Author Topic: Perpetual check in Infinite Chess (new proposed rule)  (Read 155 times)

chilipepper

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Perpetual check in Infinite Chess (new proposed rule)
« on: March 18, 2018, 02:58:55 pm »
This topic is concerning the game "Chess on an Infinite Plane" which is one type of infinite chess. For reference the game-rules can be found here:

https://chessvariantforum.createaforum.com/variant-reviews/variant-description-chess-on-an-infinite-plane/

Some game players have noted that there are theoretical end-positions in infinite chess which result in a "perpetual check". In such a position, the king is placed in serial check (checked on every move), with no prospect for conversion to another game-state. In one example, the king is forced to move in the same direction perpetually and without the involved pieces being prevented from changing their move pattern. The pieces move without repetition of absolute position, but there is "pattern" or "translational" repetition.

To my knowledge, this game state has not been seen in any actual games, but is theoretically possible. One example is shown in this example (simplified on an 8x8 board):

[FEN "8/8/K7/8/8/6k1/8/5Q1R w - - 0 1"][SetUp "1"][CurrentPosition "8/8/K7/8/8/6k1/8/5Q1R w - - 0 1"]1.Rh3+ Kg4 2.Qf3+ Kg5 3.Rh5+ Kg6 4.Qf5+ Kg7 5.Rh7+ Kg8 6.Qf7#  *

To address this game-state, a new proposed rule is being considered, stated as follows:

A player may claim a draw if the king is forced to move one square in the same direction for six consecutive moves with the involved pieces exhibiting translational repetition, and if the involved pieces are moving in a way that the game-state will not otherwise be changed (for example the checkmating pieces are blocked by an occupied square, or to a square which is attacked in a way that resolves the check).

The infinite chess committee is accepting public comment on this proposed rule. If there is no objection, the rule will be effective for all games starting May 2 of this year and after.
« Last Edit: March 22, 2018, 10:17:04 am by chilipepper »
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chilipepper

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Elaborating on the example above:

The perpetual check can be expressed as follows (standard board notation with unbounded files and ranks):

Rh3+...Kg4
Qf3+...Kg5
Rh5+...Kg6
Qf5+...Kg7
Rh7+...Kg8
Qf7+...Kg9
Rh9+...Kg10
Qf9+...Kg11
...
(and so on)


With the proposed rule, the game could end as follows:
Rh3+...Kg4  (1st check)
Qf3+...Kg5  (2nd check - 1st instance of forced repetition)
Rh5+...Kg6  (3rd check - 2nd instance of forced repetition)
Qf5+...Kg7  (4th check - 3rd instance of forced repetition)
Rh7+...Kg8  (5th check - 4th instance of forced repetition)
Qf7+...Kg9  (6th check - 5th instance of forced repetition)
Rh9+...Kg10 (7th check - 6th instance of forced repetition, Black claims a draw)

Note: if the king makes a repeated move, but the move is not forced, then there is no forced repetition, and a draw cannot be claimed.
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John_Lewis

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May I suggest a more intuitive number of moves? Off the top of my head I would say 8, because that would push you "off" a normal sized Chess board.

chilipepper

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Thanks for thinking about this situation. When I devised this suggestion, I was first thinking three moves, but then realized the pattern repeats itself every other move. So three cycles of "two-moves" is 6 moves total. So it is basically "3 fold repetition" of the same pattern.

On a normal 8x8 board, this situation resolves itself in five moves or less. The king starts on the third rank or higher, and becomes checkmated after he moves to the 8th rank (hits the edge of the board). But other than that I didn't feel that this number carries over to infinite chess in any way.

On the other hand, I wouldn't be surprised if someone can make a similar pattern where the pattern repeats itself on every third move. There might even be other strange theoretical endings in infinite chess which haven't been thought of. I'm just trying to keep the rule-book one step ahead of the actual games. :)
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John_Lewis

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While this appears like a reasonable solution, if the player who is being forced wants to, they can change the direction of the forced move, from a straight line to a diagonal and back (for example) because there are no board edges to be forced into. So this rule is slightly exploitable by a troll who wanted to the game to last longer. (For example if the opponent's clock is running low.)

How about you stick to the simplest set of criteria:

"If one King is put into Check on eight consecutive turns, either player may claim a draw."

chilipepper

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While this appears like a reasonable solution, if the player who is being forced wants to, they can change the direction of the forced move, from a straight line to a diagonal and back (for example) because there are no board edges to be forced into. ...
In the example I provided, Black's king is always forced to stay in the same file, and move one square up on each move. There is a White queen on file-f, and a White rook on file-h, so the Black king is stuck on file-g.

fyi, unlike in normal chess, one of my goals is for this game to be mathematically "perfect". It doesn't have rules based on arbitrarily selected numbers (which is nearly the case of FIDE chess, but not totally so).

FIDE chess has the 50-move rule. Some game theorists cite this as being arbitrary, because some checkmates require more than 50 moves to perform. This means that a game with a winning position may end in a draw due to an arbitrary number. In infinite chess, I'm not using rules such as this. Games should only be called a draw if it is mathematically provable to be a drawn position.

