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A Discrete Computational Aesthetics Model for A Zero-Sum Perfect Information Game

Authors:

Abstract

One of the best examples of a zero-sum perfect information game is chess. Aesthetics is an important part of it that is greatly appreciated by players. Computers are currently able to play chess at the grandmaster level thanks to efficient search techniques and sheer processing power. However, they are not able to tell a beautiful combination from a bland one. This has left a research gap that, if addressed, would be of benefit to humans, especially chess players. The problem is therefore the inability of computers to recognize aesthetics in the game. Existing models or computational approaches towards aesthetics in chess tend to conflate beauty with composition convention without taking into account the significance of the former in real games. These approaches also typically use fixed values for aesthetic criteria that are rather inadequate given the variety of possibilities on the board. The goal was therefore to develop a computational model for recognizing aesthetics in the game in a way that correlates positively with human assessment. This research began by identifying aesthetics as an independent component applicable to both domains (i.e. compositions and real games). A common ground of aesthetic principles was identified based on the relevant chess literature. The available knowledge on those principles was then formalized as a collection of evaluation functions for computational purposes based on established chess metrics. Several experiments comparing compositions and real games showed that the proposed model was able to identify differences of statistical significance between domains but not within them. Overall, compositions also scored higher than real games. Based on the scope of analysis (i.e. mate-in-3 combinations), any such differences are therefore most likely aesthetic in nature and suggest that the model can recognize beauty in the game. Further experimentation showed a positive correlation between the computational evaluations and those of human chess players. This suggests that the proposed model not only enables computers to recognize aesthetics in the game but also in a way that generally concurs with human assessment.
A DISCRETE COMPUTATIONAL AESTHETICS MODEL
FOR A ZERO-SUM PERFECT INFORMATION GAME
MOHAMMED AZLAN BIN MOHAMED IQBAL
THESIS SUBMITTED IN FULFILMENT
OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
FACULTY OF COMPUTER SCIENCE
& INFORMATION TECHNOLOGY
UNIVERSITY OF MALAYA
KUALA LUMPUR
SEPTEMBER 2008
ii
ABSTRACT
One of the best examples of a zero-sum perfect information game is chess. Aesthetics is
an important part of it that is greatly appreciated by players. Computers are currently
able to play chess at the grandmaster level thanks to efficient search techniques and
sheer processing power. However, they are not able to tell a beautiful combination from
a bland one. This has left a research gap that, if addressed, would be of benefit to
humans, especially chess players.
The problem is therefore the inability of computers to recognize aesthetics in the game.
Existing models or computational approaches towards aesthetics in chess tend to
conflate beauty with composition convention without taking into account the
significance of the former in real games. These approaches also typically use fixed
values for aesthetic criteria that are rather inadequate given the variety of possibilities
on the board. The goal was therefore to develop a computational model for recognizing
aesthetics in the game in a way that correlates positively with human assessment.
This research began by identifying aesthetics as an independent component applicable
to both domains (i.e. compositions and real games). A common ground of aesthetic
principles was identified based on the relevant chess literature. The available knowledge
on those principles was then formalized as a collection of evaluation functions for
computational purposes based on established chess metrics.
Several experiments comparing compositions and real games showed that the proposed
model was able to identify differences of statistical significance between domains but
not within them. Overall, compositions also scored higher than real games. Based on the
iii
scope of analysis (i.e. mate-in-3 combinations), any such differences are therefore most
likely aesthetic in nature and suggest that the model can recognize beauty in the game.
Further experimentation showed a positive correlation between the computational
evaluations and those of human chess players. This suggests that the proposed model
not only enables computers to recognize aesthetics in the game but also in a way that
generally concurs with human assessment.
iv
ACKNOWLEDGEMENTS
I would like to express my appreciation to my supervisor, Prof. Dato’ Ir. Dr. Mashkuri
Hj. Yaacob, who had the foresight to accept me as his doctoral student. I have benefited
from his experience, advice and the intellectual freedom he afforded to me during the
research period.
I would also like to thank John McCarthy (Stanford University, USA) for essentially
suggesting to me what I think is possibly the best approach to this research topic;
Michael Negnevitsky (University of Tasmania, Australia) for the meaningful
discussions we had about my research; Jonathan Levitt (Grandmaster of chess, UK) for
continuously and tirelessly accommodating my questions; David Friedgood (FIDE
Master and International Master of chess solving, UK) for his feedback and willingness
to share his connections to resourceful people; Peter Lamarque (University of York,
UK) who motivated me to improve my writing; and Malcolm McDowell (British Chess
Problem Society) for supplying me with several rare manuscripts on the royal game.
I also want to thank Brian Stephenson (UK) for his collection of chess compositions
which has become integral to this work; Daniel Freeman (Chessgames.com, Florida,
USA) for his support and commitment with regard to my online surveys; and the ICGA
Journal editorial board and reviewers, for their detailed and fruitful comments over the
years on various aspects of my research. I was simultaneously impressed and humbled
by their expertise.
Other people to whom I would like to express my gratitude for their comments and
feedback include Hans Gruber (Germany), Isaac Linder (who wrote back in pen and ink
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from Russia), Michael Schlosser (University of Vienna, Austria), John Troyer
(University of Connecticut, USA), Matej Guid (FIDE Master, University of Ljubljana,
Slovenia), Muhidin Mulalic (International University of Sarajevo, Bosnia and
Herzegovina), A. C. Sukla (Sambalpur University, India) and the many unnamed
computer programmers, mathematicians and statisticians I have consulted with (and
learned from) over the Internet.
Special thanks to my colleagues, Uwe Dippel and Manjit Singh. The former for being
my (unofficial) academic mentor for several years and for his translation services
(German/French to English), and the latter for having shared with me many of his
experiences as a doctoral student. I would also like to thank the University of Malaya
staff (especially in the main library) for providing impeccable assistance and academic
resources. Even though they may never hear of it, my appreciation also goes to AT&T
Inc. for their ‘Natural Voices™’ technology, which enabled a computer to read this
entire thesis back to me in an almost human voice when it would have perhaps been too
much to ask of any human.
Very special thanks to Jaap van den Herik in the Netherlands for proofreading the final
draft of this thesis, and for his insightful comments. I wish you all the best, sir, on your
move from Universiteit Maastricht to Tilburg University, and sincerely appreciate the
time and effort you have spent on my behalf. Finally, I would like to thank my family
for their support and encouragement.
This research is supported by the University Tenaga Nasional research grant
J510050123.
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For Gamers of the Future
vii
CONTENTS
ABSTRACT ..................................................................................................................... ii
ACKNOWLEDGEMENTS ........................................................................................... iv
LIST of FIGURES ......................................................................................................... xi
LIST of TABLES ......................................................................................................... xiii
ABBREVIATIONS ...................................................................................................... xiv
CHAPTER 1: INTRODUCTION .................................................................................. 1
1.0 Preliminary ...................................................................................................... 1
1.1 Motivation ........................................................................................................ 3
1.2 Thesis Objectives ............................................................................................. 5
1.3 Thesis Scope ..................................................................................................... 6
1.4 Main Contributions of this Work .................................................................. 8
1.5 Thesis Organization ........................................................................................ 9
1.6 Summary of Research Questions ................................................................. 11
CHAPTER 2: LITERATURE REVIEW .................................................................... 13
2.0 Computational Research into Chess Aesthetics ......................................... 13
2.1 Emanuel Lasker and Aesthetics ................................................................... 15
2.2 Automatic Judging of Compositions ........................................................... 17
2.3 Principles of Beauty ...................................................................................... 21
2.4 Computer Chess Problem Composition ...................................................... 24
2.5 Elements of Beauty Classified ...................................................................... 27
2.6 Beauty Heuristics in a Game Engine ........................................................... 30
2.7 Computational Improvement of Chess Problems ...................................... 33
2.8 A Look at Methodologies Used in Other Domains ..................................... 38
2.9 Chapter Summary ......................................................................................... 42
CHAPTER 3: METHODOLOGY Aesthetics in the Game ................................... 45
3.0 Components of the Research ........................................................................ 45
3.1 The Proposed Model of Aesthetics............................................................... 45
3.2 A Conceptual Framework for Aesthetics in the Game .............................. 46
3.3 An Examination of Aesthetics ...................................................................... 49
3.3.1 Composition Conventions ........................................................................... 50
3.3.2 Brilliancy in Real Games ............................................................................ 52
3.3.3 Principles of Aesthetics ............................................................................... 54
3.4 A Selection of Aesthetic Principles and Themes......................................... 58
3.5 A Formula for Cumulative Aesthetic Assessment ..................................... 61
3.5.1 The Development of Standard Evaluation Functions ................................. 62
3.5.2 Metrics and Properties Used in the Aesthetic Assessment ......................... 67
3.5.2(a) Piece Value and Piece Count .......................................................... 69
3.5.2(b) Distance, Piece Power, Mobility and Piece Field ........................... 71
3.5.2(c) Summary of Metrics and Properties ................................................ 73
3.5.3 A Note on Benchmarks ............................................................................... 74
3.6 A General Methodology for Developing Aesthetics Formalizations ........ 74
3.7 The Scope of Analysis Explained ................................................................. 78
3.8 Points of Evaluation (POE) .......................................................................... 80
3.8.1 The Moving Piece ....................................................................................... 81
3.9 Chapter Summary ......................................................................................... 82
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CHAPTER 4: METHODOLOGY Aesthetic Principle Formalizations ................ 83
4.0 Formalizing the Seven Aesthetic Principles ................................................ 83
4.1 Violate Heuristics Successfully .................................................................... 85
4.1.1 Keep Your King Safe .................................................................................. 86
4.1.2 Capture Enemy Material ............................................................................. 87
4.1.3 Do Not Leave Your Own Pieces ‘En prise’ ................................................ 89
4.1.4 Increase Mobility of Your Pieces ................................................................ 90
4.2 Use the Weakest Piece Possible .................................................................... 91
4.3 Use All of the Piece’s Power ......................................................................... 92
4.4 Win with less Material .................................................................................. 93
4.5 Checkmate Economically ............................................................................. 94
4.5.1 Explanation of the Concept ......................................................................... 95
4.5.2 Features of Economy................................................................................... 96
4.5.3 The Economy Evaluation Function ............................................................. 97
4.5.4 The Process of Evaluation ......................................................................... 100
4.5.5 Validation .................................................................................................. 102
4.5.5(a) Compositions vs. Tournament Games .......................................... 102
4.5.5(b) Compositions vs. Tournament Games (Improved) ....................... 104
4.5.5(c) Testing against Human Assessment .............................................. 105
4.5.6 Minor Economical Differences ................................................................. 106
4.5.7 Paradoxical Economy................................................................................ 108
