| 1 | The purpose of this Chapter is to describe the algorithm used in |
| 2 | GNU Go to determine eyes. |
| 3 | |
| 4 | @menu |
| 5 | * Local Games:: Local games |
| 6 | * Eye Space:: Eye space |
| 7 | * Eye Space as Local Game:: Eye space as local game |
| 8 | * Eye Example:: An example |
| 9 | * Graphs:: Underlying graphs |
| 10 | * Eye Shape:: Pattern matching |
| 11 | * Eye Local Game Values:: Pattern matching |
| 12 | * Eye Topology:: False eyes and half eyes |
| 13 | * Eye Topology with Ko:: False eyes and half eyes with ko |
| 14 | * False Margins:: False margins |
| 15 | * Eye Functions:: Functions in @file{optics.c} |
| 16 | @end menu |
| 17 | |
| 18 | @node Local Games |
| 19 | @section Local games |
| 20 | |
| 21 | The fundamental paradigm of combinatorial game theory is that games |
| 22 | can be added and in fact form a group. If @samp{G} and @samp{H} are |
| 23 | games, then @samp{G+H} is a game in which each player on his turn |
| 24 | has the option of playing in either move. We say that the game |
| 25 | @samp{G+H} is the sum of the local games @samp{G} and @samp{H}. |
| 26 | |
| 27 | Each connected eyespace of a dragon affords a local game which yields |
| 28 | a local game tree. The score of this local game is the number of eyes |
| 29 | it yields. Usually if the players take turns and make optimal moves, |
| 30 | the end scores will differ by 0 or 1. In this case, the local game may |
| 31 | be represented by a single number, which is an integer or half |
| 32 | integer. Thus if @samp{n(O)} is the score if @samp{O} moves first, |
| 33 | both players alternate (no passes) and make alternate moves, and |
| 34 | similarly @samp{n(X)}, the game can be represented by |
| 35 | @samp{@{n(O)|n(X)@}}. Thus @{1|1@} is an eye, @{2|1@} is an eye plus a |
| 36 | half eye, etc. |
| 37 | |
| 38 | The exceptional game @{2|0@} can occur, though rarely. We call |
| 39 | an eyespace yielding this local game a CHIMERA. The dragon |
| 40 | is alive if any of the local games ends up with a score of 2 |
| 41 | or more, so @{2|1@} is not different from @{3|1@}. Thus @{3|1@} is |
| 42 | NOT a chimera. |
| 43 | |
| 44 | Here is an example of a chimera: |
| 45 | |
| 46 | @example |
| 47 | @group |
| 48 | XXXXX |
| 49 | XOOOX |
| 50 | XO.OOX |
| 51 | XX..OX |
| 52 | XXOOXX |
| 53 | XXXXX |
| 54 | @end group |
| 55 | @end example |
| 56 | |
| 57 | @node Eye Space |
| 58 | @section Eye spaces |
| 59 | |
| 60 | In order that each eyespace be assignable to a dragon, |
| 61 | it is necessary that all the dragons surrounding it |
| 62 | be amalgamated (@pxref{Amalgamation}). This is the |
| 63 | function of @code{dragon_eye()}. |
| 64 | |
| 65 | An EYE SPACE for a black dragon is a collection of vertices |
| 66 | adjacent to a dragon which may not yet be completely closed off, |
| 67 | but which can potentially become eyespace. If an open eye space is |
| 68 | sufficiently large, it will yield two eyes. Vertices at the edge |
| 69 | of the eye space (adjacent to empty vertices outside the eye space) |
| 70 | are called MARGINAL. |
| 71 | |
| 72 | Here is an example from a game: |
| 73 | |
| 74 | @example |
| 75 | @group |
| 76 | |
| 77 | |. X . X X . . X O X O |
| 78 | |X . . . . . X X O O O |
| 79 | |O X X X X . . X O O O |
| 80 | |O O O O X . O X O O O |
| 81 | |. . . . O O O O X X O |
| 82 | |X O . X X X . . X O O |
| 83 | |X O O O O O O O X X O |
| 84 | |. X X O . O X O . . X |
| 85 | |X . . X . X X X X X X |
| 86 | |O X X O X . X O O X O |
| 87 | |
| 88 | @end group |
| 89 | @end example |
| 90 | |
| 91 | Here the @samp{O} dragon which is surrounded in the center has open |
| 92 | eye space. In the middle of this open eye space are three |
| 93 | dead @samp{X} stones. This space is large enough that O cannot be |
| 94 | killed. We can abstract the properties of this eye shape as follows. |
| 95 | Marking certain vertices as follows: |
| 96 | |
| 97 | @example |
| 98 | @group |
| 99 | |
| 100 | |- X - X X - - X O X O |
| 101 | |X - - - - - X X O O O |
| 102 | |O X X X X - - X O O O |
| 103 | |O O O O X - O X O O O |
| 104 | |! . . . O O O O X X O |
| 105 | |X O . X X X . ! X O O |
| 106 | |X O O O O O O O X X O |
| 107 | |- X X O - O X O - - X |
| 108 | |X - - X - X X X X X X |
| 109 | |O X X O X - X O O X O |
| 110 | |
| 111 | @end group |
| 112 | @end example |
| 113 | |
| 114 | @noindent |
| 115 | the shape in question has the form: |
| 116 | |
| 117 | @example |
| 118 | @group |
| 119 | |
| 120 | !... |
| 121 | .XXX.! |
| 122 | |
| 123 | @end group |
| 124 | @end example |
| 125 | |
| 126 | The marginal vertices are marked with an exclamation point (@samp{!}). |
| 127 | The captured @samp{X} stones inside the eyespace are naturally marked @samp{X}. |
| 128 | |
| 129 | The precise algorithm by which the eye spaces are determined is |
