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Endgame Tablebases, (EGTBs or EGTs)

precalculated endgame tables generated by an exhaustive retrograde analysis. During its game play and/or search, recognizing a specific material composition, a chess program can probe, or in principle compute these tables to determine the outcome of positions definitively and act as an oracle providing the optimal moves. The tables are usually divided into separate files for each material configuration and side to move. What all different formats of tablebases have in common is that every possible piece placement has its own, unique index number which represents the position where the information about the position is located in the file. Positions with non-null castling rights could be represented in an EGTB but typically are not.

The term Tablebase opposed to endgame database was coined by Steven Edwards by means of a data vector and a single file access for both sides to move, when he introduced his Edwards' Tablebases, addressing several shortcomings of Ken Thompson's earlier database of positions with the weaker and hence potentially losing side to move only [2] .

Purposes

When a chess game goes to the endgame phase, it may complete soon with the result as a draw or a win/loss. Both human/chess engines can finish the endgame by continuing searching, applying some endgame rules. The game can finish in this way but it is not in a perfect way: it takes much time to make move, use more moves than necessary. Sometimes the winning side may lose the change since the rule 50 moves.

EGTB can help to solve this period. It is a kind of dictionary of all endgame positions which can answer directly or indirectly for a given position:

it is a draw or a win/loss

how far/how many move from matting/being mated

what is the best move

The main advantages of using EGTBs:

It does not search but retrieves data thus it can move instantly

The move is the perfect one, leading the shortest line to win

The main disadvantages of using EGTBs:

Data too large for downloading and storing

If use when searching, it may slow down the search

Require a complicated code for probing

Data

An EGTB is a set of endgames. Each endgame is a set of positions (they must have the same material). Each position associates with an integer number which informs how far that position is from mating/being mated or converting (depends on the type of its metrics). All numbers of an endgame are simply organized as two arrays of integers (one array for one side/color). In other words, each item in those arrays is an integer and its index on an array is mapped to a unique position.

Based on values those integer numbers can answer directly two purposes of the EGTB: for a given position it is a draw or a win/loss position and how far it is from mating/being mated or converting. From those numbers, the best move of a position can be indirectly calculated as below part mentions. Theoretically, we can add more information to each item. However, due to large involving positions, any additional information may make the whole EGTB becomes significantly larger. None of the popular EGTBs store any extra information for each item.

Those arrays of an endgame can be stored in files. Typically each array is stored in one file. However, sometimes they can be combined into one file or device into more files to be easy to copy or manage. Those files may be in the form of none or compressed. If they are compressed, they are divided and compressed by small blocks thus they don't need to decompress all to extract data of just a few positions.

The size of an endgame depends on:

The number of involving pieces: more pieces, the bigger size

Type of pieces: for standard chess, more Pawns is the smaller size

Type of metric

The largest value: the integer number should be large enough to cover the largest value. Different endgames will have different largest values. Thus some endgames need only one byte per item but some need 2 bytes

Indexing: The algorithms to map between an array item and a position

Compress ratio

So far the size of an EGTB is the most important factor of success.

Metrics

DTM

Depth to Mate 'Mate' is the ultimate goal and ends the game. For each position that is represented, 'DTM' indicates the theoretical value, and the number of winner's moves to 'Mate' if won/lost - assuming that the winner is minimizing and the loser is maximizing DTM.

DTC

Depth to Conversion 'Conversion' is the changing of the force on the board, by capture and/or Pawn-conversion. For each position that is represented, 'DTC' indicates the theoretical value, and the number of winner's moves to the goal of 'Won Conversion' if won/lost - assuming that the winner is minimizing and the loser is maximizing DTC. DTC £ DTM.

DTZ

Depth to Zeroing (of the ply count) The ply-count is zeroed if and only if a Pawn is pushed or a capture is made. For each position that is represented, 'DTZ' indicates the theoretical value, and the number of winner's moves to the goal of 'Win and ply-count zeroed' if won/lost - assuming that the winner is minimizing and the loser is maximizing DTZ. DTZ £ DTC £ DTM.

