What a pleasure it was to be interviewed by Evelyn Lamb and Kevin Knudson for their wonderful podcast series, My Favorite Theorem, available on Apple, Spotify, and any number of other aggregators.

I had a chance to talk about one my most favorite theorems, the fundamental theorem of finite games.

Theorem.(Zermelo 1913) In any two-player finite game of perfect information, one of the players has a winning strategy, or both players have drawing strategies.

Listen to the podcast here: My Favorite Theorem. A transcript is also available.

Consider this fascinating vision of recursive chess — the etching below was created by Django Pinter, a tutorial student of mine who has just completed his degree in the PPL course here at Oxford, given to me as a parting gift at the conclusion of his studies. Django’s idea was to play chess, but in order for a capture to occur successfully on the board, as here with the black Queen attempting to capture the opposing white knight, the two pieces would themselves sit down for their own game of (recursive) chess. The idea was that the capture would be successful only in the event that the subgame was won. Notice in the image that not only is there a smaller recursive game of chess, but also a still tinier subrecursive game below that (if you inspect closely), while at the same time larger pieces loom in the background, indicating that the main board itself is already several levels deep in the recursion.

The recursive chess idea may seem clear enough initially, and intriguing. But with further reflection, we might wonder how does it work exactly? What precisely is the game of recursive chess? This is my question here.

My goal with this post is to open a discussion about what ultimately should be the precise the rules and operations of recursive chess. I’m not completely sure what the best rule set will be, although I do offer several proposals here. I welcome further proposals, commentary, suggestions, and criticism about how to proceed. Once we settle upon a best or most natural version of the game, then I shall hope perhaps to prove things about it. Will you help me decide what is the game of recursive chess?

Let me describe several natural proposals:

Naïve recursion. Although this seems initially to be a simple, sound proposal, ultimately I find it problematic. The naïve idea is that when one piece wants to capture another in the game at hand, then the two pieces themselves play a game of chess, starting from the usual chess starting position. I would find it natural that the attacking piece should play as white in this game, going first, and if that player wins the subgame, then the capture in the current game is successful. If the subgame is a loss, then the capture is unsuccessful.

There seem, however, to be a variety of ways to handle the losing subgame outcome, and since these will apply with several of the other proposals, let me record them here:

Failed-capture. If the subgame is lost, then the capture in the current game simply does not occur. Both pieces remain on their original squares, and the turn of play passes to the opponent. Notice that this will have serious affects in certain chess situations involving zugswang, a position where a player has no good move — because if one of them is a capture, then the player can aim to play badly in the subgame and thereby legally pass the turn of play to the opponent without having made any move. This situation will in effect cause the subgame players to attempt to lose, rather than win, which seems undesirable.

Failed-capture-with-penalty. If the subgame is lost, then the capture does not occur, but furthermore, the attacking piece is itself lost, removed from the board, and the turn of play passes to the opponent. In effect, under this rule, every attempt at capture is putting the life of the capturing piece at risk, which makes a certain amount of sense from a military point of view. I think perhaps this is a good rule.

Failed-capture-with-retry. If the subgame is lost, then the capture does not occur, but both pieces remain on their original squares, and the attacking player is allowed to proceed with another (different) move. Attempting the same attack from the same board position multiple times is subject to the three-fold repetition rule. This interpretation amounts in effect to the game play searching a large part of the game tree, exploring the possible capturing moves, but with the first successful option fixed as official. It invites manipulation by the opponent, who might play badly against a misguided capture attempt, causing it to be fixed as the official move.

Drawn subgame. A further complication arises from the fact that the subgame can itself be drawn, rather than won. Is this sufficient to cause the penalty or the retry? Or does this count as a failed capture?

