Distance to Capture

[Kirill Kryukov]

I am curious to explore "Distance to Capture" - another metric for endgame tables, inbetween the DTM (Distance to Mate) and DTC (Distance to Conversion).

What would be a good abbreviation? "DTCa"?

One of the reasons for my curiosity is that I discovered that DTC and DTZ are surprisingly close to WDL in compactness (in minichess), and that a large gap exists between DTM and DTC. So DTCa would nicely fill that gap. Another reason is that it would provide yet another kind of long lines, which might be interesting.

Has anyone heard of anyone using this metric?

[Kirill Kryukov]

I decided to call this metric "DTX", because 'x' is used for capture in chess notation. So now the set of metrics I use will look like this (in order of decreasing table size): DTM, DTX, DTC, DTZ, WDL.

[mbourzut]

Since the board is so crowded in minichess even with a small number of pieces, and the pawns don't have far to go before conversion, one would expect DTC and DTZ to be close together and close to WDL in terms of compactness. As the board size increases, the differences will increase. I'm not sure what we can learn from DTX.

[Kirill Kryukov]

Hi Marc, always nice to see you drop by.

Since the board is so crowded in minichess even with a small number of pieces, and the pawns don't have far to go before conversion, one would expect DTC and DTZ to be close together and close to WDL in terms of compactness.

Yes, as the board fills, DTC, DTZ and WDL converge to be the same (identical for a 100% filled board). Still somehow I was surprised to see the actual table sizes. It seemed counter-intuitive that a distance-based metric can get so close to WDL (in compactness) even with few pieces on board.

Apparently it has more to do with filled board than with minichess. Just in minichess we do fill the board, which we don't with larger board sizes.

As the board size increases, the differences will increase. I'm not sure what we can learn from DTX.

DTX is just a point inbetween DTM and DTC. It's much more compact than DTM, but perhaps produces less unnatural play than DTC (for those who consider DTC-based play unnatural, - I don't, and I rather consider DTM unnatural). Personally I'm curious about the longer lines, especially from positions with many pawns. Also, the same question can be asked about DTC - as it's just an arbitrary point between DTM and DTZ. Still people use DTC for some reason.

[guyhaw]

Kirill,

I'm puzzled.

DTC = 'Depth to Conversion of Force and/or Mate' ... and 'conversion' can be by capture and/or conversion of a Pawn.

So your DTX would seem to be >= DTC, with the disadvantage that it may be undefined when DTM is defined, e.g., in KQK, KRK or KPK.

There is a neatness about the fact that DTM >= DTC >= DTZ ... and DTX doesn't seem to fit into that.

More to the point, I can imagine players adopting 'mate', 'conversion' or 'zeroing of the move count' as their goal, but I don't see them restricting themselves to 'capture' as a goal.

g

[Kirill Kryukov]

Kirill,

I'm puzzled.

DTC = 'Depth to Conversion of Force and/or Mate' ... and 'conversion' can be by capture and/or conversion of a Pawn.

So your DTX would seem to be >= DTC, with the disadvantage that it may be undefined when DTM is defined, e.g., in KQK, KRK or KPK.

There is a neatness about the fact that DTM >= DTC >= DTZ ... and DTX doesn't seem to fit into that.

More to the point, I can imagine players adopting 'mate', 'conversion' or 'zeroing of the move count' as their goal, but I don't see them restricting themselves to 'capture' as a goal.

g

In your notation, DTX = 'Depth to Capture and/or Mate', to be precise. Same like we call DTC just DTC and not DTCM (conversion or mate), and the "mate" part is assumed to be there. Also same in DTZ. So, the following inequality always holds: DTM >= DTX >= DTC >= DTZ.

My original motivation for studying DTX was to separate the DTM-like information content of all N-piece tables from the set of (N-1)-piece ones. DTC does not do that, because it also separates tables from each other.

Since DTX sits inbetween DTM and DTC, I was also curious to see how the compressed DTX compares with compressed DTM and DTC. As it turns out, DTX is very close to DTC (more close than I expected). Obviously DTX is inefficient for practical purposes, but so is DTC and DTC is (or was) used a lot. DTZ (and DTZ50 in 8x8 chess) is superior for practical play, although in the end all three metrics (DTX,DTC,DTZ) appear to produce very similar database sizes. (Again this was surprizing for me).