4.5.8 Perfect Economy ....................................................................................... 109
4.6 Sacrifice Material ........................................................................................ 110
4.7 Spread Out the Pieces (Sparsity) ............................................................... 112
4.7.1 Explanation of the Concept ....................................................................... 112
4.7.2 A Look at Possible Approaches ................................................................ 115
4.7.3 The Sparsity Evaluation Function ............................................................. 116
4.7.4 Validation .................................................................................................. 119
4.7.4(a) Sparsity and Piece Count .............................................................. 120
4.7.4(b) Sparsity and Piece Count (Alternative Method) ........................... 121
4.7.4(c) Sparsity and Piece Configuration .................................................. 122
4.7.5 Discussion ................................................................................................. 124
4.8 Points of Evaluation for the Aesthetic Principles ..................................... 125
4.9 Chapter Summary ....................................................................................... 125
CHAPTER 5: METHODOLOGY Theme Formalizations .................................. 127
5.0 Formalizing the Ten Themes ..................................................................... 127
5.1 Fork .............................................................................................................. 128
5.2 Pin ................................................................................................................. 132
5.3 Skewer .......................................................................................................... 137
5.4 X-Ray ............................................................................................................ 139
5.5 Discovered/Double Attack .......................................................................... 142
5.6 Zugzwang ..................................................................................................... 146
5.7 Smothered Mate .......................................................................................... 149
5.8 Cross-check .................................................................................................. 150
5.9 Promotion .................................................................................................... 152
5.10 Switchback ................................................................................................... 154
5.11 Points of Evaluation for the Themes ......................................................... 155
5.12 Chapter Summary ....................................................................................... 156
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CHAPTER 6: EXPERIMENTAL RESULTS and DISCUSSIONS ....................... 158
6.0 The Six Experiments Performed ............................................................... 158
6.1 Experiment 1: Frequencies......................................................................... 160
6.1.1 Frequencies of the Themes........................................................................ 161
6.1.2 Frequencies of the Aesthetic Principles .................................................... 163
6.1.3 Discussion ................................................................................................. 164
6.2 Experiment 2: Evaluation of the Aesthetic Principles ............................. 165
6.3 Experiment 3: Evaluation of the Themes ................................................. 167
6.4 Experiment 4: Cumulative Evaluation ..................................................... 173
6.4.1 Aesthetic Principles Only .......................................................................... 178
6.4.2 Themes Only ............................................................................................. 180
6.4.3 Discussion ................................................................................................. 182
6.5 Experiment 5: Conformity to Authoritative Human Assessment .......... 183
6.6 Experiment 6: Correlation with Human Assessment .............................. 186
6.6.1 Survey 1 (Mixed) ...................................................................................... 190
6.6.2 Survey 2 (Mixed, Discrete Evaluations) ................................................... 192
6.6.2(a) Levels of Agreement ..................................................................... 194
6.6.3 Survey 3 (Tournament Games) ................................................................. 195
6.6.4 Survey 4 (Compositions) .......................................................................... 197
6.6.5 Survey Conclusions ................................................................................... 198
6.7 Chapter Summary ....................................................................................... 202
CHAPTER 7: CONCLUSION ................................................................................... 205
7.0 Preliminary .................................................................................................. 205
7.1 Thesis Summary .......................................................................................... 205
7.2 Thesis Contributions ................................................................................... 208
7.3 Implications of the Research ...................................................................... 209
7.4 Directions for Further Work...................................................................... 212
REFERENCES ............................................................................................................ 217
APPENDIX A: CHESS RULES ................................................................................ 233
1.0 Introduction to the Game ........................................................................... 233
1.1 Movement of the Pieces .............................................................................. 234
1.1.1 Rook .......................................................................................................... 235
1.1.2 Bishop ....................................................................................................... 235
1.1.3 Queen ........................................................................................................ 236
1.1.4 Knight ........................................................................................................ 236
1.1.5 King ........................................................................................................... 237
1.1.5(a) Castling ......................................................................................... 237
1.1.6 Pawn .......................................................................................................... 239
1.2 Check, Checkmate and Stalemate ............................................................. 240
1.3 Other Rules .................................................................................................. 242
1.3.1 Resignation and Draws ............................................................................. 242
1.3.2 Repetition of Positions .............................................................................. 243
1.3.3 50-Move Rule ........................................................................................... 243
1.3.4 Touching Pieces ........................................................................................ 244
1.4 Chess Notation ............................................................................................. 244
1.4.1 Board Notation .......................................................................................... 246
APPENDIX B: GLOSSARY of CHESS TERMS .................................................... 248
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APPENDIX C: EXAMPLE POSITIONS ................................................................. 253
Subsection 3.3.1 ....................................................................................................... 253
Subsection 3.3.3 ....................................................................................................... 254
Subsection 3.5.1 ....................................................................................................... 255
Section 4.2 ................................................................................................................ 256
Subsection 4.5.4 ....................................................................................................... 257
Section 5.1 ................................................................................................................ 257
Section 5.2 ................................................................................................................ 259
Section 5.4 ................................................................................................................ 260
Section 5.5 ................................................................................................................ 261
D (Refer Appendix) ................................................................................................. 263
APPENDIX D: CHESTHETICA .............................................................................. 264
APPENDIX E: PSEUDOCODE ................................................................................ 271
Subsection 4.1.2 ....................................................................................................... 271
Subsection 4.1.3 ....................................................................................................... 272
Section 4.3 ................................................................................................................ 272
Section 4.5 ................................................................................................................ 273
Section 4.7 ................................................................................................................ 274
Section 5.1 ................................................................................................................ 274
Section 5.2 ................................................................................................................ 276
Section 5.4 ................................................................................................................ 278
Section 5.5 ................................................................................................................ 279
Section 5.6 ................................................................................................................ 281
APPENDIX F: SURVEY DATA ............................................................................... 282
1.0 Overview of the Surveys ............................................................................. 282
1.1 Instruction Set (Surveys 1, 3 and 4) ........................................................... 283
1.1.1 Instruction Set (Survey 2) ......................................................................... 285
1.2 The Combinations Rated ............................................................................ 288
1.3 The Combinations Rated (PGN Compatible) ........................................... 300
1.4 Control Questions ....................................................................................... 305
1.4.1 Survey 1 .................................................................................................... 307
1.4.2 Survey 2 .................................................................................................... 308
1.4.3 Survey 3 .................................................................................................... 309
1.4.4 Survey 4 .................................................................................................... 310
1.5 Respondent Ratings .................................................................................... 311
1.6 Screen Captures .......................................................................................... 319
SELECTED PUBLICATIONS .................................................................................. 321
xi
LIST of FIGURES
Figure 3.1 Concept of Aesthetics in Chess ............................................................... 48
Figure 3.2 Layers of an Aesthetic Evaluation Function ........................................... 67
Figure 4.1 Scores for Violation of ‘Keep Your King Safe’ ..................................... 86
Figure 4.2 Maximum ‘Control Fields’ for the Chessmen ........................................ 98
Figure 4.3 Economy Scores of Checkmate Positions ............................................. 102
Figure 4.4 Economy Scores for Compositions and Tournament Games ............... 103
Figure 4.5 Economy Scores for ‘Improved’ Positions ........................................... 105
Figure 4.6 Minor Economic Improvements to a Position ...................................... 107
Figure 4.7 Economy Paradox ................................................................................. 108
Figure 4.8 Highly Economical Checkmates ........................................................... 110
Figure 4.9 Sparsity in Chess Compositions ............................................................ 113
Figure 4.10 Sufficient Sparsity (Constructed Positions) .......................................... 114
Figure 4.11 Sparsity Scores of Chess Positions from Tournament Games .............. 118
Figure 4.12 Sparsity Scores of Go Positions ............................................................ 119
Figure 4.13 Sparsity Values of 1,000 Random Game Positions .............................. 120
Figure 4.14 Sparsity Values of 1,000 Random Game Positions (Alternate) ............ 122
Figure 5.1 The Fork ................................................................................................ 129
Figure 5.2 The Pin .................................................................................................. 133
Figure 5.3 Aesthetic Assessment of the Pin ........................................................... 136
Figure 5.4 The Skewer............................................................................................ 137
Figure 5.5 Aesthetic Assessment of the Skewer..................................................... 138
Figure 5.6 The X-ray .............................................................................................. 139
Figure 5.7 Aesthetic Assessment of the X-Ray ...................................................... 142
Figure 5.8 The Discovered/Double Attack ............................................................. 143
