Some Chess-specific Techniques

1. Some Chess-specific Techniques • Chess is played on an 8x8 board • 8 is a power of two • Computers are especially good at handling powers of two • This happy accident has been exploited in chess programs in two main ways: • move generation speedups • bitboard representations http://csiweb.ucd.ie/Staff/acater/comp4031.html Artificial Intelligence for Games and Puzzles

2. Move generation: the obvious way • A natural way to represent the board is with an 8x8 array, • containing values 1..12 representing black pawn, ... white king • indexed by • rank running from 0 to 7 • file running also from 0 to 7 • When a chessman move is being considered, the new rank and file must satisfy • rank ≥ 0 and rank ≤ 7 and file ≥ 0 and file ≤ 7 • A rook can move in a ray over several empty squares, along a file {rank := rank ± n} or along a rank {file := file ± n} • A bishop can move in a ray over several empty squares, along a diagonal {rank := rank ± n; file := file ± n} • A queen can do either the above. • Knights, Kings and (usually) Pawns do not move in rays. http://csiweb.ucd.ie/Staff/acater/comp4031.html Artificial Intelligence for Games and Puzzles

3. Move generation with vector-based board - 1/3 • It is faster to access a 1-dimensional vector than a 2-dimensional array. • 8x8 board may be linearised so that 64 elements of a vector [0…63] correspond to squares a1, b1, c1, d1, e1, f1, g1, h1, a2, b2, … g8, h8 • Then a rook moves from square X to X - n (stopping after a multiple of 8) or to X + n (stopping short of multiple of 8) or to X ± 8n (stopping before going negative or going beyond 63) • A bishop moves from square X to X ± 9n to X ± 7n • A knight moves from square X to X±10, X±17, X±15, X±6 http://csiweb.ucd.ie/Staff/acater/comp4031.html Artificial Intelligence for Games and Puzzles

4. Move generation with vector-based board - 2/3 • For all piece types, the legality of moves must be checked. • Checking a proposed move does not go off the top or bottom of the board is easy, just check {X ≥ 0 and X ≤ 63} • Checking a proposed move does not go over the left or right edge of the board is not as easy as that; effectively you must decompose X into its rank & file components. • Can do this fairly cheaply since divide-by-8 and modulo-8 can be done with a rightshift operation and a bitwise-and operation, respectively. • The top/bottom check can be made cheaper by exploiting these facts: • all numbers 0-63 have a ‘0’ in bit position 7 (from least significant) • negative numbers, and also numbers 64-127, have a ‘1’ in bit position 7 • so whenever {X & 64} is non-zero then X is not valid http://csiweb.ucd.ie/Staff/acater/comp4031.html Artificial Intelligence for Games and Puzzles

5. a1,b1,c1,d1,e1,f1,g1,h1, , , , , , , , , a2,b2,c2,d2,e2,f2,g2,h2, 1st rank 8 unused elements 2nd rank Move generation with vector-based board - 3/3 • The board vector need not be exactly the right size. It can be bigger than needed. • It is useful for it have 128 elements. (Actually only 120 is enough!) • Chessboard squares can be mapped to vector elements as follows: • This allows both rank and file parts of an index to be checked in one operation: • indices with a ‘1’ in bit 8 are negative or greater than 127 • +ve indices with a ‘1’ in bit 4 are in the range 8-15, 24-31, 48-55, … • they correspond to vector elements not mapped to any square. • When (X & #x88) is nonzero X is invalid. This is the “Hex-88 Trick”. http://csiweb.ucd.ie/Staff/acater/comp4031.html Artificial Intelligence for Games and Puzzles

6. Bitboards • Rather than use a vector containing codes identifying chessmen, many programs use a collection of several 64-bit vectors, using one bit per chessboard square. • The technique was pioneered by “Kaissa”, a Russian program of the 1970s which ran on a computer with 64-bit words; as today’s “Alpha” chips. • Each “bitboard” uses ‘1’ bits to represent different things about a square. • Making a move then involves changing at least three of these bitboards. http://csiweb.ucd.ie/Staff/acater/comp4031.html Artificial Intelligence for Games and Puzzles

