Single-tape and multi-tape Turing machines through the lens of the Grossone methodology

Sergeyev, Yaroslav D.; Garro, Alfredo

doi:10.1007/s11227-013-0894-y

Cited by 48 publications

(10 citation statements)

References 39 publications

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“…Here we give the formal description of cellular automata with anticipation. First of all we follow the papers and books on CA (see for example [1][2][3][4][5][6][7]). We pose here the description of one-dimensional CA from [7, p.2]…”

Section: Cellular Automata With Anticipation 21 Classical Cellular Amentioning

confidence: 99%

Multivaluedness in Cellular Automata with Strong Anticipation and Some Prospects for Computation Theory

Makarenko¹

2020

Wseas Transactions on Information Science and Applications

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It is considered the general formulations and properties of cellular automata with cells which have the strong anticipatory property (introduced by D. Dubois). Multivalued behavior (hyperincursion) of solutions of such CA is described. It was posed new research problems of computation theory related to presumable multivaluedness of cellular automata with strong anticipation property. Extending of classical automata, Turing machine and algorithms had been proposed. Also some relation of such cellular automata and quantum mechanics are discussed.

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Section: Cellular Automata With Anticipation 21 Classical Cellular Amentioning

confidence: 99%

Multivaluedness in Cellular Automata with Strong Anticipation and Some Prospects for Computation Theory

Makarenko¹

2020

Wseas Transactions on Information Science and Applications

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“…In particular, metamathematical investigations on the new theory and its non-contradictory can be found in [17]. The x-based methodology has been successfully applied in several areas of Mathematics and Computer Science: single and multiple criteria optimization (see [21,22,23,24,25]), cellular automata (see [26,27]), Euclidean and hyperbolic geometry (see [28,29]), percolation (see [30]), fractals (see [31,32,33,34,35]), infinite series and the Riemann zeta function (see [36,37,38,39,40]), the first Hilbert problem, Turing machines, and supertasks (see [41,42,20,43]), numerical differentiation and numerical solution of ordinary differential equations (see [44,45,46,47,48]), etc. In this paper, divergent series and Ramanujan summation are studied.…”

Section: Introductionmentioning

confidence: 99%

Numerical infinities applied for studying Riemann series theorem and Ramanujan summation

Sergeyev¹

2018

AIP Conference Proceedings

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A computational methodology called Grossone Infinity Computing introduced with the intention to allow one to work with infinities and infinitesimals numerically has been applied recently to a number of problems in numerical mathematics (optimization, numerical differentiation, numerical algorithms for solving ODEs, etc.). The possibility to use a specially developed computational device called the Infinity Computer (patented in USA and EU) for working with infinite and infinitesimal numbers numerically gives an additional advantage to this approach in comparison with traditional methodologies studying infinities and infinitesimals only symbolically. The grossone methodology uses the Euclid's Common Notion no. 5 'The whole is greater than the part' and applies it to finite, infinite, and infinitesimal quantities and to finite and infinite sets and processes. It does not contradict Cantor's and non-standard analysis views on infinity and can be considered as an applied development of their ideas. In this paper we consider infinite series and a particular attention is dedicated to divergent series with alternate signs. The Riemann series theorem states that conditionally convergent series can be rearranged in such a way that they either diverge or converge to an arbitrary real number. It is shown here that Riemann's result is a consequence of the fact that symbol ∞ used traditionally does not allow us to express quantitatively the number of addends in the series, in other words, it just shows that the number of summands is infinite and does not allows us to count them. The usage of the grossone methodology allows us to see that (as it happens in the case where the number of addends is finite) rearrangements do not change the result for any sum with a fixed infinite number of summands. There are considered some traditional summation techniques such as Ramanujan summation producing results where to divergent series containing infinitely many positive integers negative results are assigned. It is shown that the careful counting of the number of addends in infinite series allows us to avoid this kind of results if grossone-based numerals are used.

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“…A number of papers connecting the new approach to the historical panorama of ideas dealing with infinities and infinitesimals (see [22,25,43]) has been published, and metamathematical investigations on the new theory and its non-contradictory can be found in [23,42]. This computational methodology has already been successfully applied in optimization and numerical differentiation (see [6,7,36,48]) and in a number of other theoretical and computational research areas such as cellular automata (see [4,5]), percolation (see [20,21,47]), fractals (see for instance [32,34,37,41,47]), Turing machines and supertasks (see [29,43,44]), numerical solution of ordinary differential equations (see [1,26,38], along with [45]). …”

Section: Introduction To the Algebra Of Grossonementioning

confidence: 99%

Planar methods and grossone for the Conjugate Gradient breakdown in nonlinear programming

2017

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1952), in the presence of degenerates on indefinite linear systems. Several approaches have been proposed in the literature to issue the latter drawback in optimization frameworks, including reformulating the original linear system or recurring to approximately solving it. All the proposed alternatives seem to rely on algebraic considerations, and basically pursue the idea of improving numerical efficiency. In this regard, here we sketch two separate analyses for the possible CG degeneracy. First, we start detailing a more standard algebraic viewpoint of the problem, suggested by planar methods. Then, another algebraic perspective is detailed, relying on a novel recently proposed theory, which includes an additional number, namely grossone. The use of grossone allows to work numerically with infinities and infinitesimals. The results obtained using the two proposed approaches perfectly match, showing that grossone may represent a fruitful and promising tool to be exploited within Nonlinear Programming. B Giovanni Fasano

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Single-tape and multi-tape Turing machines through the lens of the Grossone methodology

Cited by 48 publications

References 39 publications

Multivaluedness in Cellular Automata with Strong Anticipation and Some Prospects for Computation Theory

Multivaluedness in Cellular Automata with Strong Anticipation and Some Prospects for Computation Theory

Numerical infinities applied for studying Riemann series theorem and Ramanujan summation

Planar methods and grossone for the Conjugate Gradient breakdown in nonlinear programming

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