Figure 5.9 Aesthetic Assessment of the Discovered Attack ................................... 144
Figure 5.10 The Zugzwang ....................................................................................... 148
Figure 5.11 The Smothered Mate ............................................................................. 149
Figure 5.12 Aesthetic Assessment of the Cross-check ............................................. 151
Figure 5.13 The Saavedra Position ........................................................................... 153
Figure 6.1 Frequencies of Themes in the Combinations ........................................ 161
Figure 6.2 Frequencies of Aesthetic Principles in the Combinations..................... 163
Figure 6.3 Cumulative Aesthetic Scores for Combinations ................................... 174
Figure 6.4 Highest Scoring Combinations (a) COMP, (b) TG ............................... 175
Figure 6.5 Lowest Scoring Combinations (a) COMP, (b) TG ............................... 177
Figure 6.6 Cumulative Scores Based on Aesthetic Principles Only ...................... 179
Figure 6.7 Cumulative Scores Based on Themes Only .......................................... 181
Figure A.1 The Initial Position of the Pieces .......................................................... 234
Figure A.2 Movement of the Rook.......................................................................... 235
Figure A.3 Movement of the Bishop ....................................................................... 235
Figure A.4 Movement of the Queen ........................................................................ 236
Figure A.5 Movement of the Knight ....................................................................... 236
Figure A.6 Movement of the King .......................................................................... 237
Figure A.7 Before and after Castling ...................................................................... 238
Figure A.8 Castling Illegal for Both White and Black ............................................ 238
xii
Figure A.9 Movement of the Pawn ......................................................................... 239
Figure A.10 En passant ............................................................................................. 240
Figure A.11 Check..................................................................................................... 241
Figure A.12 Checkmate and Stalemate ..................................................................... 242
Figure A.13 The Chessboard and its Coordinates ..................................................... 244
Figure C.1 A Typical ‘Logical’ School Composition ............................................. 253
Figure C.2 J. Mintz, The Problemist, 1982, Helpmate in 3 (Black to Play) ........... 254
Figure C.3 The ‘Immortal Game’ (after 17. … Qxb2)............................................ 254
Figure C.4 Kasparov vs. Deep Junior, Game 5, New York, 2003 .......................... 255
Figure C.5 Deep Blue vs. Kasparov, Game 6, New York, 1997............................. 255
Figure C.6 Two-way Discovered Checkmate ......................................................... 256
Figure C.7 Two-Phase Piece Removal (Economy) ................................................. 257
Figure C.8 Activated Fork ....................................................................................... 257
Figure C.9 Repeated Fork ....................................................................................... 258
Figure C.10 Immobilizing the Queen with a Two-way Pin ...................................... 259
Figure C.11 A Three-way Pin (Bxd5) ....................................................................... 259
Figure C.12 Negative Evaluation for the Pin after 1. Qg2 ........................................ 260
Figure C.13 A Double X-ray with 1. Bxd4 ............................................................... 260
Figure C.14 Castling as a Discovered Attack Manoeuvre (0-0#) ............................. 261
Figure C.15 Double-Discovered Attack (after 1. Ndf4+) ......................................... 261
Figure C.16 Stalemate in 3 ........................................................................................ 263
Figure D.1 The Main Interface to CHESTHETICA ............................................... 265
Figure D.2 The ‘About Box’ ................................................................................... 266
Figure D.3 The Aesthetics Evaluation Panel ........................................................... 267
Figure D.4 The Thematic Frequency Chart ............................................................. 268
Figure D.5 Aesthetic Principle and Theme Selection ............................................. 268
Figure D.6 The Mate Solver .................................................................................... 269
Figure F.1 Survey 1: Control Question 1 ................................................................ 307
Figure F.2 Survey 1: Control Question 2 ................................................................ 307
Figure F.3 Survey 2: Control Question 1 ................................................................ 308
Figure F.4 Survey 2: Control Question 2 ................................................................ 308
Figure F.5 Survey 3: Control Question 1 ................................................................ 309
Figure F.6 Survey 3: Control Question 2 ................................................................ 309
Figure F.7 Survey 4: Control Question 1 ................................................................ 310
Figure F.8 Survey 4: Control Question 2 ................................................................ 310
Figure F.9 Survey 3: Screen Capture 1 ................................................................... 320
Figure F.10 Survey 3: Screen Capture 2 ................................................................... 320
xiii
LIST of TABLES
Table 3.1 General Aesthetic Principles, Conventions and Brilliancy Compared .... 55
Table 3.2 Refined Aesthetic Principles and Themes ............................................... 60
Table 3.3 Metrics and Properties Used .................................................................... 73
Table 3.4 Points of Evaluation in a Combination .................................................... 80
Table 4.1 Points of Evaluation for the Aesthetic Principles .................................. 125
Table 5.1 X-ray Defensive Capabilities ................................................................ 141
Table 5.2 Points of Evaluation for the Themes ..................................................... 156
Table 6.1 Average Scores for Aesthetic Principles ............................................... 165
Table 6.2 Standard Deviations of Average Aesthetic Principle Scores ................ 166
Table 6.3 Significance of Diff. between Mean Aesthetic Principle Scores .......... 167
Table 6.4 Average Scores for the Themes ............................................................. 168
Table 6.5 Standard Deviations of Average Scores for Themes ............................. 169
Table 6.6 Significance of Mean Differences of Theme Scores ............................. 170
Table 6.7 Average Cumulative Aesthetic Scores for the Combinations ............... 173
Table 6.8 Average Cumulative Aesthetic Scores for Aesthetic Principles Only .. 178
Table 6.9 Average Cumulative Aesthetic Scores for Themes Only ...................... 180
Table 6.10 Human vs. Computer Assessment (COMP+TG Combinations) .......... 190
Table 6.11 Human Assessment vs. Computer (Discrete Evaluations) .................... 193
Table 6.12 Level of Human Agreement with Computer Assessment ..................... 194
Table 6.13 Human vs. Computer Assessment (TG Only)....................................... 196
Table 6.14 Human vs. Computer Assessment (COMP Only)................................. 198
Table 6.15 Summary of Human-Computer Assessment Correlations .................... 199
Table 6.16 Summary of Positive Correlations ........................................................ 201
Table A.1 The Chessmen ....................................................................................... 233
Table A.2 Shorthand Notation ................................................................................ 246
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ABBREVIATIONS
1T One-tailed
2T Two-tailed
AI Artificial Intelligence
APP Use all of the Piece’s Power
COMP Chess Composition Combinations
DDA Discovered/Double Attack
FIDE Fédération Internationale des Échecs
FEN Forsyth-Edwards Notation
FM FIDE Master
GM Grandmaster
ICP Improver of Chess Problems, The
IM International Master
LOA Level of (Human) Agreement
LOD Level of (Computational) Distinction (between Scores)
PGN Portable Game Notation
SD Standard Deviation
SL Significance Level (of)
TG Tournament Game Combinations
TTUV Two-sample t-test assuming Unequal Variances
VH Violate Heuristics (Successfully)
WPP Use the Weakest Piece Possible
WWLM Win with Less Material
1
CHAPTER 1: INTRODUCTION
1.0 Preliminary
Research into games is important, especially in the field of AI. The following paragraph
articulates the main reasons quite well.
There are two principal reasons to continue to do research on games. . .
First, human fascination with game playing is longstanding and pervasive.
Anthropologists have catalogued popular games in almost every culture. . .
Games intrigue us because they address important cognitive functions. . . The
second reason to continue game-playing research is that some difficult games
remain to be won, games that people play very well but computers do not.
These games clarify what our current approach lacks. They set challenges for
us to meet, and they promise ample rewards.(Epstein, 1999)
A zero-sum perfect information game (sometimes with a hyphen between perfectand
information) is one in which a player gains at the equal expense of others and where
every player knows the results of all the previous moves. Examples include noughts and
crosses, checkers, chess and Go (in scientific literature, there is some common
agreement that the game be referred to using a capital letter to differentiate it from the
English verb ‘go’). These are games where it is theoretically possible to build a
computational move ‘tree’ of all the positions that could occur and thus facilitate perfect
play. However, for games such as Go and chess the number of possible positions to
examine is too large even for computers. It is estimated that there are approximately
1046 legal positions in chess and around 10170 in Go (Chinchalkar, 1996; Tromp and
2
Farnebäck, 2006). Even so, good evaluation functions and efficient search techniques
allow computers to play such games, with the current and notable exception of Go, quite
well (Levy and Newborn, 1992; Campbell et al., 2002; Walczak, 2003; Hauptman and
Sipper, 2007; Hsu, 2007).
One of the most popular research domains in this respect is chess. The beginnings of
chess are obscure but it is thought to have originated in northern India around 600 AD
and spread mainly through traders to other parts of the world (Eales, 2002; Shenk,
2006). The most widely played version today is Western or international chess and is
regulated by the Fédération Internationale des Échecs (FIDE) or World Chess
Federation. The rules of international chess have remained essentially the same since
1475 (Hooper and Whyld, 1996). Ever since Claude Shannon (1950) wrote his seminal
paper on how a computer could be programmed to play chess; researchers,
programmers and especially the public have been fascinated at the prospect of
computers playing the game at the same level of human experts or better (Grier, 2006).
The approach suggested is often also credited to Alan Turing (1953). The hope was that
whatever methods achieved this might shed some light onto the mechanics of human
thought processes because one must be thinking in order to play the game (Newborn,
1997). Computer chess programs today play at the grandmaster (GM) level and have
even beaten the world champion by relying mainly on brute-force (i.e. exhaustive or
nearly exhaustive searching of relevant parts of the game tree) which is different from
how humans play (Hsu, 2004). Even though this is not exactly what AI researchers were
hoping for (Hendler, 2006), research into the game has provided benefits and insights
into other areas (see section 2.0).
3
However, there is an important aspect of chess that computers are quite poor at. This is
aesthetics and it is one of the main reasons humans play (Kasparov, 1987; Damsky,
2002; Dossi, 2005). Computers cannot tell a beautiful move combination or game from
a regular or unattractive one the way humans can. Chess was chosen as a suitable
domain of research because its aesthetic aspect (especially prominent in chess
compositions) is well established in the literature, more so than Go or any other zero-
sum perfect information game.
Real games (e.g. in tournaments) are also known to exhibit aesthetics, and ‘brilliancy’
prizes are sometimes awarded to such games. With chess programs playing on a level
equal to - and in some cases greater than - the best human players, the time seems ripe
for focusing on the aesthetics of the game. Since humans strive for, and appreciate
beauty in chess, it would be valuable if computers could recognize it in a way that is
comparable to humans given that machines are able to analyze many more positions in
the game tree than humans.
1.1 Motivation
A zero-sum perfect information game is a good place where complex ideas can be
experimented with because in theory, such games are finite and particularly amenable to
computation. This is the reason chess has for many decades been the subject of much
research in various fields (see section 2.0). However, the main focus has always been on
how to make computers play the game on a level equal to, or exceeding that of the best
human players. This has, since the late 1990s, been demonstrated (see subsection 3.5.1).
Even commercially available computer programs today play at the grandmaster level.
Researchers have therefore, in this respect, now generally shifted to more complex
4
games like Go (Wu and Baldi, 2007). The following six reasons served as motivation
for this research.
Aesthetics in chess has received little attention in AI despite it being an
important part of the game that matters to players. Humans, unlike
computers, do not play solely to win. They also want their games and
compositions to be beautiful and fascinating. A computational model for
aesthetics in the game would therefore benefit humans and enhance the
capabilities of existing chess programs.
Research into automatic chess problem composition has not accounted for
aesthetics in a meaningful way or perhaps at all (see sections 2.2 and 2.4). It
neither separates aesthetics from composition convention (see Appendix B)
nor takes into account much of the knowledge that is available in chess
literature on the subject of aesthetics.
