# Difference between revisions of "Retrograde Analysis"

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Retrograde analysis [1]

Retrograde Analysis,
a method in game theory to solve game positions for optimal play by backward induction from known outcomes. A sub-genre of solving certain chess problems uses retrograde analysis to determine which moves were played to reach a position, and for the proof game whether a position is legal in the sense that it could be reached by a series of legal moves from the initial position.

# History

History based on Lewis Stiller, Multilinear Algebra and Chess Endgames [2]

```The mathematical justification for the retrograde analysis algorithm was already implicit in the 1912 paper of Ernst Zermelo [3]. Additional theoretical work was done by John von Neumann and Oskar Morgenstern [4].

The contemporary dynamic programming methodology, which defines the field of retrograde endgame analysis, was discovered by Richard E. Bellman in 1965 [5]. Bellman had considered game theory from a classical perspective as well [6] [7], but his work came to fruition in his 1965 paper, where he observed that the entire state-space could be stored and that dynamic programming techniques could then be used to compute whether either side could win any position.
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```Bellman also sketched how a combination of forward search, dynamic programming, and heuristic evaluation could be used to solve much larger state spaces than could be tackled by either technique alone. He predicted that Checkers could be solved by his techniques, and the utility of his algorithms for solving very large state spaces has been validated by Jonathan Schaeffer et al. in the domain of Checkers [8], Ralph Gasser in the domain of Nine Men’s Morris [9], and John Romein with Henri Bal in the domain of Awari [10]. The first retrograde analysis implementation was due to Thomas Ströhlein, whose important 1970 dissertation described the solution of several pawnless 4-piece endgames [11].
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# Algorithm

Retrograde analysis is the basic algorithm to construct Endgame Tablebases. A bijective function is used to map chess positions to Gödel numbers which index a database of bitmaps during construction and retrieval, in its simplest form based on a multi-dimensional array index. Following description is based on Ken Thompson's paper Retrograde Analysis of Certain Endgames with depth to mate (DTM) metric [12], and assumes White the winning side. Files of sets of chess positions, where a one-bit is associated with the Gödel number of a position, are successively manipulated during the iterative generation process:

• Bi set of the latest newly found Black-to-move and lose in i moves positions
• Wi set of the latest newly found White-to-move and win in i moves positions
• B set of all currently known Black-to-move and lose positions, union of all Bi so far
• W set of all currently known White-to-move and win positions, union of all Wi so far
• Ji is temporary superset of Bi not necessarily lose positions

The algorithm starts in enumerating all Black-to-move checkmate positions B0 with the material configuration under consideration, an un-move generator is used to to build predecessor or parent positions. The un-move generation is similar to move generation, with the difference that it is illegal to start in check, but legal to un-move into check, and illegal to capture, but legal to un-capture by leaving an opponent piece behind.

for (i=0; Bi; i++)

1. Every parent of a Bi position is a White-to-move won position - newly-won positions Wi+1 are parents of a Bi not (yet) in W
2. Wi+1 becomes subset of W
3. Every parent of a Wi+1 position is a Black-to-move and lose position if Black wanted to mate himself, stored in Ji+1
4. Only if all successors (by generating and making legal moves [13]) of a Ji+1 position are member of W, the Ji+1 position becomes member of Bi+1 and B

The algorithm terminates, if no more newly predecessor positions were found, that is either Wi+1 or Bi+1 stay empty. Each one-bit in W or B correspondents to a White-to-move and won or Black-to-move and lose position. Remaining zero bits indicate either a draw, White-to-move and lose, Black-to-move and won, or illegal positions.

# Selected Publications

## 1910 ...

• Ernst Zermelo (1913). Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels Proc. Fifth Congress Mathematicians, (Cambridge 1912), Cambridge Univ. Press 1913, 501–504. Translation: On an Application of Set Theory to the Theory of the Game of Chess.

# Forum Posts

## 2000 ...

Wu/Beal predates Koistinen by Guy Haworth, CCC, December 04, 2001

## 2010 ...

Re: Reverse move generation by Harm Geert Muller, CCC, December 30, 2014