| 130 | somewhat complex. Documentation of this algorithm is in the |
| 131 | comments in the source to the function @code{make_domains()} in |
| 132 | @file{optics.c}. |
| 133 | |
| 134 | The eyespaces can be conveniently displayed using a colored |
| 135 | ascii diagram by running @command{gnugo -E}. |
| 136 | |
| 137 | @node Eye Space as Local Game |
| 138 | @section The eyespace as local game |
| 139 | |
| 140 | In the abstraction, an eyespace consists of a set of vertices |
| 141 | labelled: |
| 142 | |
| 143 | @example |
| 144 | |
| 145 | ! . X |
| 146 | |
| 147 | @end example |
| 148 | |
| 149 | Tables of many eyespaces are found in the database |
| 150 | @file{patterns/eyes.db}. Each of these may be thought of as a local |
| 151 | game. The result of this game is listed after the eyespace in the form |
| 152 | @code{:max,min}, where @code{max} is the number of eyes the pattern |
| 153 | yields if @samp{O} moves first, while @code{min} is the number of eyes |
| 154 | the pattern yields if @samp{X} moves first. The player who owns the eye |
| 155 | space is denoted @samp{O} throughout this discussion. Since three eyes |
| 156 | are no better than two, there is no attempt to decide whether the space |
| 157 | yields two eyes or three, so max never exceeds 2. Patterns with min>1 |
| 158 | are omitted from the table. |
| 159 | |
| 160 | For example, we have: |
| 161 | |
| 162 | @example |
| 163 | @group |
| 164 | Pattern 548 |
| 165 | |
| 166 | x |
| 167 | xX.! |
| 168 | |
| 169 | :0111 |
| 170 | |
| 171 | @end group |
| 172 | @end example |
| 173 | |
| 174 | Here notation is as above, except that @samp{x} means @samp{X} or |
| 175 | @code{EMPTY}. The result of the pattern is not different if @samp{X} has |
| 176 | stones at these vertices or not. |
| 177 | |
| 178 | We may abstract the local game as follows. The two players @samp{O} |
| 179 | and @samp{X} take turns moving, or either may pass. |
| 180 | |
| 181 | RULE 1: @samp{O} for his move may remove any vertex marked @samp{!} |
| 182 | or marked @samp{.}. |
| 183 | |
| 184 | RULE 2: @samp{X} for his move may replace a @samp{.} by an @samp{X}. |
| 185 | |
| 186 | RULE 3: @samp{X} may remove a @samp{!}. In this case, each @samp{.} |
| 187 | adjacent to the @samp{!} which is removed becomes a @samp{!} . If an |
| 188 | @samp{X} adjoins the @samp{!} which is removed, then that @samp{X} |
| 189 | and any which are connected to it are also removed. Any @samp{.} which |
| 190 | are adjacent to the removed @samp{X}'s then become @samp{.}. |
| 191 | |
| 192 | Thus if @samp{O} moves first he can transform the eyeshape in |
| 193 | the above example to: |
| 194 | |
| 195 | @example |
| 196 | @group |
| 197 | ... or !... |
| 198 | .XXX.! .XXX. |
| 199 | @end group |
| 200 | @end example |
| 201 | |
| 202 | However if @samp{X} moves he may remove the @samp{!} and the @samp{.}s |
| 203 | adjacent to the @samp{!} become @samp{!} themselves. Thus if @samp{X} |
| 204 | moves first he may transform the eyeshape to: |
| 205 | |
| 206 | @example |
| 207 | @group |
| 208 | !.. or !.. |
| 209 | .XXX.! .XXX! |
| 210 | @end group |
| 211 | @end example |
| 212 | |
| 213 | NOTE: A nuance which is that after the @samp{X:1}, @samp{O:2} |
| 214 | exchange below, @samp{O} is threatening to capture three X stones, |
| 215 | hence has a half eye to the left of 2. This is subtle, and there are |
| 216 | other such subtleties which our abstraction will not capture. Some of |
| 217 | these at least can be dealt with by a refinements of the scheme, but |
| 218 | we will content ourselves for the time being with a simplified model. |
| 219 | |
| 220 | @example |
| 221 | @group |
| 222 | |
| 223 | |- X - X X - - X O X O |
| 224 | |X - - - - - X X O O O |
| 225 | |O X X X X - - X O O O |
| 226 | |O O O O X - O X O O O |
| 227 | |1 2 . . O O O O X X O |
| 228 | |X O . X X X . 3 X O O |
| 229 | |X O O O O O O O X X O |
| 230 | |- X X O - O X O - - X |
| 231 | |X - - X - X X X X X X |
| 232 | |O X X O X - X O O X O |
| 233 | |
| 234 | @end group |
| 235 | @end example |
| 236 | |
| 237 | We will not attempt to characterize the terminal states |
| 238 | of the local game (some of which could be seki) or |
| 239 | the scoring. |
| 240 | |
| 241 | @node Eye Example |
| 242 | @section An example |
| 243 | |
| 244 | Here is a local game which yields exactly one |
| 245 | eye, no matter who moves first: |
| 246 | |
| 247 | @example |
| 248 | @group |
| 249 | |
| 250 | ! |
| 251 | ... |
| 252 | ...! |
| 253 | |
| 254 | @end group |
| 255 | @end example |
| 256 | |
| 257 | Here are some variations, assuming @samp{O} moves first. |
| 258 | |
| 259 | @example |
| 260 | @group |
| 261 | ! (start position) |
| 262 | ... |
| 263 | ...! |
| 264 | @end group |
| 265 | |
| 266 | |
| 267 | @group |
