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In chess, there are some dead positions (FIDE Laws of Chess).

5.2.2 The game is drawn when a position has arisen in which neither player can checkmate the opponent’s king with any series of legal moves. The game is said to end in a ‘dead position’. This immediately ends the game, provided that the move producing the position was in accordance with Article 3 and Articles 4.2 – 4.7

Arbiters or players will terminate the game OTB, but is it possible to have computer solutions for dead position detection? If it is, then how?

Rewan Demontay
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    @J.Doe The position you posted in another comment (in a deleted answer) is a dead position because it is a forced stalemate. We should also consider positions in which both sides have spare moves but no possible way of checkmate, like a totally closed pawn chain with all white pawns on squares of the same color, all black pawns on squares of the opposite color, and a bishop per side running on squares of the same color as its own pawns. These are positions very difficult to automatically declare dead by software, because of the loops. – Daniel Alfredo Sottile Sep 26 '18 at 15:59
  • The rule terminates the game. Arbiters or players on the other hand may not. See Topolov-Nakamura 2016 and my comment there. – Martin Rattigan Oct 17 '21 at 21:38
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    Luckily it doesn't make a lot of difference if a dead position is missed because it will necessarily be a draw anyway. A position incorrectly diagnosed dead is a different matter. – Martin Rattigan Oct 17 '21 at 21:44
  • Conceivably it could make a difference in actual play if one side runs out of time in a dead position. – Noam D. Elkies Sep 26 '22 at 04:42

6 Answers6

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Computer detection of dead positions is much trickier than people think. It is unlikely that an algorithm exists that runs in reasonable time and is 100% accurate.

It is easy to check for a simple condition like insufficient material (K+B v K, K+N v K). It is less easy to check for cases with blocked pawns, for instance:

2b1k3/8/8/1p1p1p1p/1P1P1P1P/8/8/2B1K3 w - - 0 1

since there are a lot of legal moves to check. Computers aren't based on intuition like humans: you'd have to enumerate every single possible continuation for up to 150 ply without pawn moves/captures (those reset the counter) and check that none of them end up in checkmate for either side.

You could try other tricks, like trying to store the positions reached in a tree and checked that none of the reachable positions are mate. That's only 76176 positions for the example I gave, but you'd need to check that the tree is indeed exhaustive - i.e. every legal move in every position must be tried. That's not going to be efficient.

You could also try to invent some arbitrary heuristics to help your algorithm, like checking for blocked pawns and same coloured bishops. The problem with this approach is that sometimes the heuristics are wrong, or they aren't general enough. (Brian Tower's answer is an example of an attempted heuristic, but in the comments user17439 shows it fails to flag some dead positions and I show it incorrectly flags a non-dead position.)

And I would love to see any heuristic or algorithm that can distinguish between the following positions (problem by Andrew Buchanan, StrateGems 2002)

Dead:

Bb1k1b2/bKp1p1p1/1pP1P1P1/1P6/p5P1/P7/8/8 w - - 0 1

Alive:

Bb1k1b2/bKp1p1p1/1pP1P1P1/pP6/6P1/P7/8/8 w - - 0 1

The second position is alive since mate is possible after

1.Ka6 Ke8 2.Bb7 Kd8 3.Bc8 Ke8 4.Bd7+ Kd8 5.Be8 Kc8 6.Bf7 Kd8 7.Bg8 Ke8 8.Kb7 Kd8 9.g5 Ke8 10.Kc8 a4 11.Bf7#

In sum, I do not believe you can have an algorithm that is 100% accurate in distinguishing between dead and alive positions that runs in reasonable time. Brute force is the only 100% reliable method, but it is definitely not efficient. You can get a decent success rate for "practical" positions with heuristics, but it will not be 100% accurate, even for real game positions.

Remellion
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  • I agree that in general you can't make a simple enough rule that works 100% of time. Although a sufficient one for all these 3 problems can be pretty simple: figure out which tiles are reachable for all pieces you have left. This should be pretty manageable in terms of runtime. If there is overlap between black king and white non-king pieces, position may be still alive. To eliminate 2nd as dead, simply note that only king and bishop overlap with black's king. As bishop takes just 1 tile from king, black king needs to not move to get checkmated - which isn't possible in 2nd case. – Zizy Archer Jun 16 '21 at 11:52
  • Luckily, if the position repeats even once (you need twofold repetition, not threefold), then you can cut that branch. – Ferazhu Nov 26 '21 at 20:42
  • Is it not possible to use some slightly involved heuristic to determine whether the position is “clearly alive” or “clearly dead”, and only fall back to brute‐forcing if neither of those two match? Generally, positions that are “difficult to determine” tend to be very constrained and have very few possible moves available, so brute‐forcing (using a table) shouldn’t take too long, I imagine. – zamfofex Feb 15 '22 at 06:35
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    Miguel Ambrona's program does distinguish between these two positions, and correctly assigns all other positions that it's been given. It amazes me – Laska Apr 08 '23 at 06:24
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Miguel Ambrona’s CHA solver “Chess Unwinnability Analyzer” (GitHub repo, white paper) is an efficient solution to this problem. It uses an incomplete algorithm to analyse positions and determine whether there is a possible checkmate by either side, optionally falling back to a slower, more thorough “brute force” analysis when the incomplete algorithm isn’t able to conclude whether a checkmate is possible with certainty.