As I see it, however, the principal problem with the naïve recursion rule is that it seems to be ill-founded. After all, we can have a game with a capture, which leads to a subgame with a capture, which leads to a deeper subgame with a capture, and so on descending forever. How is the outcome determined in this infinitely descending situation? None of the subgames is ever resolved with a definite conclusion until all of them are, and there seems no coherent way to assign resolutions to them. All infinitely many subgames are simply left hanging mid-play, and indeed mid-move. For this reason, the naïve recursion idea seems ultimately incoherent to me as a game rule.

What we would seem to need instead is a well-founded recursion, one which would ultimately bottom-out in a base case. With such a recursion, every outcome of the game would be well-defined. Such a well-founded recursion would be achieved, for example, if on every subgame there were strictly fewer pieces. Eventually, the subgames would reduce to king versus king, a drawn game, and then the drawn subgame rule would be invoked to whatever affect it cause. But the recursion would definitely terminate. And perhaps most recursions would terminate because the stronger player was ultimately mating in all his attacks, without requiring any invocation of the drawn subgame rule.

We can easily describe several natural subgame positions with one fewer piece. For example, when one piece attacks another, we may naturally consider the positions that would result if we performed the capture, or if we removed the attacking piece; and we might further consider swapping the roles of the players in these positions. Such cases would constitute a well-founded recursion, because the subgame position would have fewer pieces than the main position. In this way, we arrive at several natural recursion rules for recursive chess.

Proof-of-sufficiency recursion. The motivating idea of this recursion rule is that in order for an attack to be successful, the attacking player must prove that it will be sufficient for the attack to be successful. So, when piece A attacks piece B in the game, then a subgame is set up from the position that would result if A were successfully to capture B, and the players proceed in the game in which the attack has occurred. This is the same as proceeding in the main game, under the assumption that the attack was successful. If the attacking player can win this subgame, then it shows in a sense the sufficiency of the attack as a means to win, and so on the proof-of-sufficiency idea, we allow this as a reason for the attack to have been successful.

One might object that this recursion seems at first to amount to just playing chess as usual, since the subgame is the same as the original game with the attack having succeeded. But there is a subtle difference. For a misguided attack, the attacked player can play suboptimally in the subgame, intentionally losing that game, and then play differently in the main game. There is, of course, no obligation that the players respond the same at the higher-level games as in the lower games, and this is all part of their strategic calculation.

Proof-of-necessity recursion. The motivating idea of this recursion rule, in contrast, is that in order for an attack to be successful, the attacking player must prove that it is necessary that the attack take place. When piece A attacks piece B in the main game, then a subgame is set up in which the attack has not succeeded, but instead the attacking piece is lost, but the color sides of the players are swapped. If a black Queen attacks a white knight, for example, then in the subgame position, the queen is removed, and the players proceed from that positions, but with the opposite colors. By winning this subgame, the attacking player is in effect demonstrating that if the attack were to fail, then it would be devastating for them. In other words, they are demonstrating the necessity of the success of the attack.

For the both the proof-of-sufficiency and the proof-of-necessity versions of the recursion, it seems to me that any of the three failed-capture rules would be sensible. And so we have quite a few different interpretations here for what is the game of recursive chess.

What is your proposal? Please let me know in the comments. I am interested to hear any comments or criticism.

I’d like to explain to you how to draw chessboards by hand in perfect perspective, using only a straightedge. In this post, I’ll explain how to construct chessboards of any size, starting with the size of the basic unit square.

This post follows up on the post I made yesterday about how to draw a chessboard in perspective view, using only a straightedge. That method was a subdivision method, where one starts with the boundary of the desired board, and then subdivides to make a chessboard. Now, we start with the basic square and build up. This method is actually quite efficient for quickly making very large boards in perspective view.

I want to emphasize that this is something that you can actually do, right now. It’s fun! All you need is a piece of paper, a pencil and a straightedge. I’ll wait right here while you gather your materials. Use a ruler or a chop stick (as I did) or the edge of a notebook or the lid of a box. Sit at your table and draw a huge chessboard in perspective. You can totally do this.