| 268 | ... (after @samp{O}'s move) |
| 269 | ...! |
| 270 | @end group |
| 271 | |
| 272 | |
| 273 | @group |
| 274 | ... |
| 275 | ..! |
| 276 | @end group |
| 277 | |
| 278 | |
| 279 | @group |
| 280 | ... |
| 281 | .. |
| 282 | @end group |
| 283 | |
| 284 | |
| 285 | @group |
| 286 | .X. (nakade) |
| 287 | .. |
| 288 | @end group |
| 289 | @end example |
| 290 | |
| 291 | Here is another variation: |
| 292 | |
| 293 | @example |
| 294 | |
| 295 | @group |
| 296 | ! (start) |
| 297 | ... |
| 298 | ...! |
| 299 | @end group |
| 300 | |
| 301 | |
| 302 | @group |
| 303 | ! (after @samp{O}'s move) |
| 304 | . . |
| 305 | ...! |
| 306 | @end group |
| 307 | |
| 308 | |
| 309 | @group |
| 310 | ! (after @samp{X}'s move) |
| 311 | . . |
| 312 | ..X! |
| 313 | @end group |
| 314 | |
| 315 | |
| 316 | @group |
| 317 | . . |
| 318 | ..X! |
| 319 | @end group |
| 320 | |
| 321 | |
| 322 | @group |
| 323 | . ! |
| 324 | .! |
| 325 | @end group |
| 326 | @end example |
| 327 | |
| 328 | |
| 329 | @node Graphs |
| 330 | @section Graphs |
| 331 | |
| 332 | It is a useful observation that the local game associated |
| 333 | with an eyespace depends only on the underlying graph, which |
| 334 | as a set consists of the set of vertices, in which two elements |
| 335 | are connected by an edge if and only if they are adjacent on |
| 336 | the Go board. For example the two eye shapes: |
| 337 | |
| 338 | @example |
| 339 | |
| 340 | .. |
| 341 | .. |
| 342 | |
| 343 | and |
| 344 | |
| 345 | .... |
| 346 | |
| 347 | @end example |
| 348 | |
| 349 | @noindent |
| 350 | though distinct in shape have isomorphic graphs, and consequently |
| 351 | they are isomorphic as local games. This reduces the number of |
| 352 | eyeshapes in the database @file{patterns/eyes.db}. |
| 353 | |
| 354 | A further simplification is obtained through our treatment of |
| 355 | half eyes and false eyes. Such patterns are identified by the |
| 356 | topological analysis (@pxref{Eye Topology}). |
| 357 | |
| 358 | A half eye is isomorphic to the pattern @code{(!.)} . To see this, |
| 359 | consider the following two eye shapes: |
| 360 | |
| 361 | @example |
| 362 | @group |
| 363 | XOOOOOO |
| 364 | X.....O |
| 365 | XOOOOOO |
| 366 | |
| 367 | @end group |
| 368 | and: |
| 369 | @group |
| 370 | |
| 371 | XXOOOOO |
| 372 | XOa...O |
| 373 | XbOOOOO |
| 374 | XXXXXXX |
| 375 | |
| 376 | @end group |
| 377 | @end example |
| 378 | |
| 379 | These are equivalent eyeshapes, with isomorphic local games @{2|1@}. |
| 380 | The first has shape: |
| 381 | |
| 382 | @example |
| 383 | |
| 384 | !.... |
| 385 | |
| 386 | @end example |
| 387 | |
| 388 | The second eyeshape has a half eye at @samp{a} which is taken when @samp{O} |
| 389 | or @samp{X} plays at @samp{b}. This is found by the topological |
| 390 | criterion (@pxref{Eye Topology}). |
| 391 | |
| 392 | The graph of the eye_shape, ostensibly @samp{....} is modified by replacing |
| 393 | the left @samp{.} by @samp{!.} during graph matching. |
| 394 | |
| 395 | |
| 396 | A false eye is isomorphic to the pattern @code{(!)} . To see this, |
| 397 | consider the following eye shape: |
| 398 | |
| 399 | @example |
| 400 | |
| 401 | XXXOOOOOO |
| 402 | X.Oa....O |
| 403 | XXXOOOOOO |
| 404 | |
| 405 | @end example |
| 406 | |
| 407 | This is equivalent to the two previous eyeshapes, with an isomorphic |
| 408 | local game @{2|1@}. |
| 409 | |
| 410 | This eyeshape has a false eye at @samp{a}. This is also found by the |
| 411 | topological criterion. |
| 412 | |
| 413 | The graph of the eye_shape, ostensibly @samp{.....} is modified by replacing |
| 414 | the left @samp{.} by @samp{!}. This is made directly in the eye data, |
| 415 | not only during graph matching. |
| 416 | |
| 417 | @node Eye Shape |
| 418 | @section Eye shape analysis |
| 419 | |
| 420 | The patterns in @file{patterns/eyes.db} are compiled into graphs |
| 421 | represented essentially by arrays in @file{patterns/eyes.c}. |
| 422 | |
| 423 | Each actual eye space as it occurs on the board is also |
| 424 | compiled into a graph. Half eyes are handled as follows. |
| 425 | Referring to the example |
| 426 | |
| 427 | @example |
| 428 | @group |
| 429 | XXOOOOO |
| 430 | XOa...O |
| 431 | XbOOOOO |
| 432 | XXXXXX |
| 433 | @end group |
| 434 | @end example |
| 435 | |
| 436 | @noindent |
| 437 | repeated from the preceding discussion, the vertex at @samp{b} is |
| 438 | added to the eyespace as a marginal vertex. The adjacency |
| 439 | condition in the graph is a macro (in @file{optics.c}): two |
| 440 | vertices are adjacent if they are physically adjacent, |
| 441 | or if one is a half eye and the other is its key point. |
| 442 | |
| 443 | In @code{recognize_eyes()}, each such graph arising from an actual eyespace is |
| 444 | matched against the graphs in @file{eyes.c}. If a match is found, the |
