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GP End-Chess Evolution of Chess Endgame Players Ami Hauptman & Moshe Sipper Outline Introduction The Game of Chess – a solved problem? Important differences between human and artificial chess players Chess Endgames - features & building blocks GP problem definition Experiment and Results

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gp end chess

GP End-Chess

Evolution of Chess Endgame Players

Ami Hauptman & Moshe Sipper

  • Introduction
    • The Game of Chess – a solved problem?
    • Important differences between human and artificial chess players
  • Chess Endgames - features & building blocks
  • GP problem definition
  • Experiment and Results
  • Future work
the game of chess
The game of Chess
  • First developed in India and Persia
  • Considered THE complex game of strategy and inventiveness
  • Enormous search space
    • Roughly 50 possible moves at mid-game
    • A typical game consists of a few dozen moves
    • Estimated at 1043 in 40-move game (Shannon, 1950)

Elephants don’t play Chess(?)

the game of chess ai history
The game of Chess – AI history
  • First chess AI at 1958 – novice level
  • Machine strength increasing linearly
  • 1997 – defeat of former world champion, Garry Kasparov, by IBM’s deep blue
  • Last years – performance still increasing
    • Mainly Hardware
    • Also Software
  • … The End ?
the game of chess5
The game of Chess
  • …NO!
  • Deep(er) blue use extreme brute-force, traversing several millions of boards ps
  • Very little generalization
  • Virtually no human resemblance
  • Deemed theoretically uninteresting
    • Chomsky: As interesting as a weight lifting competition between machine and man
  • Low A to I ratio; Low return
the game of chess basic concepts
The game of Chess – Basic concepts
  • 8x8 board
  • Each player starts with 16 pieces, of 6 different types, and may only move 1 piece per turn
  • A piece can only move into an empty square or into one containing an opponent’s piece (a capture)
  • Win by capturing the opponent’s king
the game of chess pieces
The game of Chess – pieces
  • Pawn: may only move forward (or capture diagonally)
  • Bishop: diagonals
  • Knight: L shaped moves. The only “unblockable” piece
  • Rook: Ranks & files
  • Queen: Bishop & Rook combined
  • King: 1 square in any direction. May not move into attacked square

Values :

1 3 3/3.5 5 9 ∞

the game of chess example
The game of Chess – example
  • White has over 30 possible moves
  • If black’s turn – can capture pawn at c3 and check (also fork)
the game of chess check and checkmate
The game of Chess – Check and Checkmate
  • “Checking” is attacking opponent’s king. Opponent must respond
  • “Mating” (Checkmate)

is when the opponent can’t avoid losing the king – and thus forfeiting the game

human artificial players ai search
Human & Artificial Players – AI search
  • AI uses search to assign a score to a board
  • Traverse the move tree from leaves - up
  • Select the best child using scores found
  • Only partial tree