Existing formalizations (usually in the form of an evaluation function
represented using a mathematical formula) on chess ‘quality’ typically use
fixed values attributed to aesthetic principles and chess themes. These do not
reflect the variations possible in such principles and themes (e.g. a different
piece configuration of the same theme) in a way that is both flexible yet
consistent. As a result, aesthetics is not accounted for reasonably in
compositions (see section 2.7), and even less in actual games.
Other similarly complex zero-sum perfect information games such as Go and
up to a thousand other chess variants also have an aesthetic dimension that
has not benefited from computational analysis (Pritchard, 2000b). An
aesthetics model for chess could, in principle, be extended or adapted to
these games as well for the benefit of humans. Chess variants for example,
5
usually vary in only one respect such as the board size or additional piece
types.
Aesthetic models have been developed with some success in less discrete
domains such as art (e.g. photographic images, paintings) and music
(Machado and Cardoso, 1998; Golub, 2000; Manaris et al., 2002a, 2002b;
Datta et al., 2006). A more reliable model could perhaps be developed for a
theoretically finite domain like chess. This in turn could inspire the
development of better models in those domains and others since chess is
often used or referenced in many areas of research (see section 2.0).
A personal interest in the game of chess and over 20 years of active playing
experience, combined with an equal interest and level of experience with
computer programming. The prospect of developing a computer program
probably the first of its kind - capable of recognizing beauty in chess based
on a viable model was therefore also a motivator.
Due to all the reasons mentioned above, a discrete computational aesthetics model for a
zero-sum perfect information game like chess was deemed worthy of investigation. The
term ‘discrete’ signifies the distinct yet synergetic components of the proposed model
(see section 3.1).
1.2 Thesis Objectives
The objectives of this thesis are as follows.
1. To study chess (as a zero-sum perfect information game) and its relevant
literature on aesthetics to identify the pertinent issues.
6
2. To propose a model that makes aesthetic evaluation in the game
computationally feasible.
3. To derive formalizations for a selection of aesthetic principles.
4. To derive formalizations for a selection of chess themes.
5. To develop a computer program incorporating those formalizations for the
purpose of performing relevant experiments.
6. To test the viability of the model through experimentation in terms of
aesthetic recognition in the game and positive correlation with human
aesthetic assessment.
1.3 Thesis Scope
The scope of this thesis is as follows.
1. Review the relevant literature on chess with emphasis on its aesthetic aspect.
2. Review and evaluate existing methodologies that have attempted to address
aesthetics in the game computationally. Include also research that did not
actually address aesthetics where it would have been pertinent. In the interest
of a general context, briefly review methodologies of aesthetic evaluation in
other domains such as art and music.
3. Propose a conceptual framework for aesthetics in the game which is a way of
thinking about it that can guide proper investigation. Contrast with the
current practice of conflating composition convention with aesthetics.
4. Investigate composition conventions and brilliancy in real games. Identify
the areas in which they overlap with established aesthetic principles, as also
described in chess literature.
7
5. Propose a selection of aesthetic principles and themes for evaluation. These
include those that apply, as far as possible, equally to both domains.
6. Propose a formula for cumulative aesthetic assessment in a move
combination. Explain the choice of metrics and properties to be used.
7. Present a general methodology for developing formalizations for aesthetic
principles and themes in the game.
8. Define and explain the chosen scope of mate-in-3 combinations and the
points of evaluation for each selected aesthetic principle and theme. This is
to facilitate experimentation.
9. Propose formalizations for the selected aesthetic principles and themes in the
game with explanations about the logic behind their design. Include
discussions about their strengths and limitations.
10. Develop a computer program that incorporates these formalizations to make
experimentation feasible. It should possess features which make a variety of
experiments involving compositions and real games possible.
11. Validate the aesthetics model through experimentation in terms of its ability
to recognize aesthetics in the game computationally.
12. Demonstrate that the computational assessment of aesthetics in the game,
within the chosen scope of mate-in-3 combinations, correlates positively
with human chess player aesthetic assessment.
This thesis explores aesthetic principles that pertain to the game of international chess in
general but limits experimental studies to mate-in-3 combinations. Aesthetic principles
that apply mainly to other domains such as art and music are beyond the scope of this
thesis.
8
1.4 Main Contributions of this Work
The key contributions of this research include the following.
1. To review and examine aesthetic principles (and themes) in the zero-sum
perfect information game of international chess as described in its relevant
literature.
2. To propose a conceptual framework for aesthetics as a way of thinking about
aesthetics in the game. This framework isolates aesthetics as a component
not exclusive to compositions or real games. It makes aesthetics more
computationally amenable and easier to apply to both domains in a way that
does not conflate with composition convention or brilliancy (in real games).
3. To propose a formula for cumulative aesthetic assessment in a move
sequence or combination. It is based on the idea that dynamic formalizations
for aesthetic principles and themes, in summation, can represent the aesthetic
content of a combination. These are in turn based on metrics and properties
inherent to the game. This permits the use of aesthetic evaluation in addition
to other forms of computational assessment in the game such as composition
convention and standard game-playing heuristics.
4. To present a general methodology for developing aesthetics formalizations
in the game. Using this approach (and the concept of benchmarks),
formalizations for other aesthetic principles and themes can be developed in
a similar way to those developed for this research.
5. To propose a set of dynamic formalizations (and explain the logic behind
their individual designs) for a selection of aesthetic principles and themes.
9
These represent the common ground of aesthetics between the domains of
chess compositions and real games.
6. To devise a few novel experiments for validating aesthetic recognition in the
game based on the proposed computational aesthetics model.
7. To develop a computer program incorporating the aesthetics model (and all
the formalizations) for experimental purposes. Manual evaluation is
complicated and prone to error, especially for bulk analysis. This program
can be used by other researchers to save time in evaluating aesthetics based
on the model.
8. To evaluate the proposed aesthetics model in terms of its ability, when
implemented, to recognize aesthetics in the game.
9. To test the computational aesthetic recognition capability of the model for
positive correlation with human chess player assessment.
The results of a variety of experiments suggest that the aesthetics model can be used to
recognize aesthetics in the game of chess within (at least) the scope of mate-in-3
combinations. The aesthetic scores produced by a computer program based on the
model also correlate well with human chess player aesthetic assessment.
1.5 Thesis Organization
The detailed structure of this thesis is as follows. Chapter 2 reviews the relevant chess
and scientific literature pertaining to aesthetics in the game. It generally illustrates the
importance of aesthetics in chess over the last century, and the attempts that have been
made to bring that concept into the computational domain. The chapter also briefly
10
reviews methodologies applied to gauge aesthetics in other domains such as art and
music.
Chapter 3 introduces the proposed model of aesthetics. The components of the model
include an examination of aesthetics from the perspective of problem composers and
players, as described in the relevant chess literature. A common ground of aesthetics is
identified as the focus of this research. A selection is made of aesthetic principles and
themes that fall within that common ground. The chapter also presents a formula for
cumulative aesthetic assessment and an explanation of the metrics and properties used.
The methodologies behind standard evaluation functions in the game are reviewed
before one is proposed for developing aesthetic evaluation functions (i.e. related to the
selected principles and themes). The scope of analysis (for experimental purposes) is
then explained, followed by a description of the points of evaluation in a combination.
Chapter 4 details the actual evaluation functions for the seven selected aesthetic
principles. A detailed description of each principle and the logic behind the design of its
function are presented. Diagrams, examples, and experimental validation are provided,
where appropriate. Chapter 5 details the formalizations for the ten selected themes,
similar to the previous chapter. Chapter 6 presents and explains all the experiments
performed to validate the proposed model. Included are analytical discussions of the
results. Chapter 7 concludes with a summary of the thesis, its contributions, a section on
the research implications, and directions for further work in the area.
Appendix A explains the rules of international chess and how to read its algebraic
notation. Appendix B is a glossary of chess-related terms found in this thesis. Appendix
C contains diagrams of example positions referenced primarily in the main text.
11
Appendix D features specifications and information about the computer program (i.e.
CHESTHETICA) developed for this research. Appendix E shows the essential
pseudocode for implementing many of the proposed aesthetic formalizations. Appendix
F contains the survey questionnaires used and the relevant raw data that was collected.
Unless inclusive of the word, ‘Appendix’, references to specific parts of this document
are given according to chapter, section or subsection; e.g. chapter 3, section 3.5,
subsection 3.5.2 and subsection 3.5.2(d). General formulas (in the form of equations)
are numbered sequentially according to section but instantiations of those equations
(such as sample calculations) are not numbered. Chess move notation and board
coordinates in line with other text in the main document are given in bold (e.g. a5).
Example positions are sometimes given in FEN (Forsyth-Edwards Notation) within the
main text (with a reference to a corresponding diagram in Appendix C). The words
‘thesis’ and ‘research’ are sometimes used interchangeably to refer to the work
presented in this document.
1.6 Summary of Research Questions
In principle, this thesis attempts to answer two research questions.
1) Can aesthetics in chess (within a specific scope) be recognized
computationally?
2) If so, do the computational evaluations correlate positively with human chess
player aesthetic assessment?
12
Six experiments were performed to answer both these questions (see chapter 6) with
promising results. Question 1 is confirmed to the extent of mate-in-3 combinations (a
reasonable scope of analysis in the game) and question 2 to the extent involving
competent (not necessarily expert) human chess players; consistent with what is
necessary for aesthetic appreciation in the game.
13
CHAPTER 2: LITERATURE REVIEW
2.0 Computational Research into Chess Aesthetics
Computational research into the aesthetics of chess is relatively scarce. This may be due
to the emphasis over the last few decades on getting computers to play the game ever
more proficiently (Adelson-Velskiy et al., 1970; Hartmann, 1987a, 1987b; Heinz, 1997;
Walczak, 2003). It may also be due to the assumption that there is no reliable way of
quantifying beauty in the game (le Grand, 1986). In the first case, computers have today
advanced to the level of world-class players and are even used by most of them for
training purposes (Muller, 2002; Ross, 2006; Sukhin, 2007); this likely includes the use
of game databases and not just chess-playing programs (Campitelli and Gobet, 2007).