| 445 | result of the local game is known. If a graph cannot be matched, its |
| 446 | local game is assumed to be @{2|2@}. |
| 447 | |
| 448 | @node Eye Local Game Values |
| 449 | @section Eye Local Game Values |
| 450 | |
| 451 | The game values in @file{eyes.db} are given in a simplified scheme which is |
| 452 | flexible enough to represent most possibilities in a useful way. |
| 453 | |
| 454 | The colon line below the pattern gives the eye value of the matched |
| 455 | eye shape. This consists of four digits, each of which is the number |
| 456 | of eyes obtained during the following conditions: |
| 457 | |
| 458 | @enumerate |
| 459 | @item The attacker moves first and is allowed yet another move because |
| 460 | the defender plays tenuki. |
| 461 | @item The attacker moves first and the defender responds locally. |
| 462 | @item The defender moves first and the attacker responds locally. |
| 463 | @item The defender moves first and is allowed yet another move because |
| 464 | the attacker plays tenuki. |
| 465 | @end enumerate |
| 466 | |
| 467 | The first case does @strong{not} necessarily mean that the attacker is |
| 468 | allowed two consecutive moves. This is explained with an example |
| 469 | later. |
| 470 | |
| 471 | Also, since two eyes suffice to live, all higher numbers also count |
| 472 | as two. |
| 473 | |
| 474 | The following 15 cases are of interest: |
| 475 | |
| 476 | @itemize @bullet |
| 477 | @item 0000 0 eyes. |
| 478 | @item 0001 0 eyes, but the defender can threaten to make one eye. |
| 479 | @item 0002 0 eyes, but the defender can threaten to make two eyes. |
| 480 | @item 0011 1/2 eye, 1 eye if defender moves first, 0 eyes if attacker does. |
| 481 | @item 0012 3/4 eyes, 3/2 eyes if defender moves first, 0 eyes if attacker does. |
| 482 | @item 0022 1* eye, 2 eyes if defender moves first, 0 eyes if attacker does. |
| 483 | @item 0111 1 eye, attacker can threaten to destroy the eye. |
| 484 | @item 0112 1 eye, attacker can threaten to destroy the eye, defender can threaten to make another eye. |
| 485 | @item 0122 5/4 eyes, 2 eyes if defender moves first, 1/2 eye if attacker does. |
| 486 | @item 0222 2 eyes, attacker can threaten to destroy both. |
| 487 | @item 1111 1 eye. |
| 488 | @item 1112 1 eye, defender can threaten to make another eye. |
| 489 | @item 1122 3/2 eyes, 2 eyes if defender moves first, 1 eye if attacker does. |
| 490 | @item 1222 2 eyes, attacker can threaten to destroy one eye. |
| 491 | @item 2222 2 eyes. |
| 492 | @end itemize |
| 493 | |
| 494 | The 3/4, 5/4, and 1* eye values are the same as in Howard Landman's paper |
| 495 | Eyespace Values in Go. Attack and defense points are only marked in |
| 496 | the patterns when they have definite effects on the eye value, |
| 497 | i.e. pure threats are not marked. |
| 498 | |
| 499 | Examples of all different cases can be found among the patterns in |
| 500 | this file. Some of them might be slightly counterintuitive, so we |
| 501 | explain one important case here. Consider |
| 502 | |
| 503 | @example |
| 504 | @group |
| 505 | Pattern 6141 |
| 506 | |
| 507 | X |
| 508 | XX.@@x |
| 509 | |
| 510 | :1122 |
| 511 | @end group |
| 512 | @end example |
| 513 | |
| 514 | which e.g. matches in this position: |
| 515 | |
| 516 | @example |
| 517 | @group |
| 518 | .OOOXXX |
| 519 | OOXOXOO |
| 520 | OXXba.O |
| 521 | OOOOOOO |
| 522 | @end group |
| 523 | @end example |
| 524 | |
| 525 | Now it may look like @samp{X} could take away both eyes by playing @samp{a} |
| 526 | followed by @samp{b}, giving 0122 as eye value. This is where the subtlety |
| 527 | of the definition of the first digit in the eye value comes into |
| 528 | play. It does not say that the attacker is allowed two consecutive |
| 529 | moves but only that he is allowed to play "another move". The |
| 530 | crucial property of this shape is that when @samp{X} plays at a to destroy |
| 531 | (at least) one eye, @samp{O} can answer at @samp{b}, giving: |
| 532 | |
| 533 | @example |
| 534 | @group |
| 535 | |
| 536 | .OOOXXX |
| 537 | OO.OXOO |
| 538 | O.cOX.O |
| 539 | OOOOOOO |
| 540 | |
| 541 | @end group |
| 542 | @end example |
| 543 | |
| 544 | Now @samp{X} has to continue at @samp{c} in order to keep @samp{O} |
| 545 | at one eye. After this @samp{O} plays tenuki and @samp{X} cannot |
| 546 | destroy the remaining eye by another move. Thus the eye value is |
| 547 | indeed 1122. |
| 548 | |
| 549 | As a final note, some of the eye values indicating a threat depend |
| 550 | on suicide to be allowed, e.g. |
| 551 | |
| 552 | @example |
| 553 | @group |
| 554 | |
| 555 | Pattern 301 |
| 556 | |
| 557 | X.X |
| 558 | |
| 559 | :1222 |
| 560 | |
| 561 | @end group |
| 562 | @end example |
| 563 | |