In practice, the thorough search only ends up being necessary for constrained positions where there are few moves available for each side, so even then it ends up being fast.

From the paper:

Definition (Unwinnability) We say that a position is unwinnable for a given player if there does not exists a sequence of legal moves that ends in a checkmate by the player.

Note that the above definition focuses on whether a specific player can still win, typically the player whose opponent ran out of time. Concluding that a position is dead would require two individual analyses, one for each player as the intended winner.

[…]

zamfofex
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I would expect that it is a good bit easier to write a program that is good at detecting dead positions than to write a program that plays chess well.

A simple strategy may be to play out a large number of games randomly to the end starting from the position given. If the position is dead, none of the playouts will have a result different from draw. If on the other hand the position is not dead, the probability that one playout discovers a helpmate (it need not be the shortest one) approaches one as the number of playouts tends to infinity.

In practical terms, I would conservatively expect a well-optimized random mover to be able to reach on the order of ten million playouts per second on a single core of a modern CPU. Under this assumption, 100 million random playouts per second are not unreasonable on a current PC. If we allow roughly a minute for resolution of a position as dead or alive, we can therefore probably do a few billion playouts and hence fairly reliably detect any helpmate that is found with likelihood more than, say, one in one billion under a random playout policy.

Edited: This simple strategy is likely not enough to practically solve cases like Remellion's example further up in this thread.

Nonetheless, the point stands that an algorithm that declares a position to be dead when no helpmate can be found within some computational budget will only return a wrong answer if the position contains a helpmate it cannot find. In that sense, the decision problem for dead positions is at most as hard as the problem of finding helpmates. There are good helpmate solvers and they have no trouble solving e.g. the "alive" position given by Remellion (see for instance this online solver ).

Polytropos
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  • When you say play out a large number of games to the end, how do you define end? E.g. in @Remellion's first position. – Martin Rattigan Oct 11 '21 at 20:24
  • Or, simpler, this position? FEN:4k3/8/8/8/8/8/8/4K3 w - - 0 1 – Martin Rattigan Oct 11 '21 at 20:54
  • Obviously a helpmate solver that will reliably run in sufficiently quick time and use sufficiently little storage would be the answer OP is looking for. To be confidently used a proof of the time and storage requirements would be necessary. You say that they cope with the alive position given by Remellion, but how long do the take to annnounce "no solution" for the two dead positions in my previous comments? – Martin Rattigan Oct 12 '21 at 14:23
  • I am not sure exactly which question the OP has asked. There are various degrees to "solving" the problem of dead position detection. The hardest might be something like decide aliveness for any legal position and output a short proof of death along with a proof of legality, while refusing (with proof) illegal positions. Easiest would be something like "decide aliveness quickly with lower error rate than a human arbiter for positions that may arise in games". – Polytropos Oct 12 '21 at 20:22
  • My argument would show something like "Ability to quickly supply short proof of liveness for many positions that are hard for humans, ability to correctly classify as dead all dead positions". I would not be too pessimistic that helpmate solvers could get to a point where it becomes infeasible to construct a legal alive position they can't solve (which is of course less than nonexistence of such difficult helpmates), but I could be wrong about that, of course. I would be pessimistic about exact proof (not necessarily short) for all dead positions, but could be wrong about that as well. – Polytropos Oct 12 '21 at 20:28
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I think the answer is probably not. In fact I would say that the introduction of the dead position rule to replace the previous draw by insufficient material was a mistake. (Btw why did FIDE excise the draw by insufficient material rule when they introduced the dead position rule but not the stalemate rule?)

The following position is dead under FIDE competition rules because there is no legal continuation that will result in mate before the game terminates under the 75 move rule.

7k/3N4/5K2/6B1/8/8/8/8 w - - 0 1

White to play. Ply count field in FEN 146.

The following position is also dead if the position (as defined in the 5 fold repetition rule) that would occur after Kf7 has already occurred four times previously.