Therefore, rather than just moving to more complex games like Go (Hsu, 2007;
McCarthy, 2007), more emphasis can now be placed on aesthetics in chess. In the
second case, there exists substantial material on the subject in chess literature (see the
following sections) to base a computational model on. Since human players and
problem composers value beauty in the game, the idea of computational recognition of
beauty is worthy of investigation. There is also room for its application in existing
research which, currently address chess aesthetics superficially (see sections 2.6 and
2.7). Computational approaches to art forms are not unfeasible and are likely to become
more common in the future (Boden, 2007).
It is said that there are more books written on chess than books on all other games
combined (Jonsson, 2006). With over 700 million players worldwide (Polgar, 2005), it
is arguably the most popular game in the world (Wolff, 2001). It is also recognized as a
14
sport by the International Olympic Committee (IOC, 2008). Benjamin Franklin wrote
about the benefits of chess as far back as 1779 in his article, ‘On the Morals of Chess
(Shenk, 2006). Investigations into chess have had many applications within and outside
of AI including molecular computing (Cukras et al., 1999; Faulhammer et al., 2000),
automated theorem proving (Newborn, 2000), computer music composition (Friedel,
2006), machine reading (Etzioni et al., 2007), cognitive development (O’Neil and Perez,
2007), treatment of psychiatric illness (Cavezian et al., 2008) and children education
(Ferreira and Palhares, 2008; AF4C, 2008). Such applications are not always predictable
so it is conceivable that this research could also be of interest or reference to researchers
in other fields or related ones.
This chapter reviews the more important and relevant contributions to computational
aesthetics in chess and scientific literature over the last century. It is a relatively recent
account, given for example, that documents featuring chess compositions date back over
1,000 years (Al-Adli, 9th century). Whole books have been written on chess since the
15th century (Axon, 1474). A review spanning the last 85 years or so is necessary to
illustrate how aesthetic principles have been described by experts and researchers prior
to and since the computer age. The reviews are arranged in chronological order for
proper perspective with a summary in section 2.9. A glossary of chess terms is provided
in Appendix B for reference. Even though it may not be directly applicable, in the
interest of a general context, section 2.8 presents a brief discussion about methodologies
relating to aesthetics used in other domains.
15
2.1 Emanuel Lasker and Aesthetics
Former world chess champion and mathematician Emanuel Lasker was one of the first
to write about aesthetics in chess explicitly. He maintained the world title for 27 years
starting in 1894, the longest ever held by a world champion. In his book, “Lasker’s
Manual of Chess” originally written in 1925 - he devoted a chapter to the subject
entitled, ‘The Aesthetic Effect in Chessand stressed on the concept of achievement
and correctness(Lasker, 1960). ‘Achievement’ means that beauty in the game had to
have some kind of positive result such as winning material, controlling more space on
the board, or checkmating the opponent, whereas ‘correctness’ implies that the method
of achievement be absolutely necessary and unequivocal. In other words, there had to be
no possible escape or defence by the opponent and no better way of attaining the same
achievement (e.g. in fewer moves). He also stated that in order to appreciate aesthetics
in chess, one need only to understand the game and not be a master himself.
Hence, the average player can just as easily derive pleasure from beautiful games and
compositions. He termed the pleasure spectators derived from the game due to
witnessing moves they would call ‘brilliantor ‘beautiful’ - their aesthetic valuation’.
This valuation was based on their immediate perception of a move’s brilliance. As a
result, some manoeuvres that were not, in fact, ‘correct’ (as would be revealed in the
post-mortem analysis of the game) elicited a high aesthetic valuation until their
incorrectness was discovered. In such cases, the valuation might diminish a little unless
it was suspected that the players had done so intentionally to fool the audience; in which
case it would diminish entirely. In tournament games, time constraints place a
considerable burden on players so all the variations of an attractive move combination
16
may not have been worked out properly (Harreveld et al., 2007). In compositions there
is no excuse.
So aesthetically, the occurrence of somewhatincorrectbrilliant moves in real games is
condoned whereas in compositions, they must withstand the most rigorous analysis.
Modern computer programs as a result have revealed weaknesses and flaws in many old
games and compositions that were once thought to be spectacular. Lasker showed many
examples of beautiful combinations from both tournament games and compositions.
Most of them were forced mates between 2 and 4 moves but others were longer and
more complex. One example in his book by an unknown composer featured a ‘forced
draw’ by White (despite having a significantly inferior army) that took 17 moves to
complete. In it, a pair of knights chased the enemy king around the board before the
initial position finally repeated itself. A game is considered drawn if the same position
recurs three times (see Appendix A, subsection 1.3.2).
Lasker was not specific about what he considered the tangible constituents of aesthetics
in chess and relied mainly on his experience with the game as opposed to
experimentation. His concepts of achievement’ and correctnessare useful precepts
for a framework for aesthetics even though he proposed no formalizations himself. This
is understandable given the period in which his book was written, i.e. well before the
advent of computers. It is quite possible that even the idea of computing aesthetics in
the game was too controversial to be taken seriously. Lasker made no distinction
between the aesthetics of real games and compositions and thus did not take into
account composition conventions. This supports the idea that aesthetics transcends
either domain, at least in cases where the rules are the same (i.e. not including chess
17
variants). Even so, it possibly overlooks situations where some conventions could also
apply aesthetically to real games.
Lasker made the important observation that aesthetic appreciation is not an experience
limited to masters even though having an understanding of the game would imply a
certain level of competence as a player. While master players may be needed to identify
the principles of aesthetics, others can also appreciate them. Assessing computational
evaluation of aesthetics in terms of positive correlation with human perception would
therefore not necessitate the involvement of experts. In fact, it would probably be more
useful to exclude or not focus on them since they form only a small minority of the
chess community.
The psychological aspects of aesthetic perception Lasker suggested are interesting but
difficult to gauge computationally because they rely upon the intentions of players and
subjective valuation by spectators that potentially change based on those intentions.
These are neither computationally amenable nor within the scope of this thesis. In
summary, Lasker provides a good starting point on how to approach the question of
aesthetics in the game but his contribution lacks the building blocks (i.e. discrete
components) required for a computational model.
2.2 Automatic Judging of Compositions
Vaux Wilson - a chess composition judge and author - researched, proposed, and
refined over the course of 20 years, a method of evaluating the aesthetic and strategic
elements of chess problems through a scientific approach (Wilson, 1959, 1969, 1978).
The intention was to provide a logical basis for compositions to be judged in
18
tournaments because many composers felt that judges were too arbitrary or subjective
when choosing a winning composition. The method was supposedly similar to a much
earlier and obscure system proposed by Harley (1919) that was limited to two-move
problems, but no reference was made to that work.
Wilson identified exclusively nine basic ways or ‘strategic elementsin which a moving
piece might influence the game. It could:
1. capture an opposing piece or sacrifice itself;
2. give check to the enemy king;
3. guard or abandon guarding a square;
4. move into a position where it could gain access to another square;
5. block or unblock a square;
6. castle;
7. move off, on or along a line;
8. open or close a line of check, or the guard of one square;
9. pin or unpin a piece.
These strategies were valued (typically between 2 and 10 points) based on the number
of pieces and squares involved. A ‘line’ was determined as consisting of 3 squares.
Since nothing else could possibly happen on the board when the pieces move in a chess
problem, the cumulative value of these strategies was considered to be inclusive of
aesthetics and any other impression one might obtain from a composition. Wilson’s
actual system incorporated several rules and exceptions that compensated for
composition conventions, e.g. with respect to keys and tries(see Appendix B). In
addition, a concept of economy was also evaluated and added for a final score.
19
Economy was calculated as the sum of strategic scores divided by the number of white
pieces. In general, the system was designed exclusively with compositions in mind, and
not real games, even though aesthetics is perceived in both (Humble, 1993, 1995;
Ravilious, 1994).
Wilson’s system was limited to chess problems but directly applicable to the many
different types such as orthodox mates, selfmates, helpmates, endgame studies, and
fairy problems (of any move length). He tested it on over 7,000 problems with
satisfactory results (i.e. compositions he perceived to be better scored higher) and the
system was used in a few composition tournaments. A significant positive correlation
with human judge evaluations was never demonstrated and probably because the system
was developed to address that very problem.
However, composers soon stopped using it mainly because human judges could not be
entirely replaced, as Wilson intended (le Grand, 1986). For example, compositions often
feature characteristic themes that could not properly be accounted for using the
strategies. At the time, Wilson was in the process of having a computer program
developed that incorporated his method and made calculations for composers even
easier. It is not known if this program was ever completed.
Wilson’s system conflated aesthetics with composition convention and this failed to
account for either one sufficiently, especially the former. His identification of chess
strategies was perhaps one of the earliest and most systematic approaches to the
problem of evaluating compositions (which nevertheless include an aesthetic
dimension) but suffered from oversimplification. There are two essential points that
illustrate this. First, the values attributed to each strategy were not adequately justified
20
and seemed to have stemmed from his experience as a composition tournament judge,
thereby limiting their applicability to little beyond compositions. These values also
failed to account for the variety of piece configurations possible within each strategy.
Second, metrics inherent to the game such as the number of pieces and distance (in
terms of squares on the board) were employed as the basis of some of the strategies but
were not incorporated in their universal form, e.g. defining a line as just 3 squares long
and not using the standard Shannon (1950) value of each piece (see subsection 3.5.2(a)).
These were apparently done to simplify calculations that, at the time, had to be
performed largely without the aid of computers.
Wilson tested his system experimentally on many contest-winning and generic problems
but his results were not very reliable because the intention was to replace the existing
paradigm of composition evaluation with a new, unbiased, and systematic one. What
seemed reasonable to him may not, in fact, have been so to the wider composition
community despite the best intentions to eliminate subjectivity. The scope of application
was also too large because it may be unreasonable to assume that human aesthetic
perception or appreciation of compositions remains constant despite the length of moves
or type of problem involved.
The rejection of his system by the composition community, however, is not necessarily
an indication of a failed approach. A more limited scope and flexible set of established
values (attributed to the strategies) would probably have improved it. In general, Wilson
introduced a scientific approach to aesthetics in the game even though not dealing with
it directly, and he showed that the evaluation of subjective aspects in chess is effective
to some degree using identified and quantifiable strategies or principles.