| 564 | We always assume suicide to be allowed in this database. It is easy |
| 565 | enough to sort out such moves at a higher level when suicide is |
| 566 | disallowed. |
| 567 | |
| 568 | @node Eye Topology |
| 569 | @section Topology of Half Eyes and False Eyes |
| 570 | |
| 571 | A HALF EYE is a pattern where an eye may or may not materialize, |
| 572 | depending on who moves first. Here is a half eye for @code{O}: |
| 573 | |
| 574 | @example |
| 575 | @group |
| 576 | |
| 577 | OOXX |
| 578 | O.O. |
| 579 | OO.X |
| 580 | |
| 581 | @end group |
| 582 | @end example |
| 583 | |
| 584 | A FALSE EYE is an eye vertex which cannot become a proper eye. Here are |
| 585 | two examples of false eyes for @code{O}: |
| 586 | |
| 587 | @example |
| 588 | @group |
| 589 | |
| 590 | OOX OOX |
| 591 | O.O O.OO |
| 592 | XOO OOX |
| 593 | |
| 594 | @end group |
| 595 | @end example |
| 596 | |
| 597 | We describe now the topological algorithm used to find half eyes |
| 598 | and false eyes. In this section we ignore the possibility of ko. |
| 599 | |
| 600 | False eyes and half eyes can locally be characterized by the status of |
| 601 | the diagonal intersections from an eye space. For each diagonal |
| 602 | intersection, which is not within the eye space, there are three |
| 603 | distinct possibilities: |
| 604 | |
| 605 | @itemize @bullet |
| 606 | @item occupied by an enemy (@code{X}) stone, which cannot be captured. |
| 607 | @item either empty and @code{X} can safely play there, or occupied |
| 608 | by an @code{X} stone that can both be attacked and defended. |
| 609 | @item occupied by an @code{O} stone, an @code{X} stone that can be attacked |
| 610 | but not defended, or it's empty and @code{X} cannot safely play there. |
| 611 | @end itemize |
| 612 | |
| 613 | We give the first possibility a value of two, the second a value of |
| 614 | one, and the last a value of zero. Summing the values for the diagonal |
| 615 | intersections, we have the following criteria: |
| 616 | |
| 617 | @itemize @bullet |
| 618 | @item sum >= 4: false eye |
| 619 | @item sum == 3: half eye |
| 620 | @item sum <= 2: proper eye |
| 621 | @end itemize |
| 622 | |
| 623 | If the eye space is on the edge, the numbers above should be decreased |
| 624 | by 2. An alternative approach is to award diagonal points which are |
| 625 | outside the board a value of 1. To obtain an exact equivalence we must |
| 626 | however give value 0 to the points diagonally off the corners, i.e. |
| 627 | the points with both coordinates out of bounds. |
| 628 | |
| 629 | The algorithm to find all topologically false eyes and half eyes is: |
| 630 | |
| 631 | For all eye space points with at most one neighbor in the eye space, |
| 632 | evaluate the status of the diagonal intersections according to the |
| 633 | criteria above and classify the point from the sum of the values. |
| 634 | |
| 635 | @node Eye Topology with Ko |
| 636 | @section Eye Topology with Ko |
| 637 | |
| 638 | |
| 639 | This section extends the topological eye analysis to handle ko. We |
| 640 | distinguish between a ko in favor of @samp{O} and one in favor of @samp{X}: |
| 641 | |
| 642 | @example |
| 643 | @group |
| 644 | .?O? good for O |
| 645 | OO.O |
| 646 | O.O? |
| 647 | XOX. |
| 648 | .X.. |
| 649 | |
| 650 | @end group |
| 651 | @group |
| 652 | .?O? good for X |
| 653 | OO.O |
| 654 | OXO? |
| 655 | X.X. |
| 656 | .X.. |
| 657 | @end group |
| 658 | @end example |
| 659 | |
| 660 | Preliminarily we give the former the symbolic diagonal value @code{a} |
| 661 | and the latter the diagonal value @code{b}. We should clearly have |
| 662 | @code{0 < a < 1 < b < 2}. Letting @code{e} be the topological eye value |
| 663 | (still the sum of the four diagonal values), we want to have the |
| 664 | following properties: |
| 665 | |
| 666 | @example |
| 667 | e <= 2 - proper eye |
| 668 | 2 < e < 3 - worse than proper eye, better than half eye |
| 669 | e = 3 - half eye |
| 670 | 3 < e < 4 - worse than half eye, better than false eye |
| 671 | e >= 4 - false eye |
| 672 | @end example |
| 673 | |
| 674 | In order to determine the appropriate values of @code{a} and @code{b} we |
| 675 | analyze the typical cases of ko contingent topological eyes: |
| 676 | |
| 677 | @example |
| 678 | @group |
| 679 | .X.. (slightly) better than proper eye |
| 680 | (a) ..OO e < 2 |
| 681 | OO.O |
| 682 | O.OO e = 1 + a |
| 683 | XOX. |
| 684 | .X.. |
| 685 | |
| 686 | @end group |
| 687 | |
| 688 | @group |
| 689 | .X.. better than half eye, worse than proper eye |
| 690 | (a') ..OO 2 < e < 3 |
| 691 | OO.O |
| 692 | OXOO e = 1 + b |
| 693 | X.X. |
| 694 | .X.. |
| 695 | |
| 696 | @end group |
| 697 | |
| 698 | @group |
| 699 | .X.. better than half eye, worse than proper eye |
| 700 | (b) .XOO 2 < e < 3 |
| 701 | OO.O |
