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17.07.2020 | Focus

Infinite games on finite graphs using grossone

verfasst von: Louis D’Alotto

Erschienen in: Soft Computing | Ausgabe 23/2020

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Abstract

In his seminal work, Robert McNaughton [see McNaughton (Ann Pure Appl Log 65:149–184, 1993) and Khoussainov and Nerode (Automata theory and its applications. Birkhauser, Basel, 2001)] developed a model of infinite games played on finite graphs. This paper presents a new model of infinite games played on finite graphs using the grossone paradigm. The new grossone paradigm provides certain advantages such as allowing for draws, which are common in board games, and a more accurate and decisive method for determining the winner.

Vorheriger Artikel New algebraic and geometric constructs arising from Fibonacci numbers

Nächster Artikel Discrete uniform and binomial distributions with infinite support

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In Sergeyev (2003), Sergeyev formally presents the divisibility axiom as saying for any finite natural number n sets ${\mathbb {N}}_{k,n}, \; 1\le k\le n$, being the nth parts of the set ${\mathbb {N}}$, have the same number of elements indicated by the numeral $\frac{{\displaystyle {{\textcircled {1}}}}}{n}$ where

$$\begin{aligned} {\mathbb {N}}_{k,n}=\{k,k+n,k+2n,k+3n,\ldots \}, \; 1\le k \le n,\; \bigcup ^n_{k=1}{\mathbb {N}}_{k,n}={\mathbb {N}} \end{aligned}$$

and illustrates this with examples of the odd and even natural numbers.

Here it is also important to note that the grossone methodology is not related to nonstandard analysis [see Sergeyev (2019)].

Here we use the notion of complete taken from Sergeyev (2003), that is the sequence contains ${\displaystyle {{\textcircled {1}}}}$ elements.

Here we take the convention, as found in Halmos (1974), that the $\subset $ symbol is reflexive.

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Titel: Infinite games on finite graphs using grossone
verfasst von: Louis D’Alotto
Publikationsdatum: 17.07.2020
Verlag: Springer Berlin Heidelberg
Erschienen in: Soft Computing / Ausgabe 23/2020
Print ISSN: 1432-7643
Elektronische ISSN: 1433-7479
DOI: https://doi.org/10.1007/s00500-020-05167-1

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