21
2.3 Principles of Beauty
The psychologist Stuart Margulies was perhaps the first person to study aesthetics in
chess experimentally. He derived eight principles of beauty in the game from the
judgement of experts (i.e. 30 players with an official Elo rating of over 2000) by
showing them pairs of positions and asking them to select the more beautiful solution
(Margulies, 1977). The Elo rating (see Appendix B), while widely used, is not
necessarily the best or most accurate measure of performance in the game (Donninger,
2003; Lopatka and Dzielinski, 2007; Elo, 2008). The eight identified principles of
beauty’ derived by Margulies are as follows.
1. Successfully violate heuristics.
2. Use the weakest piece possible.
3. Use all of the piece’s power.
4. Give more aesthetic weight to critical squares.
5. Use one giant piece in place of several minor ones.
6. Employ themes.
7. Avoid bland stereotypy.
8. Neither strangeness nor difficulty produces beauty.
The 1st principle involves making a move that goes against basic principles of ‘good
practice’ in chess. If the objective (e.g. checkmate, win material) is achieved despite
such a manoeuvre (e.g. leaving a piece exposed to capture), the move is considered a
successful heuristic violation. The 2nd principle is related to economy. The queen and
rook for example, have similar capabilities along ranks and files on the board. If a rook
22
is sufficient for the task, the solution is considered more beautiful than using a queen
since the former is a weaker piece.
The 3rd principle refers to the distance a particular piece travels. A piece’s mobility – the
number of squares it controls is a good reflection of its power. Hence, a piece moving
a greater distance across the board is more beautiful because less of its power is wasted.
The 4th principle places emphasis on the piece most involved in the objective. For
example, in a checkmate situation, the piece delivering mate would matter more,
aesthetically, than the one that moved; assuming they were not the same piece (like in a
discovered mate’, see section 5.5).
The 5th principle was tested using imaginary pieces not in the original piece set. It was
found that experts preferred positions where all the necessary resources required for the
task were concentrated into one powerful piece, instead of several weaker ones. This
principle is therefore also related to economy or efficiency. The 6th principle, i.e. using
chess themes (e.g. the pin, see section 5.2), is also important aesthetically. Margulies
determined that the more prominent a theme was in a solution, the more beautiful the
solution was considered to be. However, the themes had to be relevant or important to
chess.
The 7th principle implies originality and favours rare positions over common ones, but
the 8th principle seemingly contradicts it. Margulies found that highly unlikely positions
did not lead to judgements of beauty - the experts were actually equally divided between
them and common positions - and neither did solutions which were difficult to find. As
Margulies himself concluded, the 8th principle is rather a restriction of the 7th than its
contradiction. Rarity or originality is favoured aesthetically as long as it is not too
23
difficult to solve, or improbable, i.e. from the viewpoint of its likelihood of occurring in
a real game.
Margulies also questioned intermediate and novice players and found that the majority
of them (a higher proportion in the former) concurred with the experts as to which
solutions were more beautiful. This further supports Lasker’s contention that only
understanding - not mastery - is a prerequisite to appreciating beauty in the game
(Lasker, 1960; Belov et al., 1996). Margulies found that beautiful moves were often also
the most effective ones.
Margulies essentially identified through experimentation with experts many tangible
constituents or elements of aesthetics in the game of chess, and employed game metrics,
similar in some ways to Wilson. For example, pieces were evaluated according to their
relative values (see subsection 3.5.2(a)) and distance was measured in squares bound
only by the limits of the chessboard. The positions Margulies used in his experiments
were restricted to single moves to avoid ambiguity when interpreting the underlying
principle. However, he proposed no model or formalizations for aesthetics even though
the elements were clear. This is probably because his main intention was to investigate
traditionalaesthetic principles (e.g. economy, elegance, novelty) outside the domain of
chess. The game was simply a convenient place to experiment. He found that chess only
confirmed, rather than provided more insight, into the traditional principles. His derived
principles of beauty in the game, though not necessarily a conclusive set (Fine, 1978),
are nevertheless valuable to the research presented in this thesis.
24
2.4 Computer Chess Problem Composition
Schlosser (1988, 1991), in his approach to computer chess problem composition, built
on related work (van den Herik and Herschberg, 1985; van den Herik et al., 1988) that
had been done with regard to chess endgame databases. He outlined three steps that
were required for the process. The model is as follows.
1.Construct a complete database.
2.Eliminate all incorrect’ positions.
3.Select true chess problems.
Similar ideas were later used to compose problems in Tsume-Shogi, a Japanese game
not unlike chess (Hirose et al., 1997; Watanabe et al., 2000). The method was an
improvement over using a random algorithm (Noshita, 1996). A ‘reverse method’ has
recently been proposed (Horiyama et al., 2008) but is not directly applicable to chess
due to certain differences between the games. In Tsume-Shogi, each move of the
attacker must be a checking move; in a ‘shogimate’, only one solution exists; unlike
chess, where a mate might have more than one solution.
The 1st step was restricted to (a database of) endgame positions with a few pieces
because the number of possible positions increases exponentially with the pieces,
making computation unfeasible (Stiller, 1995). Complete databases, tablebases or
oraclesas they are known are designed through retrograde analysis. This involves
starting with say, a checkmate position (that has a specific game state, i.e. ‘won) and
working one ply or half-move backwards (Thompson, 1986). The process is repeated
25
until a seemingly uncertain position can be shown to lead to a checkmate in the shortest
number of moves and against any defence.
A complete database would therefore include all possible positions of a certain set or
number of pieces, and their inevitable result (win, loss or draw) in a given number of
moves. Presently, a complete database of 6-piece endgames including the two kings has
been achieved (Thompson, 1996). Seven pieces is estimated to be possible by the year
2015 (Hurd and Haworth, 2006). Such databases are also possible in other variations of
the game (Fang, 2006). Efforts have been made to reduce the size of tablebases but they
are still generally quite large with sizes running into gigabytes for just 5 and 6 pieces
(Thompson, 1996; Heinz, 1999a; Nalimov et al., 2000). Given the ‘omniscience of
such databases, they are useful for developing learning approaches in the game
(Sadikov and Bratko, 2006).
The 2nd step involves eliminating incorrect positions from the standpoint of
composition conventions. Most conventions are not very difficult to formalize and this
step helps to reduce, significantly, the number of positions found that inevitably lead to
checkmate (given orthodox problems). The 3rd step is where aesthetics is considered and
requires the intervention of human composers. Schlosser states the following.
“A computer, however, is not capable of composing like a human being. Creating a new
chess problem according to a given theme, which is the really creative part of a
composer's work, remains to be done by man.” (Schlosser, 1988)
26
“Formally, all positions left after step 2 are correct chess problems. To choose the
‘best’ ones from the potentially large set of correct positions, the imagination and
experience of a (human) expert is needed. According to the criteria of chess
composition, he selects what is new, artistically or aesthetically. There is still no way to
assign this task to a computer. An analogous situation exists in music composition or
painting.” (Schlosser, 1991)
Here, he acknowledges the importance of aesthetics and chess themes which typically
require the experience and expertise of a human composer. Schlosser implied that an
analogous situation exists in music and art, but these domains are more culturally
dependent, and have fewer discrete and computationally amenable components than
zero-sum perfect information games. Even so, computational models which address
aesthetics (in varying degrees) in those domains have since been developed with
reasonable success (Machado and Cardoso, 1998; Golub, 2000; Cope, 2001; Manaris et
al., 2002a, 2002b; Datta et al., 2006). These are beyond the scope of this thesis but their
methods are briefly discussed in section 2.8.
On its own, Schlosser’s first two steps were capable of finding forced checkmates that
were hard to solve by humans and occasionally featured interesting themes. This is to be
expected since themes or tactics are an integral part of how the game is won. The
problem, therefore, is in getting computers to recognize the aesthetics of a composition
or game for its own sake because somewhere in a massive tablebase certainly lie even
the most beautiful compositions that humans could conceive with those pieces and
would appreciate.
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The commercially available program ‘ChessExplorer’ uses a similar two step process to
create chess problems of the mate-in-2 and 3 varieties. However, its second step does
not filter them using any criteria except checking for a forced mate with only one
solution. Hence, the ‘created’ problems are usually not attractive; this is still evident in
the latest version of the program i.e. v6.11 (Nowakowski, 2005, 2008).
Schlosser’s model provides a clever way to emulate creativity in composing through the
use of brute-force searching but still relies on human intervention for the aesthetics
component. He therefore separates aesthetics from composition convention and
concedes to the limitation of his approach. The author hopes that this research will
address the aesthetics component in a way that can be incorporated into models like the
one proposed by Schlosser.
The automation of problem composition can then be improved so it does not need to
rely on human intervention as much or at all (more recent work is discussed in section
2.7). Schlosser’s model limits the scope of automatic composition to orthodox problems
and what are possible using available endgame databases. Any reasonable aesthetics
model would also need to be limited in this way to be consistent with available
information, and feasible in terms of required computing power. In summary,
Schlosser’s approach clearly identifies the gap a computational aesthetics model would
fill in this context.
2.5 Elements of Beauty Classified
One of the recent books that address aesthetics in chess is, ‘Secrets of Spectacular
Chess(Levitt and Friedgood, 1995, 2008). The book is currently in its expanded 2nd
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edition. In it, the authors a chess grandmaster and international master of problem
solving, respectively classify four elements of chess beauty namely paradox, depth,
geometry and flow. The book features a section entitled, ‘The Importance of Chess
Aesthetics’ and lists several reasons to support that contention. These include pleasure,
cultural or artistic value, educational and practical value.
Aesthetics in the game gives pleasure to a person and his life is considered to be more
meaningful than one who is unable to derive the same pleasure from it. Cultural or
artistic value is compared to paintings which exhibit the skill and genius of their artists.
In terms of education, good problems and pretty studies (a form of chess composition,
see ‘endgame study’ in Appendix B) are seen as an excellent teaching tool with
surprising solutions that can capture the imagination of those learning the game,
especially children. Finally, beautiful compositions and games have practical value in
actually improving a person’s – even a master’s - quality of play because they are full of
effective and original ideas.
Levitt and Friedgood write that virtually all world class players (e.g. Kasparov,
Botvinnik and Lasker) have an interest in the aesthetic aspect of chess and that it has
helped in their development. Returning to their elements of chess beauty, ‘paradox’
means a violation of heuristics or doing something that one is not usually supposed to
do, e.g. leaving a piece in a position to be captured. It is paradoxical because the move
wins despite going against general good practice’. Successful heuristic violation (the
first principle of beauty derived by Margulies, see section 2.3) comes under this.