| 702 | O.OO e = 2 + a |
| 703 | XOX. |
| 704 | .X.. |
| 705 | |
| 706 | @end group |
| 707 | |
| 708 | @group |
| 709 | .X.. better than false eye, worse than half eye |
| 710 | (b') .XOO 3 < e < 4 |
| 711 | OO.O |
| 712 | OXOO e = 2 + b |
| 713 | X.X. |
| 714 | .X.. |
| 715 | |
| 716 | @end group |
| 717 | |
| 718 | @group |
| 719 | .X.. |
| 720 | XOX. (slightly) better than proper eye |
| 721 | (c) O.OO e < 2 |
| 722 | OO.O |
| 723 | O.OO e = 2a |
| 724 | XOX. |
| 725 | .X.. |
| 726 | |
| 727 | @end group |
| 728 | |
| 729 | @group |
| 730 | .X.. |
| 731 | XOX. proper eye, some aji |
| 732 | (c') O.OO e ~ 2 |
| 733 | OO.O |
| 734 | OXOO e = a + b |
| 735 | X.X. |
| 736 | .X.. |
| 737 | |
| 738 | @end group |
| 739 | |
| 740 | @group |
| 741 | .X.. |
| 742 | X.X. better than half eye, worse than proper eye |
| 743 | (c'') OXOO 2 < e < 3 |
| 744 | OO.O |
| 745 | OXOO e = 2b |
| 746 | X.X. |
| 747 | .X.. |
| 748 | |
| 749 | @end group |
| 750 | |
| 751 | @group |
| 752 | .X... |
| 753 | XOX.. better than half eye, worse than proper eye |
| 754 | (d) O.O.X 2 < e < 3 |
| 755 | OO.O. |
| 756 | O.OO. e = 1 + 2a |
| 757 | XOX.. |
| 758 | .X... |
| 759 | |
| 760 | @end group |
| 761 | |
| 762 | @group |
| 763 | .X... |
| 764 | XOX.. half eye, some aji |
| 765 | (d') O.O.X e ~ 3 |
| 766 | OO.O. |
| 767 | OXOO. e = 1 + a + b |
| 768 | X.X.. |
| 769 | .X... |
| 770 | |
| 771 | @end group |
| 772 | |
| 773 | @group |
| 774 | .X... |
| 775 | X.X.. better than false eye, worse than half eye |
| 776 | (d'') OXO.X 3 < e < 4 |
| 777 | OO.O. |
| 778 | OXOO. e = 1 + 2b |
| 779 | X.X.. |
| 780 | .X... |
| 781 | |
| 782 | @end group |
| 783 | |
| 784 | @group |
| 785 | .X... |
| 786 | XOX.. better than false eye, worse than half eye |
| 787 | (e) O.OXX 3 < e < 4 |
| 788 | OO.O. |
| 789 | O.OO. e = 2 + 2a |
| 790 | XOX.. |
| 791 | .X... |
| 792 | |
| 793 | @end group |
| 794 | |
| 795 | @group |
| 796 | .X... |
| 797 | XOX.. false eye, some aji |
| 798 | (e') O.OXX e ~ 4 |
| 799 | OO.O. |
| 800 | OXOO. e = 2 + a + b |
| 801 | X.X.. |
| 802 | .X... |
| 803 | |
| 804 | @end group |
| 805 | |
| 806 | @group |
| 807 | .X... |
| 808 | X.X.. (slightly) worse than false eye |
| 809 | (e'') OXOXX 4 < e |
| 810 | OO.O. |
| 811 | OXOO. e = 2 + 2b |
| 812 | X.X.. |
| 813 | .X... |
| 814 | |
| 815 | @end group |
| 816 | @end example |
| 817 | |
| 818 | It may seem obvious that we should use |
| 819 | @example |
| 820 | (i) a=1/2, b=3/2 |
| 821 | @end example |
| 822 | but this turns out to have some drawbacks. These can be solved by |
| 823 | using either of |
| 824 | @example |
| 825 | (ii) a=2/3, b=4/3 |
| 826 | (iii) a=3/4, b=5/4 |
| 827 | (iv) a=4/5, b=6/5 |
| 828 | |
| 829 | @end example |
| 830 | |
| 831 | Summarizing the analysis above we have the following table for the |
| 832 | four different choices of @code{a} and @code{b}. |
| 833 | |
| 834 | @example |
| 835 | case symbolic a=1/2 a=2/3 a=3/4 a=4/5 desired |
| 836 | value b=3/2 b=4/3 b=5/4 b=6/5 interval |
| 837 | (a) 1+a 1.5 1.67 1.75 1.8 e < 2 |
| 838 | (a') 1+b 2.5 2.33 2.25 2.2 2 < e < 3 |
| 839 | (b) 2+a 2.5 2.67 2.75 2.8 2 < e < 3 |
| 840 | (b') 2+b 3.5 3.33 3.25 3.2 3 < e < 4 |
| 841 | (c) 2a 1 1.33 1.5 1.6 e < 2 |
| 842 | (c') a+b 2 2 2 2 e ~ 2 |
| 843 | (c'') 2b 3 2.67 2.5 2.4 2 < e < 3 |
| 844 | (d) 1+2a 2 2.33 2.5 2.6 2 < e < 3 |
| 845 | (d') 1+a+b 3 3 3 3 e ~ 3 |
| 846 | (d'') 1+2b 4 3.67 3.5 3.4 3 < e < 4 |
| 847 | (e) 2+2a 3 3.33 3.5 3.6 3 < e < 4 |
| 848 | (e') 2+a+b 4 4 4 4 e ~ 4 |
| 849 | (e'') 2+2b 5 4.67 4.5 4.4 4 < e |
| 850 | |
| 851 | @end example |
| 852 | |
| 853 | We can notice that (i) fails for the cases (c''), (d), (d''), and (e). |
| 854 | The other three choices get all values in the correct intervals. The |
| 855 | main distinction between them is the relative ordering of (c'') and (d) |
| 856 | (or analogously (d'') and (e)). If we do a more detailed analysis of |
| 857 | these we can see that in both cases @samp{O} can secure the eye |
| 858 | unconditionally if he moves first while @samp{X} can falsify it with ko |
| 859 | if he moves first. The difference is that in (c''), @samp{X} has to make |
| 860 | the first ko threat, while in (d), O has to make the first ko threat. |
| 861 | Thus (c'') is better for O and ought to have a smaller topological eye |
| 862 | value than (d). This gives an indication that (iv) is the better choice. |
| 863 | |
| 864 | We can notice that any value of @code{a}, @code{b} satisfying |
| 865 | @code{a+b=2} and @code{3/4<a<1} would have the same qualities as choice |
| 866 | (iv) according to the analysis above. One interesting choice is |
| 867 | @code{a=7/8, b=9/8} since these allow exact computations with floating |
| 868 | point values having a binary mantissa. The latter property is shared by |
| 869 | @code{a=3/4} and @code{a=1/2}. |