Lastly, I quite enjoy the long non-DTM lines (DTC and DTZ). So I am a bit curious to see some of the longest DTX lines, especially with positions with many pawns (where the ordinary DTC or DTZ would quickly use those pawns to terminate the line).

[byakuugan]

Useful tablebases would be "Distance to Force" where the job of one player is to try to force a specific formation of pieces (maybe only 1 or 2 pieces are regarded in a formation), and the opposing player is not allowed to get checkmated either. It would be useful to examine such tablebases for complicated endgames, to see how they work. For example: if a tablebase set to one formation shows that the defender can avoid that formation by playing into another formation, but the tablebase set to the other formation shows that the defender must play into the original formation to avoid the other formation, then you would be able to understand a lot of the different methods for deep complex endings.

[guyhaw]

Ok, thanks for the clarification that DTX means 'Depth to Capture and/or Mate' rather than just 'Depth to Capture'.

When I first came up with the systematic DTM, DTC, DTZ in 'Strategies for Constrained Optimisation' ...
see http://centaur.reading.ac.uk/view/creat ... fault.html and particularly http://centaur.reading.ac.uk/4566/ ...
I wanted to stick to three-letter-abbreviations so that I could write 'DTx' but maybe 'x' can stand for <any letter-string, preferably short>.

Admittedly, it would be more natural to say DTCM rather than DTC, but you can imagine the King being actually taken 'as part of the mating move' as the loser has not replying move ... another argument for the concept of the 'null move'

I can see my three-letter convention coming under strain if we want to start differentiating between 'Depth to Conversion only' and 'Depth to Conversion and/or Mate'.

I've also got into the habit of talking about 'Depth' rather than 'Distance' ... it's shorter and seems more graphic.

g

[guyhaw]

byakuugan's contribution prompts me to mention something similar that I've been thinking about for more years than I care to mention now.

The sub-7-man (s7m) EGTs, by my reckoning, contain values and depths for some 3,409,699,385,208 chess positions: that's 3.4*10^12 for short!

Not a set of information that can be managed in the head, so it would be useful to have some landmarks and roadways in this vast terrain.

I've therefore been thinking about how one might assess the significance of various positions, and my idea is that these are given 'notional values' which replace their actual values of 'win, draw or loss'.

For example:

1) KQKR, Philidor (1777), 1k6/1r6/2K5/Q7/8888 w/b: what if this were 'a draw', wtm or btm ... how does this impact White's chances

2) the pseudo-fortress in KBBKN: bN on b2, bK on c2 for example ... ditto ... [and your own favourite positions can go here]

3) a position won by White with wtm, but won more quickly with btm:
- - - - - a 'Squeeze' in at least the UK, and a type B zug in Bleicher/Haworth (2009), qv http://centaur.reading.ac.uk/4518/
- - - - - is it necessary for the win from the wtm position to go through the btm position
- - - - - this question is answered by setting the 'btm' position artificially to 'draw'

4) consider White about to move at some position in the mainline of a Chess Study
- - - - - White has some moves which are sub-optimal in terms of some metric, say DTM(ate) for now but DTZ would be better
- - - - - if White plays a sub-optimal move, can Black force White back to a position already visited in the study mainline, i.e.,
- - - - - can Black demonstrate that this move is strictly inferior to an optimal move
- - - - - [ the answer is 'no' if that DTx-sub-optimal move is optimal in the metric DTy !]
- - - - - again, this can be assessed by imagining the current position and all previously-visited positions to be 'draw' rather than 1-0.
- - - - - by this means, it is possible to show that a progressing move is 'essentially unique' if not 'absolutely unique' in a study.
- - - - - this is one kind of 'Holy Grail' in the world of studies

5) a computer (again, say 'White') playing otb potentially has a long phase of play of over 50 moves in its winning line, i.e., DTR > 50.
- - - - - maybe the computer it is playing has only DTM or DTZ EGTs, so it's not in the best position to force this long phase
- - - - - White has moves in hand in the current phase, and want to give the opponent the chance to reduce DTR
- - - - - however, White may not wish to revisit some or any of the positions already visited in the game
- - - - - its options can be constrained by setting any positions it doesn't want to visit to 'draw' rather than 'win' and recomputing the EGT.