7k/3N4/5K2/6B1/8/8/8/8 w - - 0 1

White to play. Ply count field in FEN 144.

The arbiter (and any software seeking to solve the dead position problem) needs to keep track of possible helpmates taking both the 75 move rule and 5 fold repetition rules into account.

How do arbiters cope? The answer of course is they don't. The following game actually terminated in a dead position with White's move 132, but was recorded as a draw under the 75 move rule after Black's move 132 (which wasn't actually part of the game).

https://www.chessgames.com/perl/chessgame?gid=1825274

For software to cope it would require helpmate EGTBs.

Since helpmates are generally shorter than forced mates these would possibly take less computation than the existing forced mate variants under FIDE basic rules (which now exclude any n move or n fold repetition rules) but the fact is there are also many more positions where helpmates are possible and forced mates are not, so the storage requirement would be much greater.

Under FIDE competition rules the EGTBs would need to store helpmate lengths under a DTZ75 metric for all possible combinations of positions that have been repeated four times and both the computation and storage requirements would greatly exceed those required for the current forced mate EGTBs.

I would also class the following position as dead according to the FIDE rules. (There is nothing in the FIDE rules that states that a position can become dead only as a result of a move being 'made' under art 4.7 and before a piece is touched as in art 4.3 by the player who then has the move - indeed on my reading the two events may be simultaneous.)

k6K/8/6PR/7P/8/8/8/1R6 w - - 0 1

White to play. White has touched the h6 rook.

Of course if the solution is required only for computer "chess" the solution could ignore such positions, because computer "chess" never properly implements art 4. But if the solution were intended as an aid to arbiters or chess players then these situations would also need to be taken into account (whether under basic or competition rules).

