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Infinite games on finite graphs using grossone

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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.

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Notes

  1. 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.

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

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

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

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Acknowledgements

This research was supported by the following Grant: PSC-CUNY Research Award: TRADA-47-445

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Authors and Affiliations

  1. York College, The City University of New York, Jamaica, Queens, New York, 11451, USA

    Louis D’Alotto

  2. The Graduate Center, The City University of New York, 356 Fifth Avenue, New York, NY, 10016, USA

    Louis D’Alotto

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  1. Louis D’Alotto
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Correspondence to Louis D’Alotto.

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Communicated by Yaroslav D. Sergeyev.

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D’Alotto, L. Infinite games on finite graphs using grossone. Soft Comput 24, 17509–17515 (2020). https://doi.org/10.1007/s00500-020-05167-1

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