All of these situations can be handled easily by inserting (into the initialisation phase of an EGT-generator) the setting of certain position to certain values [and remembering that this was done when EGT-validation is done!].

It is a case of computing EGTs for an appropriate variant of Chess, rather than for computing EGTs to a different metric.

Are there any volunteers out there willing to give this a go!?

As I hinted, the DTZ metric is best for this (as repetition cannot return to a previous phase of the game) but DTC and DTM are ok.

Also, as I've argued elsewhere, the assistance of WDL bitbases is useful for generating DTx EGTs as it can prevent the needless and expensive investigation of 'possibly lost positions' in a fully retrograde EGT-generating algorithm.

g

[kronsteen]

Always interesting to discuss new metrics. I had also another metric idea that I call “distance to positive conversion” (DTPC or DTP maybe ?).

DTC has the defect that it considers conversion moves by the losing side as goals. This results in the fact that these moves, which are generally threats (an adverse promotion or capture always results in losing material) are ignored, or even forced, when they don’t allow the defender to save the game. This is especially unfortunate when DTC is used with the idea of showing the fastest path towards a simpler and easier to win endgame.

In order to fix this, DTP considers as goals only promotions or captures (or mate) made by the winning side.

DTP first visible improvement over DTC is to suppress silly sacrifices in endings where the winning side has a heavy material advantage, for example in kqqkr where DTC shows the fastest forced outcome among mate (ok), rook capture (ok) or queen suicide (silly). But DTP is most useful for endings of the krppkr / kqppkq type where DTC is very prone to unnecessarily lose a pawn and run into a harder to win krpkr / kqpkq position.

Observe that DTM >= DTP >= DTC and DTM >= DTX >= DTC, but no relation exists between DTP and DTX.

Useful tablebases would be "Distance to Force" where the job of one player is to try to force a specific formation of pieces (maybe only 1 or 2 pieces are regarded in a formation), and the opposing player is not allowed to get checkmated either. It would be useful to examine such tablebases for complicated endgames, to see how they work. For example: if a tablebase set to one formation shows that the defender can avoid that formation by playing into another formation, but the tablebase set to the other formation shows that the defender must play into the original formation to avoid the other formation, then you would be able to understand a lot of the different methods for deep complex endings.

These are interesting ideas, and IMO they would be particularly useful for analysis of drawn positions. When EGTBs are used, draws are just a vast empty space where the side with material advantage only sees that he has no winning move and has no clue about how to force specific formation of pieces where defence is supposed to be most difficult.

2) the pseudo-fortress in KBBKN: bN on b2, bK on c2 for example ...

This one should be easy to answer. With few exceptions, positions where the losing side can reach the pseudo fortress have DTM>50 and positions where he can’t have DTM<50.

4) consider White about to move at some position in the mainline of a Chess Study
- - - - - White has some moves which are sub-optimal in terms of some metric, say DTM(ate) for now but DTZ would be better
- - - - - if White plays a sub-optimal move, can Black force White back to a position already visited in the study mainline, i.e.,
- - - - - can Black demonstrate that this move is strictly inferior to an optimal move
- - - - - [ the answer is 'no' if that DTx-sub-optimal move is optimal in the metric DTy !]
- - - - - again, this can be assessed by imagining the current position and all previously-visited positions to be 'draw' rather than 1-0.
- - - - - by this means, it is possible to show that a progressing move is 'essentially unique' if not 'absolutely unique' in a study.
- - - - - this is one kind of 'Holy Grail' in the world of studies

Moves that allow the losing side to either force a draw (or win) or a return to the initial position are mathematically suboptimal in every metric. Identifying these moves should prove extremely valuable for endings such as kqpkq. I wonder if flagging the initial position as drawn and recomputing a full EGTB is the best approach or if some kind of direct analysis of the DTZ (or maybe WDL ?) tree’s topology would be possible and give immediately the full result.

[guyhaw]

Replying to kronsteen's last contribution ...


'DTC' is computed on the basis that the winner wants the shortest path to 'conversion' and the loser wants the longest path.

