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2020 | OriginalPaper | Chapter

Infinite Games on Finite Graphs Using Grossone

Author : Louis D’Alotto

Published in: Numerical Computations: Theory and Algorithms

Publisher: Springer International Publishing

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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.

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previous chapter Grossone Methodology for Lexicographic Mixed-Integer Linear Programming Problems

next chapter A Novel Geometric Approach to the Problem of Multidimensional Scaling

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

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-40616-5_29/495240_1_En_29_IEq8_HTML.gif

where

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

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

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-40616-5_29/495240_1_En_29_IEq20_HTML.gif

elements is complete.

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

McNaughton, R.: Infinite games played on finite graphs. Ann. Pure Appl. Logic 65, 149–184 (1993)MathSciNetCrossRef

Caldarola, F.: The exact measures of the Sierpiński d-dimensional tetrahedron in connection with a Diophantine nonlinear system. Commun. Nonlinear Sci. Numer. Simul. 63, 228–238 (2018)MathSciNetCrossRef

De Leone, R., Fasano, G., Sergeyev, Ya.D.: Planar methods and grossone for the Conjugate Gradient breakdown in nonlinear programming. Comput. Optim. Appl. 71(1), 73–93 (2018)

De Leone, R.: Nonlinear programming and grossone: quadratic programming and the role of constraint qualifications. Appl. Math. Comput. 318, 290–297 (2018)MATH

D’Alotto, L.: A classification of one-dimensional cellular automata using infinite computations. Appl. Math. Comput. 255, 15–24 (2015)MathSciNetMATH

Fiaschi, L., Cococcioni, M.: Numerical asymptotic results in Game Theory using Sergeyev’s Infinity Computing. Int. J. Unconv. Comput. 14(1), 1–25 (2018)

Khoussainov, B., Nerode, A.: Automata Theory and Its Applications. Birkhauser, Basel (2001)CrossRef

Iudin, D.I., Sergeyev, Ya.D., Hayakawa, M.: Infinity computations in cellular automaton forest-fire model. Commun. Nonlinear Sci. Numer. Simul. 20(3), 861–870 (2015)

Lolli, G.: Infinitesimals and infinites in the history of mathematics: a brief survey. Appl. Math. Comput. 218(16), 7979–7988 (2012) MathSciNetMATH

10.

Lolli, G.: Metamathematical investigations on the theory of Grossone. Appl. Math. Comput. 255, 3–14 (2015)MathSciNetMATH

11.

Margenstern, M.: Using Grossone to count the number of elements of infinite sets and the connection with bijections. p-Adic Numbers Ultrametric Anal. Appl. 3(3), 196–204 (2011)MathSciNetCrossRef

12.

Montagna, F., Simi, G., Sorbi, A.: Taking the Pirahá seriously. Commun. Nonlinear Sci. Numer. Simul. 21(1–3), 52–69 (2015)MathSciNetCrossRef

13.

Rizza, D.: Numerical methods for infinite decision-making processes. Int. J. Unconv. Comput. 14(2), 139–158 (2019)

14.

Rizza, D.: How to make an infinite decision. Bull. Symb. Logic 24(2), 227 (2018)MathSciNetCrossRef

15.

Sergeyev, Y.D.: Arithmetic of Infinity. Edizioni Orizzonti Meridionali, Cosenza (2003)MATH

16.

Sergeyev, Y.D.: A new applied approach for executing computations with infinite and infinitesimal quantities. Informatica 19(4), 567–596 (2008)MathSciNetMATH

17.

Cococcioni, M., Pappalardo, M., Sergeyev, Y.D.: Lexicographic multiobjective linear programming using grossone methodology: theory and algorithm. Appl. Math. Comput. 318, 298–311 (2018)MATH

18.

Sergeyev, Y.D.: Computations with grossone-based infinities. In: Calude, C.S., Dinneen, M.J. (eds.) UCNC 2015. LNCS, vol. 9252, pp. 89–106. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21819-9_6CrossRef

19.

Sergeyev, Ya.D.: The Olympic medals ranks, lexicographic ordering and numerical infinities. Math. Intelligencer 37(2), 4–8 (2015)MathSciNetCrossRef

20.

Sergeyev, Ya.D.: Numerical infinities and infinitesimals: methodology, applications, and repercussions on two Hilbert problems. EMS Surv. Math. Sci. 4(2), 219–320 (2017)MathSciNetCrossRef

21.

Sergeyev, Ya.D., Garro, A.: Observability of Turing Machines: a refinement of the theory of computation. Informatica 21(3), 425–454 (2010)

22.

Zhigljavsky, A.: Computing sums of conditionally convergent and divergent series using the concept of grossone. Appl. Math. Comput. 218, 8064–8076 (2012)MathSciNetMATH

Title: Infinite Games on Finite Graphs Using Grossone
Author: Louis D’Alotto
Publisher: Springer International Publishing
Book: Numerical Computations: Theory and Algorithms
Print ISBN: 978-3-030-40615-8

Electronic ISBN: 978-3-030-40616-5

Copyright Year: 2020
DOI: https://doi.org/10.1007/978-3-030-40616-5_29

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