Computer is O (Max) opponent is X

human artificial players the machine13
Human & Artificial Players – The Machine
  • Millions of boards (nodes) per second
  • Little time for each board – less knowledge
  • Smart search algorithms –
    • pruning
    • Alpha-beta variants (negascout etc.)
  • Still use heuristics at end – can’t see all tree
  • Most research revolves around search
  • Human resemblance minimal – humans use little search
human artificial players humans
Human & Artificial Players – Humans
  • Humans use problem solving cognition
  • Deeply knowledge based –
    • Extensive “theory” exists
    • Numerous books and institutions
  • Massive use of pattern recognition
  • Also use search but
    • Less deep
    • Only develop “good” positions
  • More efficient – less nodes for “same” result
    • Reminiscent of greedy search
  • Not only in chess
human artificial players grand masters findings
Human & Artificial Players – Grand Masters - Findings
  • Play against several opponents at the same level they play against a single opponent
  • GMs and novices: same level of performance when memorizing a random board; differ when memorizing real game positions
  • GM eye movements show they only scan “correct” parts of board
  • Strong Amateurs use the same meta-search as GMs - equally deep, same nodes, same speed; Differ in knowledge of domain (De Groot)
endgames example
Endgames - example
  • White’s turn: mate in 5, with Qe6+
  • Features include:
    • #moves for black king minimal
    • Attacking, un-attacked
    • Checking
    • Officers same line\row
  • Black’s turn: draw with: Rc1+, then Qg5 – fork & exchange
endgames 2 features
Endgames (2) - features
  • Few pieces remain (typically: king, 0-3 officers and sometimes pawns)
  • Fewer options, but more moves for each piece
  • Trees still extremely large
endgames building blocks
Endgames - Building Blocks
  • Main goals
    • Reduce search by “smart” features of the board
    • Use more game knowledge as humans do
  • Allow more complex features to be built by supplying basic ones (terminals) and building methods (functions)
  • Schemata evolution
features example fork
Features Example - Fork
  • My piece is:
    • Attacking 2 or more pieces
    • Protected or not attacked
  • Opponent pieces:
    • Unprotected
    • OR protected but of greater value
  • Example: black must exchange Q for R because of fork
fork traditional ai search
Fork: Traditional AI search
  • Only 3 legal moves for black
  • Find that one of white’s next moves (out of 23 possible) captures black queen
  • Check all following moves for more piece exchanges
  • Sometimes, still check other moves (non capturing)
  • At end of search – compare remaining pieces
  • No fork “concept”
features example fork feature search gp
Features Example – Fork Feature Search (GP)
  • One of the features is isMyFork function – Checks all previously defined conditions
  • Also, use some smaller building blocks:
    • Is Opponent piece Attacked?
    • Is attacking piece protected?
    • Is opponent in check?
    • Value of attacked piece
gp problem definition
GP Problem Definition
  • Terminals & Functions
    • Numerous “chess terminals” and ERCs
    • Non-chess funtions
  • Fitness
    • Tournament
  • Run parameters
  • Termination
  • We will see each element in the following experiments
endgame experiments conducted
Endgame experiments conducted
  • KRKR – each player has 1 king and 1 rook
  • KRRKR – King with 2 Rook against King and Rook
  • KQKQ – Kings and Queens
  • KQRKQR – Combined
basic program architecture
Basic program architecture
  • Generate all possible moves (depth=1)
  • Evaluate each board with GP individual
  • Select board with best score (or stochastically decide between equal)
  • Perform best move
  • Repeat process with GP opponent until game ends (or until only kings left)
krkr endgame
KRKR Endgame
  • Each player has 1 King, 1 Rook
  • “Toy” problem for chess endgames
  • Theoretical draw (experts never lose this)
  • Some interesting positions exist
krkr endgame what needs to be learned 1
KRKR Endgame - what needs to be learned (1)
  • Avoid losing rook
    • Don’t move to attacked, unprotected squares
  • Vice versa - capture opponent’s rook if able

Black to move – white loses Rook

krkr endgame what need to be learned 2
KRKR Endgame - what need to be learned (2)
  • Avoid getting king stuck in edges
  • Again, take advantage if opponent does this

Black to move – mate in 1

krkr endgames terminals
KRKR Endgames - Terminals
  • Used in first runs:
  • Is My Rook Attacked, Is Opp Rook attacked
  • Is {My, Opp} Rook Protected (two as above)
  • Is {My, Opp} Rook In Play
  • Num Moves {My, Opp} king
  • {My, Opp}-King’s proximity to edges
  • Is Mate
  • ERCs: ± {0.25, 0.5, 1} * MAX
    • MAX = 1000 (empirically)
krkr endgames functions
KRKR Endgames - Functions
  • Boolean
    • OR2, OR3, OR4
    • AND2, AND3, AND4
    • NOT
  • Arithmetic - +, -, *
  • Combined - <, =, >, IF
  • STGP
  • For now, no “chess” functions, only terminals
krkr endgames fitness
KRKR Endgames - Fitness
  • Competitive, Random-2-ways
    • Each individual plays against k randomly selected opponents
    • Each game counts for both players
  • For each encounter
    • Several games (typically 4) are played
    • Short games - ~5-8 moves per player
    • Each game starts at a random legal position
    • Safe start - no piece is attacked at the beginning
krkr endgames fitness 2
KRKR Endgames – Fitness(2)
  • Scoring method:
    • Victory: 1-2 points
    • Piece count advantage (theoretical win) – ¾ point
    • Draw: ½ point
      • After advantage – 0 points
    • Loss: 0 points
krkr endgames parameters
KRKR Endgames – Parameters
  • Population size - 80
  • #Generations - 150..250
  • Operators:
    • Reproduction 0.35
    • Crossover 0.5
    • Mutation 0.15 (including ERC mutation)
  • Termination – ~10-25 hours
krkr endgames results
KRKR Endgames – Results
  • Every 10 generations, best individual played against:
    • Best of generation 0
    • An opponent performing random moves
    • Longer games: ~10-12 moves per player
  • 50-150 games
  • Games were doubled – each player staring from both positions
krkr endgames results34
KRKR Endgames – Results
  • Bad results – no distinct improvement
  • Several reasons:
    • Arithmetic operations problematic – we get large numbers
    • Mate not distinct enough (traditionally terminates the search)
    • Boolean functions not clear enough
    • Slow Runs due to large trees with repeating functions
krkr endgames improvements
KRKR Endgames –Improvements
  • Boolean functions
    • Divided to good and bad
    • Example: Is-My-King-In-Check changed to Is-My-King-Not-In-Check
    • Mate changed to 1000*Mate
    • Added Not-My-Rook-Attacked-Unprotected and Opp-Rook-Attacked-Unprotected
krkr endgames results improvements
KRKR Endgames – Results - Improvements
  • Also consulted Chess Experts – added more:
  • Is-Opp-King-Behind-Rook
  • Split to
    • Opp-King-Prox-Rook
    • Opp-King-Behind-Rook
  • Is-Stalemate (only kings left)