So 'conversion' is a goal for both sides (and 'DTC' is an abbreviation for 'DTC-DTC' or 'DTCok/DTCnot-ok') .. but it is a goal that the defender seeks to avoid.


In the study scenario, I'm only considering sub-optimal moves which still preserve the value of the position. Obviously, moves that sac a half- or whole-point are sub-optimal, but I'm using 'sub-optimal' in a metric sense v some DTx.


Re 'making sure a position is not visited (again), or discovering that it has to be revisited':

'Following the tree' is in some cases infeasible, and even when feasible, is hardly communicable to a reader. The only solution is to recompute the EGT for the variant of chess with some set of positions P = {pi} which are actually won for White but considered to be draws.

The ideas that accelerate the creation of this EGT for variant chess are:

1) the computation needed does not have to consider anything other than positions with the Pawns where they are
- - - - - hence, the DTZ metric is a 'natural' for this
2) values and DTZ depths need not be computed for positions with a DTZ-depth less than the minimum depth of a position pi in P
3) position in P are set to 'draw' but all others are 'unknown',
4) WDL bitbase information facilitates the computation of any EGT, and parallelisation is possible though hardly necessary here,
5) once the values/depths of the destination positions of the suboptimal moves are determined, the EGT-generation can stop.
- - - - - I can think of situations in which this is valuable - when there is a trivial 'undo last move, remake last move' 4-ply loop as is common.

g

[Kirill Kryukov]

Useful tablebases would be "Distance to Force" where the job of one player is to try to force a specific formation of pieces (maybe only 1 or 2 pieces are regarded in a formation), and the opposing player is not allowed to get checkmated either. It would be useful to examine such tablebases for complicated endgames, to see how they work. For example: if a tablebase set to one formation shows that the defender can avoid that formation by playing into another formation, but the tablebase set to the other formation shows that the defender must play into the original formation to avoid the other formation, then you would be able to understand a lot of the different methods for deep complex endings.

Reliably identifying more intermediate goals (other than capture and pawn move) is very interesting question. I don't know if it's easy or even possible to do general recognition of such intermediate goals. If this can work then it will further reduce the storage requirement for DTZ-like tables, which will be very nice.

[Kirill Kryukov]

Always interesting to discuss new metrics. I had also another metric idea that I call “distance to positive conversion” (DTPC or DTP maybe ?).

DTC has the defect that it considers conversion moves by the losing side as goals. This results in the fact that these moves, which are generally threats (an adverse promotion or capture always results in losing material) are ignored, or even forced, when they don’t allow the defender to save the game. This is especially unfortunate when DTC is used with the idea of showing the fastest path towards a simpler and easier to win endgame.

In order to fix this, DTP considers as goals only promotions or captures (or mate) made by the winning side.

DTP first visible improvement over DTC is to suppress silly sacrifices in endings where the winning side has a heavy material advantage, for example in kqqkr where DTC shows the fastest forced outcome among mate (ok), rook capture (ok) or queen suicide (silly). But DTP is most useful for endings of the krppkr / kqppkq type where DTC is very prone to unnecessarily lose a pawn and run into a harder to win krpkr / kqpkq position.

Observe that DTM >= DTP >= DTC and DTM >= DTX >= DTC, but no relation exists between DTP and DTX.

Yes, an interesting idea. It's curious that someone else mentined exactly the same idea last week in Russian-language chess forum (link to the message). Although the idea itself is very natural.

I don't like to call it "DTP", because the same idea can also be applied to DTX and DTZ (only tracking the progress of the winning side). My initial response was to call it DTCG, DTXG and DTZG ('G' for "greedy" - as these new metrics are greedy enough to not sacrifice all extra queens of the winning side). But perhaps even better naming is possible.

[kronsteen]

Re 'making sure a position is not visited (again), or discovering that it has to be revisited':

'Following the tree' is in some cases infeasible, and even when feasible, is hardly communicable to a reader. The only solution is to recompute the EGT for the variant of chess with some set of positions P = {pi} which are actually won for White but considered to be draws.

Yes, right. I just realize that my idea is slightly different.