Martin Rattigan
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    To my best knowledge (and I only recently made my national arbiter license), assume in your last position you add a bPh7 and White loses on time before completing the forced Rxh7. Still 0-1, touch-move is NOT relevant for the "future". – Hauke Reddmann Oct 10 '21 at 16:50
  • @Hauke Reddmann Art 5.2.2 is The game is drawn when a position has arisen in which neither player can checkmate the opponent’s king with any series of legal moves. The game is said to end in a ‘dead position’. This immediately ends the game, provided that the move producing the position was in accordance with Article 3 and Articles 4.2 – 4.7.* Immediately White touches the h6 rook the rule applies (whether or not there is a black pawn on h7 as far as I can see) and the position is dead at that point. So I'm struggling to follow your comment. – Martin Rattigan Oct 10 '21 at 21:06
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    @Hauke Reddmann Time doesn't enter into it. The rule says neither player can checkmate the opponent's king, time is not involved (not even under competition rules). – Martin Rattigan Oct 10 '21 at 21:17
  • @Hauke Reddmann I don't have an arbiter's license, but I would still question your evaluation of the score should the flag fall in the circumstances you posit. Art 6.9 says Except where one of Articles 5.1.1, 5.1.2, 5.2.1, 5.2.2, 5.2.3 applies, if a player does not complete the prescribed number of moves in the allotted time, the game is lost by thatplayer. However, the game is drawn if the position is such that the opponent cannot checkmate the player’s king by any possible series of legal moves. The last sentence would appear to apply even ignoring the dead position rule. – Martin Rattigan Oct 10 '21 at 21:25
  • The first position is a bad example because in all realistic situations it would have already been drawn via the 50-move rule. The current rules are the best rules that apply properly in most reasonable situations. The previous rulesets that used to be in place also came with their (worse) shortcomings. For instance, under United States Chess Federation rules, there are some positions where a player has forced mate but you can get a draw by letting the clock run out – David Oct 10 '21 at 21:58
  • @David In fact games very often continue many moves past the point where a draw could have been claimed under the 50 move rule even if one side is obviously losing. – Martin Rattigan Oct 10 '21 at 22:36
  • @David The dead position rule says nothing about realism and realism would probably not be taken into account in any software solution. But you might prefer this FEN:2n5/8/8/4N3/8/3K4/1k4B1/2b5 w - - 148 1 The reason I think the dead position rule was a mistake is that there is no simple algorithm given for determining if a position is dead or not. That strikes me as a non rule. Of course what I said doesn't apply under USCF rules anyway because the dead rule is claimable under those rules. I'm assuming that the question is aimed at internationally accepted rules rather than local variants. – Martin Rattigan Oct 10 '21 at 23:28
  • @MartinRattigan All FIDE-sanctioned events must have a human arbiter, so how well computers can implement the rules is not a matter of consideration when making the rulebook. The types of situations that can arise in practice are though, and a position that has played for 70+ moves without captures or pawn moves is something that will never happpen in a tournament game. – David Oct 11 '21 at 11:17
  • @David I agree that how well computers can implement the rules should not be a consideration in formulating the rules (whether for formal or informal games). For an examples of games that went on for 70+ moves without captures or pawn moves see https://www.chessgames.com/perl/chessgame?gid=1268705 or https://www.chessgames.com/perl/chessgame?gid=1861167 or https://www.chessgames.com/perl/chessgame?gid=1861167 or https://www.chessgames.com/perl/chessgame?gid=1825274 – Martin Rattigan Oct 11 '21 at 12:51
  • The first of those games was played before the 75 move rule was introduced, so perhaps not relevant. The last three were adjudicated draws under the 75 move rule, but the last reached a dead position one half move previously without the arbiter noticing. – Martin Rattigan Oct 11 '21 at 12:57
  • Sorry, I put Yudasin-Erenburg in twice in my first post. The third game should have been https://www.chessgames.com/perl/chessgame?gid=1959810 . – Martin Rattigan Oct 11 '21 at 13:02
  • @David It's my fault for commenting on the dead position rule in my answer, but a discussion of what the rules should be is getting a little off topic. OP asked if a computer solution could be found for the "dead position problem". – Martin Rattigan Oct 11 '21 at 13:31
  • @MartinRattigan I still fail to see what's so "wrong" about those examples. All I see is the rules being applied as they should, while the previous ruleset was a mess and sometimes led to controversy – David Oct 12 '21 at 09:07
  • @David I didn't say there was anything wrong about my examples nor the games I posted with the exception of Topalov-Nakamura where the dead position rule was not being applied as it should. – Martin Rattigan Oct 12 '21 at 13:00
  • Let us continue this discussion in chat. – Martin Rattigan Oct 12 '21 at 13:03
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    Am I missing something about the KBNvK endgame positions? I think they're mate in half a dozen moves... – Glorfindel May 13 '22 at 15:54
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    @Glorfindel 7k/3N4/5K2/6B1/8/8/8/8 is a #3 (1. Kf7 Kh7 2. Sf8+ Kh8 3. Bf6#) as well as h#3 ( 1. Kg8 Ke6 2. Kg7 Nf6 3. Kf8 Bh6# ) ... so there certainly seems to be paths to mate. I would have expected at least one of the commenters to catch that. But position 1 and position 2 seem to be expected to be different ... ? So perhaps they are not the original positions? –  Jul 03 '22 at 08:18
  • IMO when you lose on time, you can challenge the arbiter (or your opponent), and they have to prove that the position is winnable from their side. – Ferazhu Sep 18 '22 at 22:33
  • @Glorfindel You're probably missing the fact that a game under FIDE competition rules terminates when the ply count reaches 150 or before. Both positions are White to play so your second example doesn't apply. Your first example needs 5 ply so the game would terminate without mate from ply count 146. Mate in 3 can be achieved only with 1.Kf7 so if the position with that move had already occurred four times mate again could not be achieved because 1.Kf7 would terminate the game under the 5-fold repetition rule. The positions are not dead under FIDE basic rules. – Martin Rattigan Sep 25 '22 at 18:53
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    Ah, now do I understand the meaning of 'ply count' - it might be worthwhile to [edit] the answer to make that a little bit clearer. – Glorfindel Sep 25 '22 at 19:15
  • @Ferazhu Depends on whether you're playing FIDE or USCF rules, but in neither case does the player make the challenge; it's the arbiter's task to decide. Under FIDE rules he has to decide if the position is half dead and mostly the same under USCF rules but in some cases e.g. KNNvKNN the arbiter is tasked with deciding if there is a forced mate. In that case if you don't know how to play the ending and your opponent does, you do best to go for coffee and eventually let the arbiter play it for you. – Martin Rattigan Sep 25 '22 at 19:26
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    @Glorfindel Yes, will do. Second time in a week or so my use has been misunderstood, so I should be more specific. – Martin Rattigan Sep 25 '22 at 19:36
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Detecting for insufficient materials for checkmate is super easy. Just get the FEN position, and look for the number of pieces remaining and what they are.

The second solution requires enumerating all possibilities until a certain depth. If there is really no legal way to make progress, the variations will quickly repeat. A modern laptop should have the memory to do it.

SmallChess
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There are two classes of dead position (positions from which helpmate is impossible)

  • Insufficient material

  • Positions where both pawn moves are impossible and future captures are also impossible

The first is trivial to program. The second less so but I think still possible. Checking for possible pawn moves is straightforward. Checking for possible future piece captures is less straightforward but should still be possible.

Brian Towers
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