Black moves and White loses Rook

krkr endgames results improvements37
KRKR Endgames – Results - Improvements
  • Arithmetic functions canceled
    • Although Still using floats for terminals
    • Also divide to good and bad: NumNotMovesOppKing
    • Theoretically justified – more “logical” search in literature
    • Empirically - need more logical rules, and not : ( > (+ (#moves-k #moves-opp-k) 5.5))
  • Memoization – saves more than ½ the time
krkr endgames final results
KRKR Endgames – Final Results
  • Improvement
    • Above 75% of games against random end in advantage or mate
    • Still, too few mates, even when score for win is increased – difficult to learn move sequence
    • Same against best of generation 0 (almost random)
    • The main thing that was learned was avoiding getting the rook captured
krrkr endgames
KRRKR Endgames
  • Example (right)
  • Very good for white
    • Black king exposed
    • 2 rooks close
    • Next move – captures rook
    • (mate in 5)
krrkr endgames goals
KRRKR Endgames - goals
  • One player has 2 rooks, the other – 1
  • Not theoretically drawn
  • We want one generalized individual for all endgames and positions (Not one for each endgame):
    • Each player needs to play both advantage, draw (KRKR) and disadvantage
    • Terminals need to be more general
krrkr endgames changes
KRRKR Endgames - changes
  • Terminals - changed and added to cope with changing state
    • Material-Count (recall each rook = 5)
    • Num-My-Pieces-Not-Attacked, since now there are more than 1
    • Is-My-King-Protecting-Piece and My-Officers-Same-Line to allow more complex considerations
  • Functions
    • If-Adv-Then-(left child)-Else-(right child)
    • Eventually divided to 3 trees
krrkr endgames changes42
KRRKR Endgames - changes
  • Also added – comparing differences to parent node
    • Boolean Is-Material-Increase, which compares to the parent node (board)
    • Material decrease is not needed since considering only my move
    • Not-My-King-Moves-Decreaseto further use number of moves for king
krrkr endgames opponents
KRRKR Endgames – Opponents
  • Random forsaken; Best-of-0 still used but less
  • Added new opponent – MASTER
    • a program we wrote based on consultation with experts, highest being InternationalMaster Boris Gutkin, ELO 2400 (only about 3000 of those…)
    • Used ~50 general positions and rules derived from them, together with scores for each
    • Defined a strategy (“Expert”) accordingly
  • Tested evolved programs against it
    • Human competitive?
krrkr endgames fitness
KRRKR Endgames – Fitness
  • Test were conducted by assigning each player both roles for each position
  • Fitness was refined – score effected by:
    • Starting position (advantage or disadvantage)
    • End result – win, loss or draw
    • Adv position ending in draw receives a score of near zero
    • Dis-adv ending in a draw will receive better than 0.5
krrkr endgames results
KRRKR Endgames – Results
  • Expert-defined performed extremely well against Random and Best0
  • Evolved programs performed generally as well as expert defined, sometimes better

