The Mateinn Problem of Infinite Chess Is Decidable
 Dan Brumleve,
 Joel David Hamkins,
 Philipp Schlicht
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Abstract
The mateinn problem of infinite chess—chess played on an infinite edgeless board—is the problem of determining whether a designated player can force a win from a given finite position in at most n moves. Although a straightforward formulation of this problem leads to assertions of high arithmetic complexity, with 2n alternating quantifiers, the main theorem of this article nevertheless confirms a conjecture of the second author and C. D. A. Evans by establishing that it is computably decidable, uniformly in the position and in n. Furthermore, there is a computable strategy for optimal play from such mateinn positions. The proof proceeds by showing that the mateinn problem is expressible in what we call the firstorder structure of chess \(\mathord{\frak{Ch}}\) , which we prove (in the relevant fragment) is an automatic structure, whose theory is therefore decidable. The structure is also definable in Presburger arithmetic. Unfortunately, this resolution of the mateinn problem does not appear to settle the decidability of the more general winningposition problem, the problem of determining whether a designated player has a winning strategy from a given position, since a position may admit a winning strategy without any bound on the number of moves required. This issue is connected with transfinite game values in infinite chess, and the exact value of the omega one of chess \(\omega_1^{\rm chess}\) is not known.
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 Evans, C.D., Hamkins, J.D., Woodin, W.H.: Transfinite game values in infinite chess (in preparation)
 Fraenkel, A.S., Lichtenstein, D. (1981) Computing a perfect strategy for n×n chess requires time exponential in n. J. Combin. Theory Ser. A 31: pp. 199214 CrossRef
 Khoussainov, B., Minnes, M. (2010) Three lectures on automatic structures. Logic Colloquium 2007. Assoc. Symbol. Logic, La Jolla, pp. 132176 CrossRef
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 Stanley (mathoverflow.net/users/2807), R.: Decidability of chess on an infinite board. MathOverflow, http://mathoverflow.net/questions/27967 (version: July 20, 2010)
 Title
 The Mateinn Problem of Infinite Chess Is Decidable
 Book Title
 How the World Computes
 Book Subtitle
 Turing Centenary Conference and 8th Conference on Computability in Europe, CiE 2012, Cambridge, UK, June 1823, 2012. Proceedings
 Pages
 pp 7888
 Copyright
 2012
 DOI
 10.1007/9783642308703_9
 Print ISBN
 9783642308697
 Online ISBN
 9783642308703
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 7318
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
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 Editors

 S. Barry Cooper ^{(16)}
 Anuj Dawar ^{(17)}
 Benedikt Löwe ^{(18)}
 Editor Affiliations

 16. School of Mathematics, University of Leeds
 17. Computer Laboratory, University of Cambridge
 18. Institute for Logic, Language and Computation, University of Amsterdam
 Authors

 Dan Brumleve ^{(19)}
 Joel David Hamkins ^{(20)} ^{(21)} ^{(22)}
 Philipp Schlicht ^{(23)}
 Author Affiliations

 19. Topsy Labs, Inc., 140 Second Street, 6th Floor, San Francisco, CA, 94105, United States of America
 20. Department of Philosophy, New York University, 5 Washington Place, New York, NY, 10003, United States of America
 21. Mathematics, CUNY Graduate Center, The City University of New York, 365 Fifth Avenue, New York, NY, 10016, United States of America
 22. Mathematics, College of Staten Island of CUNY, Staten Island, NY, 10314, United States of America
 23. Mathematisches Institut, Rheinische FriedrichWilhelmsUniversität Bonn, Endenicher Allee 60, 53115, Bonn, Germany
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