cs/0509042v1 fatcat:m3sdhpoggrakdiklvi7yryiifa

Locally Computable Structures [chapter]

Russell G. Miller
2007 Lecture Notes in Computer Science  
We introduce the notion of a locally computable structure, a natural way of generalizing the notions of computable model theory to uncountable structures S by presenting the finitely generated substructures  ...  First, our notion of mextensional local computability of S ensures that the Σn-theory of S will be Σn for all n ≤ m+1.  ...  These include the bitmap model, detailed in [2] , and the Blum-Shub-Smale model (see [1] ).  ... 
doi:10.1007/978-3-540-73001-9_60 fatcat:rqzoaa2s45hfvbtixxaqmi4fui

Complexity of Linear Problems with a Fixed Output Basis

Erich Novak, Henryk Woźniakowski
2000 Journal of Complexity  
We study the case when the elements g i are fixed and compare it with the case where the g i can be chosen arbitrarily.  ...  It seems reasonable to look for an approximation as a linear combination of some elements g i from the space G and compute only the coefficients c i of this linear combination.  ...  For a formal description see Blum, Shub, and Smale (1989) and Blum et al. (1997) , as well as Novak (1995) and Novak and Woz niakowski (1996) , and also Woz niakowski (1997) for a discussion of this  ... 
doi:10.1006/jcom.1999.0538 fatcat:dmjny3qxrjgwtdnpemzpie6v7q

Beyond the universal Turing machine

B. Jack Copeland, Richard Sylvan
1999 Australasian Journal of Philosophy  
According to the nonclassicist, the set of well-defined computations is not exhausted by the computations that can be carried out by a Turing machine.  ...  We provide an overview of the field and a philosophical defence of its foundations  ...  Interest in Abramson's paper has been stimulated by its rediscovery by Blum, Shub and Smale (1989) .  ... 
doi:10.1080/00048409912348801 fatcat:3g35dt4cfvg6joya3r6ohmkbly

A survey of recursive analysis and Moore's notion of real computation

Walid Gomaa
2011 Natural Computing  
The theory of analog computation aims at modeling computational systems that evolve in a continuous manner.  ...  In 1996 Moore introduced a function algebra that captures his notion of real computation; it consists of some basic functions and their closure under composition, integration and zerofinding.  ...  A dedicated survey about the class of Blum-Shub-Smale (BSS) models was written by K. Meer and C. Michaux [24] ; this survey essentially focuses on the complexity-theoretic aspects of these models.  ... 
doi:10.1007/s11047-011-9278-5 fatcat:2ssd2bqnnrdtzmaacoywcjfnhi

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B. Jack Copeland
2012 Minds and Machines  
A survey of the field of hypercomputation, including discussion of a variety of objections.  ...  (1989) Blum, Shub and Smale described abstract machines for performing sequential computations with arbitrary real numbers (Blum et al., 1989 ; see also Blum et al., 1998) .  ...  BLUM, SHUB AND SMALE Part of their motivation was to 'bring the theory of computation into the domain of analysis, geometry and topology' (1989, p. 1).  ... 
doi:10.1023/a:1021105915386 fatcat:xqjhsdvrpbgd5oiytqwtpeorke

Are Computational Transitions Sensitive to Semantics?

Michael Rescorla
2012 Australasian Journal of Philosophy  
But it does not distinguish vision from numerous noncomputational processes, such as planetary motion, that can also be modelled on a computer.  ...  The calculator's semantic properties inform which number-theoretic function it computes. In that sense, its semantic properties are bona fide computational properties.  ...  Similarly, Blum-Shub-Smale machines are basically register machines that can store arbitrary real numbers.  ... 
doi:10.1080/00048402.2011.615333 fatcat:5wpokkomfzfrlnzfje2geujfmq

A Computational Foundation for the Study of Cognition

David J. Chalmers
2011 Journal of Cognitive Science  
See Blum, Shub and Smale (1989) for a careful working-out of some of the relevant theory of computability.  ...  Some have disputed the premise: for example, Penrose (1989) has speculated that the effects of quantum gravity are noncomputable, and that these effects may play a role in cognitive functioning.  ... 
doi:10.17791/jcs.2011.12.4.325 fatcat:xqahq2erynenjcyhvezuizjiqq
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