Start with a horizon, having two points at infinity (orange), at left and right, and a third point midway between them (brown), which we will call the diagonal infinity. Also, mark the front corner of your chess board (blue).

Extend the front corner to the points at infinity. And then mark off (red) a point that will be a measure of the grid spacing in the chessboard. This will the be size of the front square.

You can extend that point to infinity at the right. This delimits the first rank of the chessboard.

Next, extend the front corner of the board to the diagonal infinity.

The intersection of that diagonal with the previous line determines a point, which when extended to infinity at the left, produces the first square of the chessboard.

And that line determines a new point on the leading rank edge. Extend that point up to the diagonal infinity, which determines another point on the second rank line.

Extend that line to infinity at the left, which determines another point on the leading rank edge.

Continuing in this way, one can produce as many first rank squares as desired. Go ahead and do that. At each step, you extend up to the diagonal infinity, which determines a new point, which when extended to infinity at the left determines another point, and so on.

If you should now reflect on the current diagram, you may notice that we have actually determined many further points in the grid than we have mentioned — and thanks to my daughter Hypatia for noticing this simplification — for there is a whole triangle of further intersection points between the files and the diagonals.

We can use these points (and we do not need them all) to construct the rest of the board, by drawing out the lines to infinity at the right. Thus, we construct the whole chessboard:

One can construct a perspective chessboard of any size this way, and one can simply continue with the construction and make it larger, if desired.

It will look a little better if you add a point at infinity down below (and do so directly below the diagonal point at infinity, but a good distance down below the board), and extend the board downward one level. The corresponding diagram on yesterday’s post might be helpful.

You can now color the tile pattern, and you’ll have a chessboard in perfect perspective view.

If you keep going, you can make extremely large chessboards. In time, I hope that you will come to learn how to complete an infinite chess board in finite time.

Abstract. We present a position in infinite chess exhibiting an ordinal game value of $\omega^4$, thereby improving on the previously largest-known values of $\omega^3$ and $\omega^3\cdot 4$.

This is a joint work with Cory Evans and Norman Perlmutter, continuing the research program of my previous article with Evans, Transfinite game values in infinite chess, namely, the research program of finding positions in infinite chess with large transfinite ordinal game values. In the previous article, Cory and I presented a position with game value $\omega^3$. In the current paper, with Norman Perlmutter now having joined us accompanied by some outstanding ideas, we present a new position having game value $\omega^4$, breaking the previous record.

A position in infinite chess with value $\omega^4$

In the new position, above, the kings sit facing each other in the throne room, an uneasy détente, while white makes steady progress in the rook towers. Meanwhile, at every step black, doomed, mounts increasingly desperate bouts of long forced play using the bishop cannon battery, with bishops flying with force out of the cannons, and then each making a long series of forced-reply moves in the terminal gateways. Ultimately, white wins with value omega^4, which exceeds the previously largest known values of omega^3.

In the throne room, if either black or white places a bishop on the corresponding diagonal entryway, then checkmate is very close. A key feature is that for white to place a white-square white bishop on the diagonal marked in red, it is immediate checkmate, whereas if black places a black-square black bishop on the blue diagonal, then checkmate comes three moves later. The bishop cannon battery arrangement works because black threatens to release a bishop into the free region, and if white does not reply to those threats, then black will be three steps ahead, but otherwise, only two.

The throne room

The rook towers are similar to the corresponding part of the previous $\omega^3$ position, and this is where white undertakes most of his main line progress towards checkmate. Black will move the key bishop out as far as he likes on the first move, past $n$ rook towers, and the resulting position will have value $\omega^3\cdot n$. These towers are each activated in turn, leading to a long series of play for white, interrupted at every opportunity by black causing a dramatic spectacle of forced-reply moves down in the bishop cannon battery.

The rook towers

At every opportunity, black mounts a long distraction down in the bishop cannon battery. Shown here is one bishop cannon. The cannonballs fire out of the cannon with force, in the sense that when each green bishop fires out, then white must reply by moving the guard pawns into place.