| 870 | |
| 871 | When there are three kos around the same eyespace, things become |
| 872 | more complex. This case is, however, rare enough that we ignore it. |
| 873 | |
| 874 | |
| 875 | @node False Margins |
| 876 | @section False Margins |
| 877 | |
| 878 | The following situation is rare but special enough to warrant separate |
| 879 | attention: |
| 880 | |
| 881 | @example |
| 882 | @group |
| 883 | OOOOXX |
| 884 | OXaX.. |
| 885 | ------ |
| 886 | @end group |
| 887 | @end example |
| 888 | |
| 889 | Here @samp{a} may be characterized by the fact that it is adjacent |
| 890 | to O's eyespace, and it is also adjacent to an X group which cannot |
| 891 | be attacked, but that an X move at 'a' results in a string with only |
| 892 | one liberty. We call this a @dfn{false margin}. |
| 893 | |
| 894 | For the purpose of the eye code, O's eyespace should be parsed |
| 895 | as @code{(X)}, not @code{(X!)}. |
| 896 | |
| 897 | @node Eye Functions |
| 898 | @section Functions in @file{optics.c} |
| 899 | |
| 900 | The public function @code{make_domains()} calls the function |
| 901 | @code{make_primary_domains()} which is static in @file{optics.c}. It's purpose |
| 902 | is to compute the domains of influence of each color, used in determining eye |
| 903 | shapes. @strong{Note}: the term influence as used here is distinct from the |
| 904 | influence in influence.c. |
| 905 | |
| 906 | For this algorithm the strings which are not lively are invisible. Ignoring |
| 907 | these, the algorithm assigns friendly influence to |
| 908 | |
| 909 | @enumerate |
| 910 | @item every vertex which is occupied by a (lively) friendly stone, |
| 911 | @item every empty vertex adjoining a (lively) friendly stone, |
| 912 | @item every empty vertex for which two adjoining vertices (not |
| 913 | on the first line) in the (usually 8) surrounding ones have friendly |
| 914 | influence, with two CAVEATS explained below. |
| 915 | @end enumerate |
| 916 | |
| 917 | Thus in the following diagram, @samp{e} would be assigned friendly influence |
| 918 | if @samp{a} and @samp{b} have friendly influence, or @samp{a} and @samp{d}. It |
| 919 | is not sufficent for @samp{b} and @samp{d} to have friendly influence, because |
| 920 | they are not adjoining. |
| 921 | |
| 922 | @example |
| 923 | uabc |
| 924 | def |
| 925 | ghi |
| 926 | @end example |
| 927 | |
| 928 | The constraint that the two adjoining vertices not lie on the first |
| 929 | line prevents influence from leaking under a stone on the third line. |
| 930 | |
| 931 | The first CAVEAT alluded to above is that even if @samp{a} and @samp{b} have |
| 932 | friendly influence, this does not cause @samp{e} to have friendly influence if |
| 933 | there is a lively opponent stone at @samp{d}. This constraint prevents |
| 934 | influence from leaking past knight's move extensions. |
| 935 | |
| 936 | The second CAVEAT is that even if @samp{a} and @samp{b} have friendly influence |
| 937 | this does not cause @samp{e} to have influence if there are lively opponent |
| 938 | stones at @samp{u} and at @samp{c}. This prevents influence from leaking past |
| 939 | nikken tobis (two space jumps). |
| 940 | |
| 941 | The corner vertices are handled slightly different. |
| 942 | |
| 943 | @example |
| 944 | +--- |
| 945 | |ab |
| 946 | |cd |
| 947 | @end example |
| 948 | |
| 949 | We get friendly influence at @samp{a} if we have friendly influence |
| 950 | at @samp{b} or @samp{c} and no lively unfriendly stone at @samp{b}, @samp{c} |
| 951 | or @samp{d}. |
| 952 | |
| 953 | Here are the public functions in @file{optics.c}, except some simple |
| 954 | access functions used by autohelpers. The statically declared functions |
| 955 | are documented in the source code. |
| 956 | |
| 957 | @itemize @bullet |
| 958 | @item @code{void make_domains(struct eye_data b_eye[BOARDMAX], struct eye_data w_eye[BOARDMAX], int owl_call)} |
| 959 | @findex make_domains |
| 960 | @quotation |
| 961 | This function is called from @code{make_dragons()} and from |
| 962 | @code{owl_determine_life()}. It marks the black and white domains |
| 963 | (eyeshape regions) and collects some statistics about each one. |
| 964 | @end quotation |
| 965 | @item @code{void partition_eyespaces(struct eye_data eye[BOARDMAX], int color)} |
| 966 | @findex partition_eyespaces |
| 967 | @quotation |
| 968 | Find connected eyespace components and compute relevant statistics. |
| 969 | @end quotation |
| 970 | @item @code{void propagate_eye(int origin, struct eye_data eye[BOARDMAX])} |
| 971 | @findex propagate_eye |
| 972 | @quotation |
| 973 | propagate_eye(origin) copies the data at the (origin) to the |
| 974 | rest of the eye (invariant fields only). |
| 975 | @end quotation |