Percent of favorable results in game outcomes

main experiment kqrkqr
Main Experiment – KQRKQR
  • Most complex endgame we worked with
  • Still theoretical draw
  • Highly position dependant – “noisy”
  • Larger trees
    • 2 officers
    • Queens
  • Easier to mate
kqrkqr endgames changes
KQRKQR Endgames - changes
  • Added – more “heavy” terminals (and components)
    • Boolean Is-Not-Mate-in-one, most time consuming but necessary
    • Boolean Is-My-King-Not-Trapped
      • Not all king’s moves lead closer to edges
      • Important but vague – usually happens with complex terminals
    • My-Officers-Same-Line
kqrkqr endgames new opponent
KQRKQR Endgames – New Opponent
  • CRAFTY, second in the 2004 Computer Chess Championship (held at Bar-Ilan)
  • Uses brute force methods; State-of-the-art search algorithms
  • Specializes in Blitz games (typically 3 minutes per game)
  • We limited to 5 secs per move, enough to scan ~1.5 Million boards with pruning
kqrkqr endgames our parameters
KQRKQR Endgames – Our parameters
  • Used lookahead of depth 2
    • Typically ~5 secs per move
    • Simple Minimax search, but not Alpha-Beta
  • Played 5-6 moves per game
  • Never cancelled a game, even if it started with mate-in-4 (which CRAFTY easily saw)
    • Played each position 2 times, switching places
    • ~100 games - reduce noises in starting positions
multiple endgames
Multiple Endgames
  • Aim for general-purpose strategies
  • All endgames used during evolution
  • Results:
sample gp endchess
Sample GP-Endchess

Tree 0:

(If3 (Or2 (Not (Or2 (And2 OppPieceAttUnprotected NotMyKingInCheck) (Or2 NotMyPieceAttUnprotected 100*Increase))) (And2 (Or3 (And2 OppKingStuck NotMyPieceAttUnprotected) (And2 OppPieceAttUnprotected OppKingStuck) (And3 -1000*MateInOne OppKingInCheckPieceBehind NotMyKingStuck)) (Or2 (Not NotMyKingStuck) OppKingInCheck))) NumMyPiecesUNATT (If3 (< (If3 (Or2 NotMyPieceAttUnprotected NotMyKingInCheck) (If3 NotMyPieceAttUnprotected #NotMovesOppKing OppKingInCheckPieceBehind) (If3 OppKingStuck OppKingInCheckPieceBehind -1000*MateInOne)) (If3 (And2 100*Increase 1000*Mate?) (If3 (< NumMyPiecesUNATT (If3 NotMyPieceAttUnprotected -1000*MateInOne OppKingProxEdges)) (If3 (< MyKingDistEdges #NotMovesOppKing) (If3 -1000*MateInOne -1000*MateInOne NotMyPieceATT) (If3 100*Increase #MovesMyKing OppKingInCheckPieceBehind)) NumOppPiecesATT) (If3 NotMyKingStuck -100.0 OppKingProxEdges))) (If3 OppKingInCheck (If3 (Or2 NotMyPieceAttUnprotected NotMyKingInCheck) (If3 (< MyKingDistEdges #NotMovesOppKing) (If3 -1000*MateInOne -1000*MateInOne NotMyPieceATT) (If3 100*Increase #MovesMyKing OppKingInCheckPieceBehind)) NumOppPiecesATT) (If3 (And3 -1000*MateInOne NotMyPieceAttUnprotected 100*Increase) (If3 (< NumMyPiecesUNATT (If3 NotMyPieceAttUnprotected -1000*MateInOne OppKingProxEdges)) (If3 (< MyKingDistEdges #NotMovesOppKing) (If3 -1000*MateInOne -1000*MateInOne NotMyPieceATT) (If3 100*Increase #MovesMyKing OppKingInCheckPieceBehind)) NumOppPiecesATT) -1000*MateInOne)) (If3 (< (If3 100*Increase MyKingDistEdges 100*Increase) (If3 OppKingStuck OppKingInCheckPieceBehind -1000*MateInOne)) -100.0 (If3 (And2 NotMyPieceAttUnprotected -1000*MateInOne) (If3 (< NumMyPiecesUNATT (If3 NotMyPieceAttUnprotected -1000*MateInOne OppKingProxEdges)) (If3 (< MyKingDistEdges #NotMovesOppKing) (If3 -1000*MateInOne -1000*MateInOne NotMyPieceATT) (If3 100*Increase #MovesMyKing OppKingInCheckPieceBehind)) NumOppPiecesATT) (If3 OppPieceAttUnprotected NumMyPiecesUNATT MyFork)))))

Tree 1:

(If3 NotMyPieceAttUnprotected #NotMovesOppKing 1000*Mate?)

Tree 2:

(If3 1000*Mate? NumMyPiecesUNATT -1000*MateInOne)

  • Draw and better against Master-defined
  • Draw against a world class opponent
    • On limited conditions (not a full game, time ,etc.)
  • Shows deep search may have an alternative
  • Fast, pattern-oriented search suggests more human resemblance
  • Search and lookahead are still important
future work
Future Work
  • Add more pieces
  • Improve evolution speed
    • Parallel nets
    • Stronger board representations
  • Develop more cognitive models using evolution
  • Search scheme space as well as game space
  • Tackle beyond endgames
    • Openings and mid-game
  • General game concept schemas (?)