Under perfect knowledge and assuming that both sides use a DTx EGTB (i.e. play a DTx optimal strategy), moves can be categorized as follows :

- Optimal : a move that decreases DTx by 1 move (if the side to move is winning), or maximizes DTx (if the side to move is losing), or holds the draw (if the position is drawn)
- Suboptimal : a non-optimal move that doesn’t change the evaluation of the position, i.e preserves the win but doesn’t decrease DTx (for the winning side), or doesn’t maximize DTx (for the losing side). There are no suboptimal moves if the initial position is drawn
- Losing : a move that turns a win into a draw, or a draw into a loss (half point losing move), or a win into a loss (full point losing move).

It may be possible to go farther and identify a very special category of suboptimal moves : these are suboptimal moves played by the winning side that allow the defender to force a return to the position from which they are played (or else a draw or a win). Theses moves can be called “no progress moves” as they are mathematically suboptimal in every possible metric, when for any other DTx-suboptimal move it should mathematically be possible to build another metric in which this move is optimal. The “no progress move” information is intrinsic to the move itself and doesn’t depend on the moves previously played like the scheme you suggest, so perhaps it could be easier to compute, to store and to handle. And when all “no progress moves” are identified, it should also be possible to exhibit “unique progress moves” or “bottleneck moves”, i.e. moves played from a position from which all moves, except this one, are no progress (or losing). Unique progress moves are mathematically optimal in every possible metric, and they must be present in any practical learning scheme for human players, so they definitely are of main interest.

Just an example (breaking 3rd rank defence in kqkr) : in 3k4/5Q2/1r6/3K4/8/8/8/8 w - - 0 1, is the well known counter-intuitive move Qf4 a unique progress move ? It is DTM and DTC/Z/Z50 optimal, but is a player who ignores this move really unable to win ?

[guyhaw]

To kronsteen and others of course ...

Thank you for your interest in the 'uniqueness' and 'looping' challenges. I think we are using different words but converging rapidly in terms of the ideas.


In my 'Strategies for Constrained Optimisation', http://centaur.reading.ac.uk/4566/ , I defined subsets of moves as:

'V' = 'Value Preserving' ... and within that as
'O' = metric-optimal (either minimal for the attacker, or maximal for the defender)

Therefore, a move-subsetting strategy S could be defined, e.g., SVM-C- ... which is 'consider only value-preserving moves which minimise DTM and DTC in that order.'

The word 'Losing' is meant to mean 'losing value' but unfortunately could be interpreted as 'letting go of both the win and/or the draw'. Anyway, I prefer to think in terms of subsetting via a sequence of filters (see above), each defining a smaller set of moves from those available.

As you say, there's a subset of the non-DTx-optimal moves that allow the defender to force the attacker (presuming they still want to win) back to the current position. This set is independent, again as you say, of any previous positions visited in a game or study.

[ However, I was considering the slightly more general (but not more difficult) problem of making sure that the defender cannot force White to visit any previously-visited position in a Chess Study solution. This is probably a distraction at the moment. ]

Your 'no progress' moves could be called 'time wasting' or preferably, 'move wasting' moves: they don't halt progress indefinitely but certainly delay it.


You say that if a move is NOT a 'no progress move', it should be mathematically possible to define a metric in which it is optimal. I think that's speculation, and I'm not sure what purpose those metrics would serve, but you may be confusing your speculation with the opposite statement which is a fact, namely ...

'If a move is optimal in any metric, it is not a move-wasting move.'

I have found it useful to use, in sub-6-man chess, to use my DTC, DTM and DTZ EGTs concurrently - to establish that certain moves are not move-wasting moves. Obviously, if the defender cannot force a revisit to a position 'in one metric', they cannot do so 'in any metric'

If a move is a 'unique progress move', what I call an 'effectively unique move', it is optimal in all metrics, sure. However, there can be multiple progress moves which are each 'unique optimal moves' in different metrics. Easy to set up a position in which one has a 3-way choice of pushing a Pawn 'in one', converting force in 2 or mating in three.


In a study context, one would not want to play a sub-optimal move that allowed Black to force a White (intent on winning) to revisit ANY previous position. It may be that you are pushing your wKing from a1 via b1 to c2, and you identify an alternative move from b1 that forces a return to a1 but not necessarily to b1 and Kb2 is also available.