Depthrefers to the point of the key move being obscured or unclear at the beginning
but realized later. It is the sort of foresight in a move that does not make much sense at
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first and is not necessarily paradoxical, but makes perfect sense by the end of the
combination.
Geometryis the chance or planned formation of shapes on the chessboard that
resemble say, alphabets. While this is very rare in real games, some compositions
feature it. Geometry also includes other visual effects on the board such as symmetry
and the relevance of particular lines (i.e. ranks, files, diagonals) to a solution. Flow
describes a move sequence that is basically forced instead of complicated with many
side variations. Flow is therefore more common in real games than in compositions
where side variations may even be laudable. Themes though not explicitly classified
as an element of beauty are treated as a given rather than examined individually. This
is probably because not all implementations of chess themes are noteworthy from an
aesthetics standpoint even though most examples of exemplary beauty tend to feature
some theme or other.
Levitt and Friedgood succeed in presenting to a modern audience the aesthetic aspect of
chess using contemporary examples and styles of play. This is important because in the
computer age, it is sometimes thought that little is now left to the imagination,
especially for a zero-sum perfect information game. Their examples (of beauty in the
game) illustrate that there are still useful tactics and strategies that our current
computational approach to playing is unable or slow to recognize. This remains true
even today. In other words, we are still better at solving certain problems in the game
using our creativity than computers, despite their brute-force approach. Levitt and
Friedgood, however, do not themselves propose any models or formalizations for
aesthetics. Yet, their broad classification of the elements of beauty and well-chosen
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examples of the finer aspects within them elucidate the principles of aesthetics proposed
by others (e.g. Margulies, though he is not cited) and extend them even further.
For instance, some of their examples suggest the importance of mobility (the number of
squares controlled by a piece in a particular position) as an additional property for
aesthetic computation (in addition to piece value and distance). Other examples
demonstrate the wide variety of possibilities within individual chess themes, which
Margulies identified as his 6th principle but only briefly explained. Similar to Lasker,
their contributions stem from experience rather than experiment (although it is notable
they had the distinct advantage of computer analysis) and they make no significant
distinction between compositions and real games in terms of aesthetics. They also do
not suggest a way of thinking about aesthetics as an independent component that is not
necessarily exclusive to composition conventions or real games. Nevertheless, their
examination of aesthetics in the game contributes significantly to the literature on the
subject.
2.6 Beauty Heuristics in a Game Engine
Aesthetics in chess has also been applied as a viable alternative to traditional game-
playing heuristics. Walls (1997) proposed using some of the principles of beauty
derived by Margulies (see section 2.3) in a chess engine to see if it performed better
than one that used standard heuristics. It was found that the engine using beauty
heuristics was 25% faster and needed to analyze 33% fewer nodes (i.e. positions) than
the one using standard heuristics, for solving direct-mate chess problems between 2 and
5 moves long.
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There are two important implications of Walls’ research. First, it supported his position
that humans make good moves in chess (at least in part) based on their ‘sense of
beauty’. Second, his findings also suggest a correlation between effectiveness and
aesthetics in the game. This is similar to using beauty as a measurement of
performance’, and it has been suggested not only in chess but also in economics
(Katsenelinboigen, 1990, 1997). It is not known, however, if the results obtained by
Walls remain true in a full game. He did not apply all of the principles derived by
Margulies because not all of them were applicable to the scope of his research.
The first principle (see section 2.3), successfully violate heuristics was adopted but
excluded standard heuristics that did not apply to mating problems. These included
those that (the violation of which) made it difficult to find the forced mate. The second,
third and fourth principles were summarized as ‘do not waste any power. They
encapsulate traditional aesthetic principles such as economy, parsimony and simplicity.
Walls modified the fifth principle to ‘use all of the piecesbecause the original one used
imaginary pieces and could also be interpreted as wasting less power (or using all
available pieces). He eventually rejected this modified principle due to computational
overhead. Themes were not included for the same reason and because they were thought
to distract from finding the quickest solution to checkmate. The seventh and eighth
principles by Margulies were also not included.
The heuristics implemented in the standard game engine were limited to those
applicable to mating problems and included the following.
1. Place the enemy king in check.
2. Attack the squares surrounding the enemy king.
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3. Sacrifice pieces if they lead to checkmate.
4. Gain enemy material.
The ‘beauty’ version of the engine included the first three standard game engine
heuristics and the following additional ones.
1. Violate the ‘gain enemy materialheuristic.
2. Use the weakest piece possible (to check).
3. Use all of the piece’s power.
4. Give aesthetic weight to the critical piece.
The first additional heuristic was limited to violating just the 4th standard heuristic
because violation of any of the others would impede finding the checkmate solution.
Walls implemented these beauty heuristics in a rather rudimentary manner. For
example, in order to encourage the engine into making sacrificial moves, the standard
evaluation function which calculates the material balance of a position (typical in all
chess programs) was disabled. Weaker pieces (pertaining to the second additional
heuristic) were determined using the standard Shannon values but limited to checking
moves. The third additional heuristic (i.e. use all of the piece’s power) involved
counting the number of squares made by the checking piece. There was no difference
therefore, between a queen moving across 5 squares and a bishop moving the same
distance. While this speeds computation, it may not correlate well with human aesthetic
perception of the manoeuvres since they involve pieces of different value.
The fourth additional heuristic awarded extra points to the move based on the number of
higher valued pieces in the piece set than the one performing check. Weaker pieces
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therefore, were considered more critical than stronger ones. In general, Walls may have
compromised his experiments to a degree by oversimplifying the aesthetic principles he
used. The reason he did this was because the main focus was not aesthetics itself but an
improvement in game-playing heuristics which requires fast computation for efficient
searching.
A complex representation or formalization of the aesthetic principles, however
necessary, would have slowed down the beauty heuristics engine considerably. It is
difficult to say if any realistic measure or identification of aesthetics was attained
through his experiments. Initial tests with human players on the aesthetics component
alone would have established this. Even so, Walls demonstrated - with some degree of
success - a computational implementation of chess aesthetic evaluation that can
potentially improve game-playing heuristics.
2.7 Computational Improvement of Chess Problems
Some of the relatively recent research in the area has sought to improve the composing
ability of computers with regard to chess because unlike playing, computers are quite
poor in problem composition. The Improver of Chess Problems (ICP) was presented as
a model to improve the quality of two-move mate problems (HaCohen-Kerner et al.
1999). A significant proportion of the knowledge required to evaluate the quality of
compositions was formalized for this purpose through consultation with two
international masters of chess composition. Based on the model, a chess problem is
typically put through several ‘transformations’ in order to improve it. These
transformations include the following.
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1. Deletion, addition or replacement of a piece on the board.
2. Transfer of a piece to another square within the same rank or file.
3. Transfer of a set of pieces using a particular movement (e.g. 3 files to
the right).
Each transformation is tried on a specific piece on the board (in a sequence of
importance) and applied only if the new position satisfies the three criteria mentioned
below.
1. It is legal.
2. It is a two-mover with only one key move.
3. It has a higher quality score.
The new position does not need to include the themes of the original problem or the best
quality score of the best transformation found so far. Thematic considerations were
considered restrictive to the number of improvements possible while weaker initial
transformations were seen as possibly leading to better overall improvement. The
quality evaluation function they used is shown below. V denotes the value function, Ti
the set of all themes in the position, Bj the set of all bonuses granted and Pk the set of all
penalties imposed.
( ) ( ) ( )
severe deficiency
otherwise
0
m
ij k
ij k
qVT VB VP
=+−
∑∑ ∑
Ten themes common to compositions (e.g. direct battery, Grimshaw) were attributed
relative but fixed values ranging from 10 to 45. Bonuses included desirable practices in
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composition such as placing the black king in the centre of the board (10 points) and
certain manoeuvres like sacrifices in the key move (5 × piece’s value). Penalties ranged
from severe deficiencies (e.g. illegal position, not a mate-in-2) to smaller things such as
a check (-50 points) or pinning a black piece (5 × piece’s value) in the key move.
The ICP was tested on 36 orthodox mate-in-2 chess problems taken from composition
books and managed to improve 10 or 27.7% of them. Eight of the ten were improved
after a single transformation and the remainder after two transformations. Mate-in-2
miniatures (no more than 7 pieces on the board) were used so that improvements could
be achieved within a reasonable amount of time. The low proportion of improvements
obtained from the 36 problems was attributed to the fact that these were known
compositions and mostly already optimized.
Aesthetics was not explicitly accounted for in the ICP model even though some of the
knowledge for evaluating the ‘quality’ of compositions included certain principles of
aesthetics such as themes and sacrifices. There are two significant issues. First, the
relative (fixed) values of themes, bonuses and penalties seemed to have been
determined arbitrarily by just two master composers, including the final determination
of perceived improvement over the original problems in the experiment. It is
noteworthy that the flow of information from expert to non-expert in complex domains
like chess often results in a bottleneck which affects the quality of the knowledge
formalized (Michie, 1986; Guid et al., 2008). Aesthetically, this approach does not
account for varying configurations of particular themes (e.g. using different pieces in
different places for a similar purpose) and the perception of composers in general; the
majority of whom are not masters.
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Second, this model is not easily applicable to real chess games where aesthetics is also
perceived. The ICP was designed specifically with chess problems in mind.
Nevertheless, the ICP is an improvement over Schlosser’s model (see section 2.4)
because there is an attempt to deal with the aesthetics component, albeit in a way that
conflates it with convention (comparable to Wilson’s system, see section 2.2).
An improved model called, Chess Composerused a similar approach to the ICP but
had fewer types of transformations (Fainshtein and HaCohen-Kerner, 2006a, 2006b).
This reduced the branching factor and increased the depth of applied transformations.
Chess Composer used brute-force searching to find a global maximum and did not
suffer from ICP’s limitations in terms of pruning the search tree, 1) wherever the
position was not legal, 2) not a two-mover with one key or, 3) with a lower quality score
than the original problem. Much of the domain knowledge was taken from the ICP
model with some additions, but the quality function remained the same.
Chess Composer was tested on 100 mate-in-2 chess problems and managed to improve
the quality of 97 of them. Despite its relative slowness, the model’s higher success rate
can be attributed to using a better search technique and greater depth of transformations
(3 levels instead of 2). Most of the improvements were, in fact, achieved after various
sequences of three transformations. The authors recognized Chess Composer’s
limitations in terms of aesthetics and state the following, before alluding to the use of
ICP’s formalized knowledge to compensate.
