# Infinite chess and the theory of infinite games, Dartmouth Mathematics Colloquium, January 2014

This will be a talk for the Dartmouth Mathematics Colloquium on January 23rd, 2014.

Abstract. Using infinite chess as a central example—chess played on an infinite edgeless board—I shall give a general introduction to the theory of infinite games. Infinite chess is an example of what is called an open game, a potentially infinite game which when won is won at a finite stage of play, and every open game admits the theory of transfinite ordinal game values. These values provide a measure of the distance remaining to an actual victory, and when they are known, the game values provide a canonical winning strategy for the winning player. I shall exhibit

several interesting positions in infinite chess with high transfinite game values. The precise value of the omega one of chess, however, the supremum of all such ordinal game values, is an open mathematical question; in the case of infinite three-dimensional chess, meanwhile, Evans and I have proved that every countable ordinal arises as a game value. Infinite chess also illustrates an interesting engagement with computability issues. For example, there are computable infinite positions in infinite chess that are winning for white, provided that the players play according to a computable procedure of their own choosing, but which is no longer winning for white when non-computable play is allowed. Also, the mate-in-n problem for finite positions in infinite chess is computably decidable (joint work with Schlicht, Brumleve and myself), despite the high quantifier complexity of any straightforward representation of it. The talk will be generally accessible for mathematicians, particularly those with at least rudimentary knowledge of ordinals and of chess.

# Playful paradox with large numbers, infinity and logic, Shanghai, June 2013

This will be a talk at Fudan University in Shanghai, China, June 12, 2013, sponsored by the group in Mathematical Logic at Fudan, for a large audience of students.

Abstract: For success in mathematics and science, I recommend an attitude of playful curiosity about one’s subject. We shall accordingly explore a number of puzzling conundrums at the foundations of mathematics concerning issues with large numbers, infinity and logic. These are serious issues—and we’ll have serious things to say—while still having fun. Can one complete a task involving infinitely many steps? Are there some real numbers that in principle cannot be described? Is every true statement provable? Does every mathematical problem ultimately reduce to computational procedure? What is the largest natural number that can be written or described in ordinary type on an index card? Which is bigger, a googol-bang-plex or a googol-plex-bang? Is every natural number interesting? Is every sentence either true or false or neither true nor false? We will explore these and many other puzzles and paradoxes involving large numbers, logic and infinity, and along the way, learn some interesting mathematics and philosophy.   The Largest-Number Contest.  In preparation for the talk, and with a nod to Douglas Hofstadter, we shall be holding a contest:  Who can describe the largest number on an ordinary index card?   See the contest announcement poster.