The other point to make in a study context is that a dual move which is not a 'move-wasting move' may also be ignorable. If it necessarily converges with the mainline of the study only a few plies 'downstream', it is maybe relatively ignorable. My example of wK moving from a1 to c2 via either b1 or b2 is a case in point. These 'duals' can in some cases be eliminated merely by having a more expressive notation for 'allowable moves', e.g. 1.Ka1 ... 2.Kb1/Kb2 ... 3.Kc2.


I don't see a clearer demonstration that a move is 'effectively unique' than to create an EGT for the variant of chess with the 'current position' set to draw rather than win. However, I have shown that such EGT-generations need not take long, given WDL information, the DTZ metric, the Pawn-slice constraint and parallelisation.


Apologies: I'll have to consider your question about the 3rd-rank defence ... or just refer you to the relevant JNunn book (which will probably be more effective).

Good thinking anyway: I'll put up some positions today with very few men where there are questions about the 'alternative winning moves'.

g

[Arpad Rusz]

This discussion reminds me the following study by Siegfried Hornecker:

Mat Plus, 2009
7k/6p1/3p4/p2P2P1/P7/8/8/5K2


White wins

1. g6 Kg8 2. Ke2 Kf8 3. Kd3 Ke7 4. Kc4 Kf6 5. Kb5 Kxg6 6. Kxa5 Kf5 7. Kb5 $1 g5 8. a5 g4 9. a6 g3 10. a7 g2 11. a8=Q g1=Q 12. Qa3 Ke4 13. Kc6 $1 Qd4 14. Qa5 Kf5 15. Qb5 Kf6 16. Qb6 Qxb6+ 17. Kxb6 Kf5 18.Kc7 Ke5 19. Kc6 +-

Siegfried showed me this in 2007, with the remark that sadly, 16.Qf1+ also wins...
The QP/QP tablebase shows the following:

Before move 15. we have:
15.Qb5...#69
15.Qa3...#71
and there are no more winning moves.

15.Qa3 can be easily spoted as a time losing dual (15.Qa3 Ke4 16.Ka5! only move to win), so we have no problem here.

15...Kf6 was a suboptimal move:

15...Qe3...#68
15...Qf4...#61
15...Qe5...#60
15...Kf6...#59
15...Qe4...#59
................

On move 16:
16.Qb6...#59
16.Qa5...#70
16.Qf1+...#74
and there are no more winning moves.

16.Qa5 again is a simple time losing move but what about 16.Qf1+?
After 16.Qf1+ we have:
16...Ke7...#73
16...Kg6...#65
16...Kg7...#65
16...Kg5...#64
and so on.

I have found that the suboptimal 16...Kg5! is the only move that forces white back after 17.Qg2 Kf6 18.Qf1! Kg5 19.Qc1+ Kf5 20.Qb1+ Kg5 21.Qb5 Kf6. This saves the study!

************************************
I prefer the following version but he didn't like it:

6k1/6p1/3p2P1/p7/P2P4/K7/8/8


1. d5 Kf8 2. Kb3 Ke7 3. Kc4 Kf6 4. Kb5 Kxg6 5. Kxa5 Kf5 6. Kb5 g5 7. a5 g4 8.a6 g3 9. a7 g2 10. a8=Q g1=Q 11. Qa3 Ke4 12. Kc6 Qd4 13. Qa5 Kf5 14. Qb5 Kf6 15. Qb6 Qxb6+ 16. Kxb6 Kf5 17. Kc7 Ke5 18. Kc6 +-

8/8/2Kp4/3Pk3/8/8/8/8

[guyhaw]

Arpad Rusz’s contribution is excellent and timely.

Siegfried Hornecker’s study is HH#75649 in Harold van der Heijden’s Study Database, http://www.hhdbiv.nl/ for those who want to explore.

In discussing it, I will use the following notation:
- - - - - °: stm has only one ‘physical’ move on the board
- - - - - "': stm has only one move which preserves the theoretical value of the position
- - - - - ": stm has only one move after the move-filtering strategy has been applied
- - - - - ': stm is playing one of the optimal moves indicated by the strategy nominated
- - - - - Chess(P): a variant of chess in which the ‘one side to move’ positions in set P are deemed drawn rather than won.

Arpad is correct that White’s path to the win is absolutely unique until move 16w. 15.Qa3 concedes 2m in DTM terms as after 15...Ke4", White must play 16.Qa5"' and then 16...Kf5" returns to position 15w at move 17w.