Bishop cannon

Upon firing, each bishop will position itself so as to attack the entrance diagonal of a long bishop gateway terminal wing. This wing is arranged so that black can make a series of forced-reply threats successively, by moving to the attack squares (marked with the blue squares). Black is threatening to exit through the gateway doorway (in brown), but white can answer the threat by moving the white bishop guards (red) into position. Thus, each bishop coming out of a cannon (with force) can position itself at a gateway terminal of length $g$, making $g$ forced-reply moves in succession. Since black can initiate firing with an arbitrarily large cannon, this means that at any moment, black can cause a forced-reply delay with game value $\omega^2$. Since the rook tower also has value $\omega^2$ by itself, the overall position has value $\omega^4=\omega^2\cdot\omega^2$.

Bishop gateway terminal wing

With future developments in mind, we found that one can make a more compact arrangement of the bishop cannon battery, freeing up a quarter board for perhaps another arrangement that might lead to a higher ordinal values.

Alternative compact version of bishop cannon battery

Read more about it in the article, which is available at the arxiv (pdf).

Abstract. I shall give a general introduction to the theory of infinite games, with a focus on the theory of transfinite ordinal game values. These ordinal game values can be used to show that every open game — a game that, when won for a particular player, is won after finitely many moves — has a winning strategy for one of the players. By means of various example games, I hope to convey the extremely concrete game-theoretic meaning of these game values for various particular small infinite ordinals. Some of the examples will be drawn from infinite chess, which is chess played on a chessboard stretching infinitely without boundary in every direction, and the talk will include animations of infinite chess positions having large numbers of pieces (or infinitely many) with hundreds of pieces making coordinated attacks on the chessboard. Meanwhile, the exact value of the omega one of chess, denoted $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$, is not currently known.

Abstract. I shall give a general introduction to the theory of infinite games, using infinite chess—chess played on an infinite chessboard stretching without bound in every direction—as a central example. Since chess, when won, is always won at a finite stage of play, infinite chess is an example of what is known technically as an open game, and such games admit the theory of transfinite ordinal game values, which provide a measure in a position of the distance remaining to victory. I shall exhibit several interesting positions in infinite chess with very high transfinite ordinal game values. Some of these positions involve large numbers of pieces, and the talk will include animations of infinite chess in play, with hundreds of pieces (or infinitely many) making coordinated attacks on the board. Meanwhile, the precise ordinal value of the omega one of chess is an open mathematical question.

Abstract. I shall give a general introduction to the theory of infinite games, using infinite chess—chess played on an infinite edgeless chessboard—as a central example. Since chess, when won, is won at a finite stage of play, infinite chess is an example of what is known technically as an open game, and such games admit the theory of transfinite ordinal game values. I shall exhibit several interesting positions in infinite chess with very high transfinite game values. The precise value of the omega one of chess is an open mathematical question.

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.

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.

C.~D.~A.~Evans and J. D. Hamkins, “Transfinite game values in infinite chess,” Integers, vol. 14, p. Paper No.~G2, 36, 2014. [Bibtex]

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In this article, C. D. A. Evans and I investigate the transfinite game values arising in infinite chess, providing both upper and lower bounds on the supremum of these values—the omega one of chess—denoted by $\omega_1^{\mathfrak{Ch}}$ in the context of finite positions and by $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$ in the context of all positions, including those with infinitely many pieces. For lower bounds, we present specific positions with transfinite game values of $\omega$, $\omega^2$, $\omega^2\cdot k$ and $\omega^3$. By embedding trees into chess, we show that there is a computable infinite chess position that is a win for white if the players are required to play according to a deterministic computable strategy, but which is a draw without that restriction. Finally, we prove that every countable ordinal arises as the game value of a position in infinite three-dimensional chess, and consequently the omega one of infinite three-dimensional chess is as large as it can be, namely, true $\omega_1$.