1. A submission entry consists of the description of a positive integer written on an ordinary index card, using common mathematical operations and notation or English words and phrases.
2. Write legibly, and use at most 100 symbols from the ordinary ASCII character set.  Bring your submission to the talk.
3. Descriptions that fail to describe a number are disqualified.
4. The submission with the largest number wins.
5. The prize will be $1 million USD divided by the winning number itself, rounded to the nearest cent, plus another small token prize. Example submissions: 99999. 10*(10*99)+5 The population of Shanghai at this moment. # The theory of infinite games, with examples, including infinite chess This will be a talk on April 30, 2013 for a joint meeting of the Yeshiva University Mathematics Club and the Yeshiva University Philosophy Club. The event will take place in 5:45 pm in Furst Hall, on the corner of Amsterdam Ave. and 185th St. Abstract. I will give a general introduction to the theory of infinite games, suitable for mathematicians and philosophers. What does it mean to play an infinitely long game? What does it mean to have a winning strategy for such a game? Is there any reason to think that every game should have a winning strategy for one player or another? Could there be a game, such that neither player has a way to force a win? Must every computable game have a computable winning strategy? I will present several game paradoxes and example infinitary games, including an infinitary version of the game of Nim, and several examples from infinite chess. # The mate-in-n problem of infinite chess is decidable, Cambridge, June 2012 This will be a contributed talk at the Turing Centenary Conference CiE 2012 held June 18-23, 2012 in Cambridge, UK. Abstract. The mate-in-$n$problem of infinite chess—chess played on an infinite edgeless board—is the problem of determining whether a designated player can force a win from a given finite position in at most$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 mate-in-$n$positions. The proof proceeds by showing that the mate-in-$n$problem is expressible in what we call the first-order structure of chess$\frak{Ch}$, 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 mate-in-$n$problem does not appear to settle the decidability of the more general winning-position 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. # Infinite chess: the mate-in-n problem is decidable and the omega-one of chess, Cambridge, March 2012 I have just taken up a visiting fellow position at the Isaac Newton Institute for mathematical sciences in Cambridge, UK, where I am participating in the program Syntax and Semantics: the legacy of Alan Turing. I was asked to give a brief introduction to some of my current work, and I chose to speak about infinite chess. Infinite chess is chess played on an infinite edgeless chessboard. The familiar chess pieces move about according to their usual chess rules, and each player strives to place the opposing king into checkmate. The mate-in-$n$problem of infinite chess is the problem of determining whether a designated player can force a win from a given finite position in at most$n$moves. A naive formulation of this problem leads to assertions of high arithmetic complexity with$2n$alternating quantifiers—there is a move for white, such that for every black reply, there is a countermove for white, and so on. In such a formulation, the problem does not appear to be decidable; and one cannot expect to search an infinitely branching game tree even to finite depth. Nevertheless, in joint work with Dan Brumleve and Philipp Schlicht, confirming a conjecture of myself and C. D. A. Evans, we establish that the mate-in-$n$problem of infinite chess is computably decidable, uniformly in the position and in$n$. Furthermore, there is a computable strategy for optimal play from such mate-in-$n$positions. The proof proceeds by showing that the mate-in-$n$problem is expressible in what we call the first-order structure of chess, which we prove (in the relevant fragment) is an automatic structure, whose theory is therefore decidable. Unfortunately, this resolution of the mate-in-n problem does not appear to settle the decidability of the more general winning-position 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. I will also discuss recent joint work with C. D. A. Evans and W. Hugh Woodin showing that the omega one of infinite positions in three-dimensional infinite chess is true$\omega_1$: every countable ordinal is realized as the game value of such a position. # The mate-in-n problem of infinite chess is decidable • D. Brumleve, J. D. Hamkins, and P. Schlicht, “The Mate-in-$n$Problem of Infinite Chess Is Decidable,” in How the World Computes, S. Cooper, A. Dawar, and B. Löwe, Eds., Springer Berlin Heidelberg, 2012, vol. 7318, pp. 78-88. @incollection{BrumleveHamkinsSchlicht2012:TheMateInNProblemOfInfiniteChessIsDecidable, year= {2012}, isbn= {978-3-642-30869-7}, booktitle= {How the World Computes}, volume= {7318}, series= {Lecture Notes in Computer Science}, editor= {Cooper, S.~Barry and Dawar, Anuj and L{\"o}we, Benedikt}, doi= {10.1007/978-3-642-30870-3_9}, title= {The Mate-in-$n$Problem of Infinite Chess Is Decidable}, url= {}, publisher= {Springer Berlin Heidelberg}, author= {Brumleve, Dan and Hamkins, Joel David and Schlicht, Philipp}, pages= {78-88}, eprint = {1201.5597}, archivePrefix = {arXiv}, primaryClass = {math.LO}, } Infinite chess is chess played on an infinite edgeless chessboard. The familiar chess pieces move about according to their usual chess rules, and each player strives to place the opposing king into checkmate. The mate-in-$n$problem of infinite chess is the problem of determining whether a designated player can force a win from a given finite position in at most$n$moves. A naive formulation of this problem leads to assertions of high arithmetic complexity with$2n$alternating quantifiers—*there is a move for white, such that for every black reply, there is a countermove for white*, and so on. In such a formulation, the problem does not appear to be decidable; and one cannot expect to search an infinitely branching game tree even to finite depth. Nevertheless, the main theorem of this article, confirming a conjecture of the first author and C. D. A. Evans, establishes that the mate-in-$n$problem of infinite chess is computably decidable, uniformly in the position and in$n$. Furthermore, there is a computable strategy for optimal play from such mate-in-$n$positions. The proof proceeds by showing that the mate-in-$n$problem is expressible in what we call the first-order structure of chess, which we prove (in the relevant fragment) is an automatic structure, whose theory is therefore decidable. Unfortunately, this resolution of the mate-in-$n$problem does not appear to settle the decidability of the more general winning-position 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^{\frak{Ch}}\$ is not known.

Richard Stanley’s question on mathoverflow: Decidability of chess on infinite board?