Position 16w = ‘P1’ = 8/8/2Kp1k2/1Q1P4/3q4/8/8/8 w.

He states that, here, “16.Qa5 again is a simple time losing move” but this is a good example of the difficulty of assessing whether moves are time-wasting or not. I have not been able to convince myself that it is. How does one explore the tree of moves systematically to prove this, especially when White is not constrained to make progress in metric terms on every move?! The trees are potentially large as 16.Qa5 concedes 11m of depth and 16.Qf1+ concedes 15m. I would very much welcome an exposition of why 16.Qa5 is merely a time-wasting move.

A fuller explanation of why 16.Qf1+ is also a time-wasting move would be even more welcome.

How much simpler it would be to say, assuming it’s the case, that the endgame table for Chess(P1), i.e. Chess with position P1 deemed to be a draw, indicates that 16.Qa5 and 16.Qf1+ only draw.

The computation of the relevant Chess(P1) EGT is equivalent to computing an EGT for a 4-man endgame as the pawns do not move.

g

[Arpad Rusz]

He states that, here, “16.Qa5 again is a simple time losing move” but this is a good example of the difficulty of assessing whether moves are time-wasting or not. I have not been able to convince myself that it is. How does one explore the tree of moves systematically to prove this, especially when White is not constrained to make progress in metric terms on every move?! The trees are potentially large as 16.Qa5 concedes 11m of depth and 16.Qf1+ concedes 15m. I would very much welcome an exposition of why 16.Qa5 is merely a time-wasting move.
A fuller explanation of why 16.Qf1+ is also a time-wasting move would be even more welcome.

There is nothing complicated here with 16.Qa5 or even with 16.Qf1+.

I think we should start with this position (A):
8/8/2Kp4/3P4/3qk3/Q7/8/8


The only winning move is 14.Qa5. Let the black move be 14...Kf5 (as in the study).
We arrive to position B:

8/8/2Kp4/Q2P1k2/3q4/8/8/8

As 15.Qa3 leads to the previously visited position A after black's swithback, we have 15.Qb5 as "only" move. After 15...Kf6!? we have the critical position C:

8/8/2Kp1k2/1Q1P4/3q4/8/8/8


16.Qa5 leads to position B after black's swithback 15...Kf5(!).
16.Qb6 is the best move leading to a "winning position".
16.Qf1+ Kg5(!) (position D):

8/8/2Kp4/3P2k1/3q4/8/8/5Q2


Now we have three winning moves: 17.Qb5 (Switchback), 17.Qg2+ Kf6 18.Qf1+ (forced) 18...Kg5 (position D), or 17.Qc1.
After 17.Qc1 Kf5 we have position E:

8/8/2Kp4/3P1k2/3q4/8/8/2Q5


Again there are 3 winning moves: 18.Qf1 (Switchback), 18.Qa3 Ke4 (position A) and 18.Qb1+ Kf6 (position F):

8/8/2Kp1k2/3P4/3q4/8/8/1Q6


And now only 19.Qb6 and 19.Qf1+ wins. As we have seen 19.Qf1+ Kg5 (position D), there is no more doubt that we should play Qb6 already from the critical position C.

The "winning position":
8/8/1QKp1k2/3P4/3q4/8/8/8


If we set this position in the tablebase as a draw, all positions A to F will be also drawn positions.

[guyhaw]

Arpad ... thank you for your explanation of the Hornecker study which I will ponder.

The study is a neat example of something I'm writing about at the moment, so I may use it (and your info) with acknowledgement for your contribution.

Basically, if one could calculate EGTs for variants of chess like Chess(P), there could be lots of uses for composers I guess:

- - - a) confirming that all the wtm positions in a study go to 'draw' if the end position is deemed to be a draw,
- - - b) seeing if alternative winning moves are actually just time-wasters (as discussed),
- - - c) seeing if a non-metric-optimal move in a game allows the opponent to force repetition of a position (as for 'b'),
- - - d) seeing if an equi-optimal move gives a path which must necessarily converge with the given study mainline after a few plies,
- - - e) just seeing what the impact would be of a position being deemed a draw (e.g. Philidor 1777 KQKR or the KBBKN pseudo-fortress)
- - - f) seeing if the winning line from a wtm 'squeeze position' (easier to win from btm) necessarily goes through the btm position
- - - g) your hypothetical use of such a facility goes here

Thanks - g

[Arpad Rusz]

g)Identifying the "key positions" in a tablebase. (Positions whose evaluation affects the value of lot of other positions.)
Examples are the "critical position" and the "winning position" from the previous study.
It would be nice to have a number attached to each position (= the number of positions which gets affected if we change its value).