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“The concept of a ‘high-quality mate problem’ in chess is hard to define, especially if
an automatic program is involved. It is not simple to define concepts such as beauty,
originality, uniqueness of the solution, and difficulty to solve.”
(Fainshtein and HaCohen-Kerner, 2006a)
Both the ICP and Chess Composer models are practical methods of ‘composing’ or at
least improving existing chess problems within a limited scope (e.g. mate-in-2) from the
standpoint of composition conventions. Longer problems are possible but would require
more effective search techniques in order to be computationally feasible (Fainshtein and
HaCohen-Kerner, 2006b). They admit that improving the quality function might
contribute to that (Fainshtein and HaCohen-Kerner, 2006a). In general, Chess
Composer improves more in terms of performance rather than technique, when
compared to the ICP, and does not look any deeper into the question of properly
accounting for aesthetics.
The conflation between composition convention, and aesthetics, still exists and as a
result the latter is improperly accounted for. The attribution of fixed but relative values
to themes and conventions with the aid of masters in composition (for both the ICP and
later supplemented in Chess Composer) is a step closer toward complete automation of
the process but nonetheless fails to account for aesthetic variety within each theme and
convention. Even though some conventions are described using simple formulas, e.g.
bonus for X number of pieces on the board = 3 × (18-X), the constants used are not
explained and therefore presumably have little basis in chess literature.
Given the brute-force searching required in both of these models, complexity with
regard to aesthetics was perhaps rightfully avoided. The research presented in this
38
thesis, however, deals exclusively with aesthetics and a model which should enable
computers to recognize it in the game like humans do. Since brute-force searching for
larger purposes (e.g. automatic composition) is beyond the scope of this thesis,
formalization of the available knowledge we have on chess aesthetics need not be
compromised. This means that the variety of configurations possible within the
established aesthetic principles and themes can be better accounted for.
2.8 A Look at Methodologies Used in Other Domains
Even though the methodologies used in other domains are not quite applicable to zero-
sum perfect information games (the domains are of different natures), this section
presents a brief discussion of a select few for a general context. One of the earliest
attempts to formalize aesthetics is the mathematician Birkhoff’s model of, M = O/C
where M = aesthetic value, O = order and C = complexity (Birkhoff, 1933; Scha and
Bod, 1993). ‘Order’ was identified with factors like symmetry and repetition, and
‘complexity’ with the amount of effort required to attend to a pattern and assimilate it.
These enabled similarities and relations between elements to be discerned using
numerical values. Even though Birkhoff specified procedures for attributing such values
to certain things such as polygons, vase outlines, melodies and lines of verse, his model
was not successful mainly due to oversimplification. Maximum aesthetic satisfaction
was simply not obtained using the most order and least complexity in patterns, as his
model suggested. Experimental results proved this (Berlyne, 1972).
Later attempts have limited such models to just one domain and identified the measures
used to factors strictly within that domain. Machado and Cardoso (1998) for example,
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equated the aesthetic value of artwork (i.e. images) as IC/PC where IC = image
complexity and PC = processing complexity. IC was estimated as the amount of error
during lossy compression (e.g. JPEG) divided by the compression ratio. PC was
estimated using fractal compression because they had reason to believe it was closer to
how humans process images in their mind. Experimental results with users were
encouraging.
Datta et al. (2006) derived aesthetic factors for photographs from rules of thumb in
photography, common intuition, and observed trends in ratings. These were compared
against aesthetic ratings of over 3,000 random photos by members of an online photo
community to identify the features that correlated well (Physorg, 2006). A classification
and regression model was then built using a subset of relevant features. Regression
models can be used to obtain absolute quantitative results for aesthetic features whereas
classification models work better for qualitative results (i.e. high and low thresholds).
They found the latter to be more suitable because it was difficult to differentiate
between the aesthetics of photos that scored closely.
Manaris et al. (2002a, 2002b) used discrete representations (e.g. frequency of notes,
intervals) of particular attributes in music (e.g. pitch, volume) to recognize beautiful
compositions. They found that the aspects of beauty in music may be algorithmically
classifiable and identifiable based on an experiment which compared ‘quality’ MIDI
renderings of musical pieces (by known composers from various music genres) against
random pieces (i.e. noise) by testing for conformity with the ZipfMandelbrot Law.
This law essentially states that phenomena generated by complex social and natural
systems tend to follow a statistically predictable structure. Examples include human
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language and music. They acknowledge, however, that their statistical approach has
limitations; notably that minor changes to a particular musical piece (e.g. one note to
another) may be statistically insignificant but highly significant (aesthetically) to the
listener.
A more recent approach combined statistical, connectionist and evolutionary
components with the assumption that popularity of music correlates with aesthetics
(Manaris et al., 2007). There is potential in evolutionary algorithms for both the
aesthetics of music and art (similar to biological evolution) but it challenges what we
even classify as ‘art’ because ordinarily it is ‘created’ by humans, not evolved
(McCormack, 2006). Methods used for larger purposes such as composing and
classifying musical styles (Golub, 2000; Cope, 2001; Miranda et al., 2007) often do not
address aesthetics directly so they are not discussed here.
The computational approach to aesthetics in other domains is generally problematic
because aesthetic features or principles are often loosely defined and culturally
dependent. This means that the experts themselves find it difficult to be specific enough
about the constituents of beauty within a particular domain. Researchers therefore, find
it easier to extract and weight features on their own through the analysis of domain-
related resources such as photographs, music compositions or prose. There are many
ways this analysis can be done and justified. The extracted features may or may not
have anything to do with aesthetics in the domain but are then usually tested against
human perception to identify which ones are most likely related to beauty.
Computer systems (usually after some training) can then use formalizations of those
features to recognize aesthetics (to a limited degree) in other objects within that domain.
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It is difficult to apply features or methods from one domain to another because in many
cases, they are significantly dissimilar and perceived through different senses. The
objects themselves (e.g. literature and photographs) may use different ‘channels’ even
in cases where the modality or sensory organ (e.g. the eye) is the same (Vaughan,
2006). For that matter, the aesthetic criteria used to evaluate novels are likely different
from poetry or plays, despite the same channel and modality (Weyhrauch, 1997).
There are two main advantages - in terms of aesthetic evaluation - to a zero-sum perfect
information game like chess. First, it is theoretically finite and particularly amenable to
computation. Second, because the rules and pieces are the same anywhere (at least for
international chess), it is not subject to cultural influence and marked difference of
opinion, even aesthetically. While cultural influence or upbringing may affect the
general style of a person’s play, the objectives of the game are the same and must be
adhered to. Losing a queen unnecessarily for example, is inadvisable regardless of
where you are from.
This makes aesthetic principles within that theoretically finite domain arguably more
reliable and consistent than in any other. The typical approach of performing regression
analysis to determine aesthetic weights, and neural networks to train the system, is
possible; but there is no inherent subjectivity in chess that calls for it. Reliable,
aesthetically-rated chess combinations required for such an approach are also scarce.
Just as chess programs tend to simulate the abstract intelligence of human playing
ability through the use of discrete brute-force techniques, it is possible that human
aesthetic perception in the game can also be simulated using an analogous approach. It
may, in fact, be the most suitable one.
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Given that chess literature has sufficient information relating specifically to the
aesthetics of the game, it was not deemed necessary to conduct further experiments to
derive more principles. Additionally, the resources necessary for that (e.g. a large
number of master players) were not available to the author. The approach taken in this
research can therefore be considered unique compared to how computational aesthetics
is addressed in other domains. The details are in the following chapter.
2.9 Chapter Summary
Research into chess aesthetics started with the recognition by master players and
composers that it is a significant aspect or dimension to the game. They showed that it is
not limited to compositions and also extends to real games. Since aesthetics is more
prominent in compositions, it is often conflated with composition convention, and not
treated separately in a way that would make computational application of it easier. This
is especially true when looking at real games where composition conventions do not
necessarily apply, yet beauty is also perceived. In the case of orthodox problems, there
is no difference between compositions and tournament games except in the way the
positions are configured to reflect certain principles and themes.
Attempts to classify and quantify aesthetics have been made through the systematic
identification of principles and themes that are present in what players consider
beautiful positions and combinations. These have been essential to the development of
models and formalizations which try to account for aesthetics as a by-product while
focusing on conventions that are more easily defined and recognized on the board.
Applications of this approach therefore focus on automatic composition of problems in
general (within a limited scope) but produce relatively poor results in comparison to
43
human composers. Part of the reason lies in the rudimentary adaptation of the
established aesthetic principles and themes by using typically fixed, but relative values;
even though this is sometimes done through consultation with master composers.
The variety of configurations possible in each principle and theme (which is what
makes a combination beautiful) is greater than what can be realistically accounted for
with fixed values, even relative ones. The number and suitability of principles and
themes identified that apply to both domains (i.e. compositions and real games) are
equally important. This is a compromise which researchers have often made because
their models rely on brute-force searching that works best by minimizing computational
load. They also have no conceptual framework for aesthetics that enables it to be
described independently of composition conventions. In most models, aesthetics is not
explicitly accounted for so the quality functions designed to evaluate it often suffer from
oversimplification. The principles and themes typically used are also limited and often
restricted to just a selection within the domain of compositions, which neglects the
portion of aesthetics that also exists in real games.
The repercussions of all this are significant because an inadequate or poor account of
aesthetics will affect any subsequent application of such a model. This might include the
following.
1. Versatile chess database search engines (e.g. that are able to
locate beautiful games and combinations automatically).
2. Programs that aid judges of composition tournaments and
brilliancy prizes.
3. Chess program personality modules (for more human-style
play).
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4. Game-playing engines (based on beauty heuristics).
5. Automatic chess annotators (Guid et al., 2008).
The issues apply not only to international chess but possibly hundreds of other chess
variants and similar zero-sum perfect information games with an aesthetic dimension
that could benefit from such a model. The next chapter presents the research
methodology and proposed aesthetics model.
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CHAPTER 3: METHODOLOGY – Aesthetics in the Game
3.0 Components of the Research
This chapter details the aesthetics model (for the game of chess) proposed in this thesis.
The model was designed to answer