7. “Ordinary” moves of knight and king • These pieces move fixed distances in any of 8 directions. • They capture any enemy piece occupying their target square. • There is no need to consider any squares other than source and target. • For each type of piece, • and for each source bit position, • up to eight bitmaps can be constructed and stored • to represent the up-to-8 possible targets for that piece type starting from that source square • If say the “d5” bit is set in the “white” and “knight” bitboards, • then consider the eight d5-knight bitmaps in turn • check that {WhiteBitboard & currentbitmap} is zero, if so • zero the “d5” bit in the occupied, white, & knight bitboards • zero target bit in the black, pawn, bishop, rook, queen, & king bitboards • set to one the target bit in the occupied, white, & knight bitboards http://csiweb.ucd.ie/Staff/acater/comp4031.html Artificial Intelligence for Games and Puzzles

8. Rook moves • Rooks (and Queens) can move in rays, along ranks or along files. • all squares they pass over must be unoccupied • the final square may be unoccupied or occupied by an enemy piece • For each square, a bitmap can specify the other squares a rook (or queen) could reach if unimpeded. • To see if a rook (queen) could actually reach a particular square involves checking the occupied bitboard against a substring of bits. (If a bitwise AND returns nonzero, the square cannot be reached) • It is better then to use four separate bitmaps - attack bitboards - for the four different directions of movement. http://csiweb.ucd.ie/Staff/acater/comp4031.html Artificial Intelligence for Games and Puzzles

9. Flipped bitboards • Checking substrings of bits involves forming substrings of bits. • This is easily done by bit-shifting when the bits in a bitstring are consecutive in memory. For the bitboard organisation shown before, that makes it easy to manipulate attack bitboards for attacks along a rank, but not along a file. • By using additional bitboards, flipped on their sides, this too is made easy. http://csiweb.ucd.ie/Staff/acater/comp4031.html Artificial Intelligence for Games and Puzzles

10. Bishop moves • Bishops (and Queens) can move in rays, along diagonals. Again, • all squares they pass over must be unoccupied • the final square may be unoccupied or occupied by an enemy piece • For each square, a bitmap can specify the other squares a bishop (or queen) could reach if unimpeded. • As before, in order to generate substrings of bits easily it would be necessary to transform the bitboards. This (complex) transformation has the name “rotated bitboard”. • The term “rotated bitboard” has come to apply to flipped bitboards as well as the two transformations to follow: http://csiweb.ucd.ie/Staff/acater/comp4031.html Artificial Intelligence for Games and Puzzles

11. Rotating a bitboard for a1…h8 1. Cut bitboard along a diagonal http://csiweb.ucd.ie/Staff/acater/comp4031.html Artificial Intelligence for Games and Puzzles

12. Rotating a bitboard for a1…h8 1. Cut bitboard along a diagonal 2. Rearrange halves to make parallelogram http://csiweb.ucd.ie/Staff/acater/comp4031.html Artificial Intelligence for Games and Puzzles

13. Rotating a bitboard for a1…h8 1. Cut bitboard along a diagonal 2. Rearrange halves to make parallelogram 3. Squash to make new square http://csiweb.ucd.ie/Staff/acater/comp4031.html Artificial Intelligence for Games and Puzzles

14. Rotating a bitboard for a8…h1 1. Cut bitboard along a diagonal 2. Rearrange halves to make parallelogram 3. Squash to make new square http://csiweb.ucd.ie/Staff/acater/comp4031.html Artificial Intelligence for Games and Puzzles

15. Using rotated bitboards • Programs that use rotated bitboards do so for the sake of efficiency of the logic for move generation and attack detection. • They use many precomputed bitboards, using memory to gain speed. • Whenever a move is made, a few more bitboards must be kept up to date: • Regular file-oriented bitboards, for occupied, white, black, pawn etc. • often separating further: blackpawn, whitepawn, blackknight, … • Flipped, rank-oriented versions of all of those. • Rotated a1-h8 versions, and • Rotated a8-h1 versions • With true 64-bit machine words, the bitwise operations are very fast. • With 32-bit words, even if a language allows 64-bit integers, benefits are less. http://csiweb.ucd.ie/Staff/acater/comp4031.html Artificial Intelligence for Games and Puzzles

16. Some online references • supertech.lcs.mit.edu/~heinz/dt/node8.html • (beware, rank & file confused in 1st paragraph) • www.onjava.com/pub/a/onjava/2005/02/02/bitsets.html?page=last&x-showcontent=text • atlanchess.com/html/rotated_bitboards.html • www.fzibi.com/cchess/bitboards.htm http://csiweb.ucd.ie/Staff/acater/comp4031.html Artificial Intelligence for Games and Puzzles