Draws can also happen if the players agree to a draw, but this is related to a player's skill, and not the game rules. :)
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Martin0

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If the goal is to make a mathematical perfect game, then this sounds like a terrible rule. If the exact same position is not reached, then you can't really say that no progress has been made or not. Let's say black has a pawn on g15 for example. Then the sequence of
Rh3+...Kg4
Qf3+...Kg5
Rh5+...Kg6
Qf5+...Kg7
Rh7+...Kg8
Qf7+...Kg9
Rh9+...Kg10
Qf9+...Kg11

is really not just repeating the position. Each step the black king takes is one step closer to his pawn on g15. So the game should continue:

Rh11+...Kg12
Qf11+...Kg13
Rh13+...Kg14
Qf13#

The same argument could be made if instead of a pawn on g15, black had a rook on g999999. Eventually the sequence would end with:
Rh999997+...Kg999998
Qf999997#.
« Last Edit: March 22, 2018, 04:54:05 am by Martin0 »

chilipepper

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...Let's say black has a pawn on g15 for example. ...
Hi Martin0, good to see you here. :)

I believe the rule was written to address the situation you describe. The rule has this statement:

...and if the involved pieces are moving in a way that the game-state will not otherwise be changed (for example the checkmating pieces are blocked by an occupied square, or to a square which is attacked in a way that resolves the check).

I'm not sure if I phrased it in the best way, but I intended for the rule to not apply if the moving pieces "run into" some other pieces, or attacked squares which changes the situation.
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chilipepper

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I should also point out that in the example provided (with only the pieces shown), White actually has a forced mate. He could simply move the king to the g-file (such as in your example), and White can win.

So the example represents a situation where Black has other pieces, and White believes he is inferior, but White plays for the draw by performing the perpetual check.

Thanks for your analysis of this game ending. I'm pretty sure there are situations that I haven't addressed. But I hope to get them in the rule book before they are ever seen in actual play. :)
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Martin0

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The easiest assumption is that white must keep checking or else he is getting mated. Playing for a draw (perpetual) while your opponent has only a king left does not sound like a good strategy.

Anyway, "game-state will not otherwise be changed" is something hard to define or understand exactly what it means. Pieces "run into some other pieces, or attacked squares" is something that does not need to change within the moves the draw is declared, but much later. To me it sounded like you referred to moves made when the draw was declared, not any number of moves in the same direction.

But really "game-state will not otherwise be changed" is something you can't really define properly since the relative position between all pieces can matter in a position. If I just do a lot of checks to make my opponents king inactive and then wants to do something else (maybe my opponent having an inactive king will decide the game!), then what is your definition of "game-state will not otherwise be changed" in order to make my strategy valid? If you include king activity in the equation, then no position could be declared a draw this way.

So in my eyes the rule is flawed and should not exist. If a player makes "no effort to win", then an arbiter could declare the game a draw. It doesn't matter which of the players refused the draw offer.

One side note is that a "fast forward" could help in some cases. Requiring a player to answer if he will check indefinitely or do something else x moves into the line, then the game could be declared a draw if indefinitely is the answer. That rule can include any repeated moves in the position, not just checks.
« Last Edit: March 22, 2018, 12:47:32 pm by Martin0 »

chilipepper

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The easiest assumption is that white must keep checking or else he is getting mated...

Yes, I agree. That is the potential dilemna with perpetual check. A player might also employ it as a strategy when he is inferior, but wants to play for a guaranteed draw (which is basically the same thing).

...One side note is that a "fast forward" could help in some cases. Requiring a player to answer if he will check indefinitely or do something else x moves into the line, then the game could be declared a draw if indefinitely is the answer...

This idea is interesting, but in general, I think chess should normally be able to be played with little or no (verbal) communication between the players. The only exception I believe is resigning, and draw offers. (excluding communicating for the sake of etiquette).

While I am sure that there are theoretical positions that can be played with a perpetual check, I agree with you that there may be situations where defining the class of these positions can be difficult.

Since adding this rule might add problems or cause ambiguities, I will not add it to the rulebook at this time. But it is an area that might need more exploration for this game to be completely "buttoned up" to address all possible game endings.
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HGMuller

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Verbal communication is usual for draw offers, and repeating positions is equivalent to an automatic non-verbal draw offer, which the opponent can take (= claim), or decline. So it doesn't seem very much out of line with existing custom to allow a player to explicitly state that a 'pseudo-repeat' is NOT a draw offer (so it cannot be claimed). OTOH, this is sort of implied by the fact that he does not claim the draw himself.

It seems that between non-malicious players no rule is needed at all; if one of the players can force a draw this way, and expresses his intention to do so by offering a draw, the other would sooner or later see there is no way avoiding the draw, and accept. So a rule would only have to address the problem of a malicious player, which insists on prolonging the game forever without a chance of winning. The type of player that doesn't resign in a lost position, but just abandons the game, forcing the opponent to wait out the time forfeit out of spite.

