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International Conference on Numerical Computations: Theory and Algorithms

NUMTA 2019: Numerical Computations: Theory and Algorithms pp 346–353Cite as

Infinite Games on Finite Graphs Using Grossone

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11974))

Abstract

In his seminal work, Robert McNaughton (see [1] and [7]) 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 model 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 when a game is played to infinite duration.

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

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Notes

  1. 1.

    In [15], 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 where

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

    .

  2. 2.

    Here we use the notion of complete taken from [15], that is the sequence containing elements is complete.

  3. 3.

    It is noted here that, as is usual, the \(\subset \) symbol can also imply equality.

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Author information

Authors and Affiliations

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

    Louis D’Alotto

  2. The Graduate Center, The City University of New York, 356 Fifth Avenue, New York City, 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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Editors and Affiliations

  1. University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  2. University of Calabria, Rende, Italy

    Dmitri E. Kvasov

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D’Alotto, L. (2020). Infinite Games on Finite Graphs Using Grossone. In: Sergeyev, Y., Kvasov, D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science(), vol 11974. Springer, Cham. https://doi.org/10.1007/978-3-030-40616-5_29

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  • DOI: https://doi.org/10.1007/978-3-030-40616-5_29

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  • Online ISBN: 978-3-030-40616-5

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