DTZ50

Depth to Zeroing (of the ply count) in the context of the 50-Move Rule The DTC, DTM and DTZ metrics do not consider the current FIDE 50-move rule whereby a game in which no irreversible moves have been played for 100 plies can be claimed as a draw by the player 'on move'. In this metric, in addition to the DTC/DTM/DTZ draws, positions requiring more than 50 winner's moves to zero the ply-count are defined as draws. DTZ £ DTZ50 if DTZ50 ¹ draw. 'DTZ50' is one of a set of 'DTZk' rules: a k-move rule effectively applies if there are only k moves left before a draw-claim might be made. k1 £ k2 implies DTZk2 £ DTZk1 if DTZk1 ¹ draw.

Note that, because depth is measured in winner's moves rather than plies, this metric does not define as draw a position in which 50 winner's moves and 51 loser's moves are required to zero the ply-count, despite the fact that the loser would claim a draw after 100 plies had been played in that endgame.

DTR

Depth by the (k-move) Rule 'The Rule' referred to here is a notional set of k-move rules. For each position that is represented, 'DTR' indicates the theoretical value and, for won/lost positions, the least k for which DTZk is not draw - assuming that the winner is minimising and the loser is maximising DTR. DTZ £ DTR. No DTR EGTs have been created to date [3] .

DTZR

Depth to Zeroing move, give that DTR is being minimaxed If DTR = draw, then DTZR = draw. Otherwise, Otherwise DTRZR indicates the theoretical value, and the number of winner's moves to the zeroing of the ply count, given that the two sides are minimaxing first DTR and then DTZR. DTZR would be used in conjunction with DTR to progress a deep win in the context of a k-move rule because DTR can remain unchanged over a sequence of moves. DTZR £ DTR but it is not always the case that DTZ £ DTZR as may seem intuitively obvious: the loser may zero the move-count to maximise DTR.

Summary

To summarize the different metrics of tablebases, each metric has its strengths and weaknesses. Nalimov's DTM EGTs are widely and commercially available, and used by many chess engines or chess GUIs. DTC EGTs are more easily computed and stored as they involve smaller depth-ranges and are more compressible. For endgames with less or equal than 5 pieces on the board, KNNKP comes closest to requiring more than one Byte per position to store DTM: maxDTM = 115 (1-0) and 73 (0-1) necessitating 190 values in the EGT. For KPPKP, maxDTM = 127 (1-0) but only 42 (0-1). But for certain 6 piece endgames, one Byte is not enough: maxDTM = 262 for a KRNKNN position. The DTC and DTZ metrics increasingly postpone the need for two Bytes per position in an uncompressed EGT, but KRBNKQN has maxDTC = maxDTZ = 517 (0-1), both wtm and btm.

In extremis, he unmoderated use of DTM, DTC or DTZ EGTs will let slip a winnable position in the context of the 50-move rule ... hence the marginal need for the DTR and DTZR metrics, coupled with the ability to recompute DTR/DTZR EGTs in the context of a limited move-budget.

Bitbases

see main article Endgame Bitbases

Inside the search much denser tables with value {won, draw, loss, invalid} or boolean {win, not_win} information are sufficient with respect to bounds. Denser tables are more efficiently used, given that memory and cache are limited resources. Boolean Win/Not_Win Bitbases require only one bit per position (and if necessary, further enquiry if draw is to be distinguished from loss with respect to bounds). Value Bitbases require two bits per position.

Indexing

Every position is assigned to a unique index to specify the location of the information stored about it. The main purpose of indexing is to locate and read information (from databases) such as distance to mate for a given position based on its index.

There are two main aims when generating the index number. The easier the algorithm to generate the index from any given position, the faster will be the probing of the tablebase as well as the generation of it. But the second aim is to get a relatively small range of index-numbers, to keep the size of the file as small as possible.