Making a rule that specifies an automatic draw will alter the game result for non-malicious players, in some cases. In general, I think that this is too high a price for dealing with malicious players; a good rule would only penalize the latter, without affecting the former. A player should always be allowed to play on if something is to be achieved, within the limits of the 50-move rule. (And even the latter is dubious on an infinite board.) To prevent abuse of that right, it seems reasonable to force a player that wants to exercise it to mention how many more pseudo-repeats he needs before he is going to break the pattern, and penalize him by a loss instead of a draw when he pseudo-repeats the position after that. This still would allow the possibility of abuse, by mentioning an impossibly large number, say a billion. But this can be solved by allowing the game to 'fast-forward' the number of mentioned repeats, and continue from there. This would in fact be a rule that would also benefit normal play.

A draw offer in a pseudo-repeat position can be seen as expressing the intent to keep repeating forever. The opponent can then decline, (like usual for draw offers), but in the case of a pseudo-repeat he must decline by mentioning a number. (Which then puts a limit on how often he can keep pseudo-repeating that position without forfeiting.) Or by doing a move silently, implying the number 0 (i.e. not being allowed any future pseudo-repeats of the position). When he mentioned a number he must then move the translating group of pseudo-repeating pieces by the mentioned number of steps before pressing the clock, and the game will continue from there with any further pseudo-repeat of the position now being a loss for him. In fact there is no verbal communication needed at all, as the declining player shows on the board how often he intends to repeat, by moving the translating sub-set of pieces.
« Last Edit: April 14, 2018, 01:30:49 am by HGMuller »

chilipepper

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It seems that between non-malicious players no rule is needed at all; if one of the players can force a draw this way, and expresses his intention to do so by offering a draw, the other would sooner or later see there is no way avoiding the draw, and accept.
I agree. In such a situation either player can still offer a draw, and any reasonable player would agree once it is known there is no intention (or ability) to break out of the perpetual check. If it's a non-tournament game, then nobody else would care about the outcome anyway.

One of the few situations where not addressing this, and it could lead to a problem, would be as follows:

1. The game is in a tournament, and other players are waiting for the result, so the contest can proceed. Also consider the game has a long time-control, such as 3 days per move.

2. The checking player continues to play the pseudo-repeating pattern, occasionally offering a draw.

3. The checked player refuses to agree to a draw, continuing to move out of check.

4. The tournament decides the World Champion.

5. Both players are healthy and young, whereas other players in the tournament have shorter life-expectancies.

...This still would allow the possibility of abuse, by mentioning an impossibly large number, say a billion...
If such a rule can be developed where players are asked to name a theoretical number, and the moves can be predicted by player's intentions, wouldn't it just be easier to call the game result right away? (i.e. don't make a rule where players are required to communicate with each other and name theortical figures to call a draw. Just call the draw).

Making a rule that specifies an automatic draw will alter the game result for non-malicious players, in some cases. In general, I think that this is too high a price for dealing with malicious players.
I agree, and for this reason I don't plan to add anything to the rule book for now. But with time and special care I might be able to make a sensible and not too confusing rule to be added later.
« Last Edit: April 14, 2018, 11:08:26 pm by chilipepper »
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HGMuller

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If such a rule can be developed where players are asked to name a theoretical number, and the moves can be predicted by player's intentions, wouldn't it just be easier to call the game result right away? (i.e. don't make a rule where players are required to communicate with each other and name theortical figures to call a draw. Just call the draw).
Wouldn't it be easier still to declare the game a draw in the start position? Then they wouldn't have to play at all!

The purpose is not to make rules that are easy, but rules that are good. And arbitrarily declaring a draw in a position that is a forced win is just badly interfering with the spirit of the game.

The point is that people would only 'call out' a number when they intend to break the pattern at some point, because if they don't they will eventually reach that number of repeats and (according to te proposed rule) forfeit. So if a forced win is possible, it would never get out of reach, no matter how many pseudo-repeats are needed to execute the winning plan.

A famous situation in infinite chess is when there is an unavoidable mate threat against you, but you can stall by checking with a Rook from a distance. Eventually that will not work, because the checked King will steadily walk towards the checking Rook (which is supposed to be unprotected), so that the latter evantually will have to refrain from checking or get captured (after which the mate unrolls). But by deciding how much distance the Rook will take on the initial move, the number of moves needed to break out of the checking can be made arbitrarily high. This is the famous 'mate-in-infinite'.

Many of the problems you mention with duration are intrinsic to having an infinite board. But the rule I proposed solves many of them. If in the Rook-checking example the defending player moves the Rook 1000 squares away, the King would need 1000 moves to reach it to break the checking. Allowing the King to move there in one step, based on the pseudo-repeat rule (where the King is the only translating piece) would be a great time saver.

 As I explained, no communication is required by this. Just move the King next to the Rook from the pseudo-repeated position.
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chilipepper

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...arbitrarily declaring a draw in a position that is a forced win is just badly interfering with the spirit of the game.
I don't believe that the rule that I proposed does that, nor does the example that I provided. However, since I'm uncertain if the rule has ambiguous interpretations for some undiscovered positions, it has not yet been added to the rule book.

Many of the problems you mention with duration are intrinsic to having an infinite board.
My intention with this thread is only to address perpetual check. But I'm sure there are other interesting and unique situations that merit special discussion. Other threads and more discussion are possible in the future. :)
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