The article is 38 pages, with 18 figures detailing many interesting positions of infinite chess. My co-author Cory Evans holds the chess title of U.S. National Master.

Let’s display here a few of the interesting positions.

First, a simple new position with value $\omega$. The main line of play here calls for black to move his center rook up to arbitrary height, and then white slowly rolls the king into the rook for checkmate. For example, 1…Re10 2.Rf5+ Ke6 3.Qd5+ Ke7 4.Rf7+ Ke8 5.Qd7+ Ke9 6.Rf9#. By playing the rook higher on the first move, black can force this main line of play have any desired finite length. We have further variations with more black rooks and white king.

Next, consider an infinite position with value $\omega^2$. The central black rook, currently attacked by a pawn, may be moved up by black arbitrarily high, where it will be captured by a white pawn, which opens a hole in the pawn column. White may systematically advance pawns below this hole in order eventually to free up the pieces at the bottom that release the mating material. But with each white pawn advance, black embarks on an arbitrarily long round of harassing checks on the white king.

Here is a similar position with value $\omega^2$, which we call, “releasing the hordes”, since white aims ultimately to open the portcullis and release the queens into the mating chamber at right. The black rook ascends to arbitrary height, and white aims to advance pawns, but black embarks on arbitrarily long harassing check campaigns to delay each white pawn advance.

Next, by iterating this idea, we produce a position with value $\omega^2\cdot 4$. We have in effect a series of four such rook towers, where each one must be completed before the next is activated, using the “lock and key” concept explained in the paper.

We can arrange the towers so that black may in effect choose how many rook towers come into play, and thus he can play to a position with value $\omega^2\cdot k$ for any desired $k$, making the position overall have value $\omega^3$.

Another interesting thing we noticed is that there is a computable position in infinite chess, such that in the category of computable play, it is a win for white—white has a computable strategy defeating any computable strategy of black—but in the category of arbitrary play, both players have a drawing strategy. Thus, our judgment of whether a position is a win or a draw depends on whether we insist that players play according to a deterministic computable procedure or not.

The basic idea for this is to have a computable tree with no computable infinite branch. When black plays computably, he will inevitably be trapped in a dead-end.

In the paper, we conjecture that the omega one of chess is as large as it can possibly be, namely, the Church-Kleene ordinal $\omega_1^{CK}$ in the context of finite positions, and true $\omega_1$ in the context of all positions.

Our idea for proving this conjecture, unfortunately, does not quite fit into two-dimensional chess geometry, but we were able to make the idea work in infinite **three-dimensional** chess. In the last section of the article, we prove:

Theorem. Every countable ordinal arises as the game value of an infinite position of infinite three-dimensional chess. Thus, the omega one of infinite three dimensional chess is as large as it could possibly be, true $\omega_1$.

Here is a part of the position. Imagine the layers stacked atop each other, with $\alpha$ at the bottom and further layers below and above. The black king had entered at $\alpha$e4, was checked from below and has just moved to $\beta$e5. Pushing a pawn with check, white continues with 1.$\alpha$e4+ K$\gamma$e6 2.$\beta$e5+ K$\delta$e7 3.$\gamma$e6+ K$\epsilon$e8 4.$\delta$e7+, forcing black to climb the stairs (the pawn advance 1.$\alpha$e4+ was protected by a corresponding pawn below, since black had just been checked at $\alpha$e4).

The overall argument works in higher dimensional chess, as well as three-dimensional chess that has only finite extent in the third dimension $\mathbb{Z}\times\mathbb{Z}\times k$, for $k$ above 25 or so.

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 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. An equivalent account of the result arises from the realization that the structure of chess is interpretable in the standard model of Presburger arithmetic $\langle\mathbb{N},+\rangle$. 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.

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.

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, 2012, vol. 7318, pp. 78-88. [Bibtex]

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title= {The Mate-in-$n$ Problem of Infinite Chess Is Decidable},
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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.