The most straight forward way would be to for every piece on the board use 6 bit (1-64 squares)

index = position_white_King + position_black_King * 64 for each piece { index = index * 64 + position }

Almost all work on this field is to reduce index spaces and reduce data sizes (e.g., by compressing). The above indexing can be optimised in several ways.

Remove redundant symmetrical indexes:

A chess board can be rotated and mirrored (if pawns are on the board only a vertical reflection is possible).

For pieces of the same type, different arrangements can be left away.

Remove illegal indexes:

Pieces may not stand on the same square.

The side to move may not give check.

The distance between the two Kings must be greater than one.

Retrieving

Probe

When a chess program (e.g., a chess engine or a chess GUI) works with an EGTB, it needs to retrieve data (integer numbers) for some given chess positions. The process named probe and has below steps:

convert a given chess position into an index via indexing algorithms

access data (typically organized as an integer array) using the index to retrieve the integer of that chess position

If data of that endgame is not ready in memory, usually the probe process has to do some extra work:

From data index calculate to block index

Read data of that block from storage into memory

Uncompress the block (if it is compressed)

Convert the endgame index into block index and retrieve the right data

Search

After probing the chess engine may have the result as an integer number for the queried position. Depending on the metric, the number may be a distance to mate or distance to conversion, the position is a draw or a win or loss for the querying side. Sometimes that value (state of draw/win/loss) may be enough for searching. However, sometimes that state is not enough and it needs to know which is the best move to make. The solution is to generate all legal moves from that position, make one by one, probe that EGTB for values of each new position. After all, compare the values of all children's positions to find which one is the best to make.

Speed

Typically the probe process requires some computing and it may access storage which is usually so slow, compared with one stored in memory. All may make it be a slow one. An engine that probes EGTB when searching may make the search be slower. The good point is that the engine can stop searching on the branch probed the EGTB.





Brief Info

of some popular Endgame tablebases:

Format Metric 1st published 5 men 6 men 7 men Thompson DTC 1991 2.5 GiB (not completed) - - Edwards DTM 1994 56 GiB (estimated) - - Nalimov DTM 1998 7.1 GiB 1.2 TiB - Scorpio WDL 2005 214 MiB 48.1 GiB - Gaviota DTM 2008 6.5 GiB - - Lomonosov DTM 2012 - - 140 TiB Syzygy WDL + DTZ50 2013 / 2018 939 MiB 150.2 GB 17 TB

Formats

Generators

7-man EGTBs

2005

In 2005, Yakov Konoval started to collaborate with Marc Bourzutschky on constructing 7-man EGTBs. Yakov wrote the generator and Marc a verification program and some utilities for extracting the data. In October 2005, Yakov Konoval and Marc Bourzutschky announced that a position in the ending of a KRRNkrr requires 290 moves to convert to a simpler winning endgame [4] . The old record was 243 moves from a position in a rook and knight versus two knights endgame, discovered by Lewis Stiller in 1991 [5] .

2006

In March 2006, the wizards of 7-men endgames, Marc Bourzutschky and Yakov Konoval found a 330 moves win in KQBNkqb [6] and in May 2006 a 517 moves win in KQNkrbn [7] .

2012

In July 2012, Vladimir Makhnychev and Victor Zakharov from Moscow State University completed 4+3 DTM-tablebases (525 endings including KPPPKPP). The next set of 5+2 DTM-tablebases was completed during August 2012 [8] . The tablebases are named Lomonosov Tablebases, using the Lomonosov Supercomputer.

2013

Convekta Ltd. announced the release of Lomonosov Tablebases. Until December 31, 2013, the tablebases will be available online through ChessOK products [9].

2018

Starting in March 2018, seven piece Syzygy Bases, originally created for up to six pieces by Ronald de Man in 2013, were generated by Bojun Guo, finished in August 2018 [10].

See also

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