[guyhaw]

Agreed about 'key positions' in endgames. I'm open to suggestions, and I'm going through J.Nunn's first trilogy now - and maybe the 2nd trilogy which I also have has something to say.

My list at the moment is basically Philidor(1777) for KQKR and the 'Roycroft' Kc2Nb2 pseudo-fortress for KBBKN.

g

[guyhaw]

Something simpler, even trivial:

8/8/8/pP6/4R3/3K4/8/1k6 w ... DTC = 2, DTM = 4, DTZ = 1

Minimising DTC, 1.Ra4 ... minimising DTM, 1.Kc3 ... minimising DTZ, 1.b6 (obviously not a time-wasting move )

Maybe this can be demonstrated with less men and/or less plies?

g

[guyhaw]

Took me a while to get enough 'space' to work through Arpad's explanation which is logical enough ... there were a number of positions with the wK on 'x' and the bK on 'y' and these were accounted for.

Ok: I guess we're doing what the chess engines do. Once they see a position they've seen before in this line of the tree, they say 'no progress' and rightly deem the line a draw.

[ I think the Graph-History Interaction problem is something to do with seeing a position before and incorrectly calling it a draw. ]

If we call position p16b in the mainline position C', then - after all other options (given what Black chooses to reply with which is the difficult bit to find), White will play 19.Qb6 and finish up at position C' which it could have been at in position p16b in the mainline.

Agreed: if position C' is set to 'draw' in the Chess(P) EGT, all of positions A-F would be 'draw'. What a pity we can't do it yet!


So the question is ... is the proof that says 'the EGT for Chess(P)' says all other moves are time-wasters' preferable to the tree-based proof (with some 26 moves and positions) that I now have in my chess engine? This study and position 16w is a good one to use for the discussion.

A proof that says: position C' is a draw in Chess(C') ==> alternative moves at p16w only draw
is lexically neater than a tree of moves, where everywhere one would have to check that all White's options (using the EGT) have been accounted for.

Of course, one would have to trust the creator of the EGT for Chess(C') but one is trusting the creator of the Chess EGT when checking out White's and Black's options anyway. Counter-argument: the main Chess EGT _has_ been verified as correct.

g

[guyhaw]

This is a fascinating study but I wonder if Black's 16...Qxb6+ (hoping that White would get on the wrong foot with regard to the Trebuchet position) qualifies as 'best defence' as Study Protocol requires.

DTM was -58 but after 17.Kxb6 was -15. Black surely could have made White work harder. Maybe the idea was to bring the study to some sort of clear win for White but should Black throw in the towel to achieve that?


An EGT-based 'proof' that the non-DTM-optimal moves at position 16w are time-wasters has the economy of a one-liner:
- - - - - 'Moves 16.Qa5 and 16.Qf1+ are time-wasters'

We then have to put our faith in this 'oracle' statement, so it should come from a trusted EGT generator.

Demonstrations of moves being time-wasters which are based on a tree of lines and moves obviously require the reader to do a lot more work, and this time to trust that the author of the tree has, on this occasion, explored all the options for both White and Black - which could be tricky to near-impossible.

[ However, in this case, Arpad was totally thorough. ]

g

[Arpad Rusz]

This is a fascinating study but I wonder if Black's 16...Qxb6+ (hoping that White would get on the wrong foot with regard to the Trebuchet position) qualifies as 'best defence' as Study Protocol requires.

It is the best defence! (In a game it would be a bad move but chess studies are using different logic.)
There is no other move that forces White to find unique winning moves. All the other Black defences are not part of the study: we could say that White simply captures the pawn and has a "theoretical win".

(The thematic try 15.Qa5-b6? is also followed by 15...Qxb6! -+)