| 976 | @item @code{int find_eye_dragons(int origin, struct eye_data eye[BOARDMAX], int eye_color, int dragons[], int max_dragons)} |
| 977 | @findex find_eye_dragons |
| 978 | @quotation |
| 979 | Find the dragon or dragons surrounding an eye space. Up to |
| 980 | max_dragons dragons adjacent to the eye space are added to |
| 981 | the dragon array, and the number of dragons found is returned. |
| 982 | @end quotation |
| 983 | @item @code{void compute_eyes(int pos, struct eyevalue *value, int *attack_point, int *defense_point, struct eye_data eye[BOARDMAX], struct half_eye_data heye[BOARDMAX], int add_moves)} |
| 984 | @findex compute_eyes |
| 985 | @quotation |
| 986 | Given an eyespace with origin @code{pos}, this function computes the |
| 987 | minimum and maximum numbers of eyes the space can yield. If max and |
| 988 | min are different, then vital points of attack and defense are also |
| 989 | generated. If @code{add_moves == 1}, this function may add a move_reason for |
| 990 | @code{color} at a vital point which is found by the function. If |
| 991 | @code{add_moves == 0}, set @code{color = EMPTY.} |
| 992 | @end quotation |
| 993 | @item @code{void compute_eyes_pessimistic(int pos, struct eyevalue *value, int *pessimistic_min, int *attack_point, int *defense_point, struct eye_data eye[BOARDMAX], struct half_eye_data heye[BOARDMAX])} |
| 994 | @findex compute_eyes_pessimistic |
| 995 | @quotation |
| 996 | This function works like @code{compute_eyes()}, except that it also gives |
| 997 | a pessimistic view of the chances to make eyes. Since it is intended |
| 998 | to be used from the owl code, the option to add move reasons has |
| 999 | been removed. |
| 1000 | @end quotation |
| 1001 | @item @code{void add_false_eye(int pos, struct eye_data eye[BOARDMAX], struct half_eye_data heye[BOARDMAX])} |
| 1002 | @findex add_false_eye |
| 1003 | @quotation |
| 1004 | turns a proper eyespace into a margin. |
| 1005 | @end quotation |
| 1006 | @item @code{int is_eye_space(int pos)} |
| 1007 | @item @code{int is_proper_eye_space(int pos)} |
| 1008 | @findex is_proper_eye_space |
| 1009 | @findex is_eye_space |
| 1010 | @quotation |
| 1011 | These functions are used from constraints to identify eye spaces, |
| 1012 | primarily for late endgame moves. |
| 1013 | @end quotation |
| 1014 | @item @code{int max_eye_value(int pos)} |
| 1015 | @findex max_eye_value |
| 1016 | @quotation |
| 1017 | Return the maximum number of eyes that can be obtained from the |
| 1018 | eyespace at @code{(i, j)}. This is most useful in order to determine |
| 1019 | whether the eyespace can be assumed to produce any territory at |
| 1020 | all. |
| 1021 | @end quotation |
| 1022 | @item @code{int is_marginal_eye_space(int pos)} |
| 1023 | @item @code{int is_halfeye(struct half_eye_data heye[BOARDMAX], int pos)} |
| 1024 | @item @code{int is_false_eye(struct half_eye_data heye[BOARDMAX], int pos)} |
| 1025 | @findex is_marginal_eye_space |
| 1026 | @findex is_halfeye |
| 1027 | @findex is_false_eye |
| 1028 | @quotation |
| 1029 | These functions simply return information about an eyeshape that |
| 1030 | has already been analyzed. (They do no real work.) |
| 1031 | @end quotation |
| 1032 | @item @code{void find_half_and_false_eyes(int color, struct eye_data eye[BOARDMAX], struct half_eye_data heye[BOARDMAX], int find_mask[BOARDMAX])} |
| 1033 | @findex find_half_and_false_eyes |
| 1034 | @quotation |
| 1035 | Find topological half eyes and false eyes by analyzing the |
| 1036 | diagonal intersections, as described in the Texinfo |
| 1037 | documentation (Eyes/Eye Topology). |
| 1038 | @end quotation |
| 1039 | @item @code{float topological_eye(int pos, int color, struct eye_data my_eye[BOARDMAX],struct half_eye_data heye[BOARDMAX])} |
| 1040 | @findex topological_eye |
| 1041 | @quotation |
| 1042 | See Texinfo documentation (Eyes:Eye Topology). Returns: |
| 1043 | @itemize @bullet |
| 1044 | @item 2 or less if @code{pos} is a proper eye for @code{color}; |
| 1045 | @item between 2 and 3 if the eye can be made false only by ko |
| 1046 | @item 3 if @code{pos} is a half eye; |
| 1047 | @item between 3 and 4 if the eye can be made real only by ko |
| 1048 | @item 4 or more if @code{pos} is a false eye. |
| 1049 | @end itemize |
| 1050 | Attack and defense points for control of the diagonals are stored |
| 1051 | in the @code{heye[]} array. |
| 1052 | @code{my_eye} is the eye space information with respect to @code{color}. |
| 1053 | @end quotation |
| 1054 | @item @code{int obvious_false_eye(int pos, int color)} |
| 1055 | @findex obvious_false_eye |
| 1056 | @quotation |
| 1057 | Conservative relative of @code{topological_eye()}. Essentially the same |
| 1058 | algorithm is used, but only tactically safe opponent strings on |
| 1059 | diagonals are considered. This may underestimate the false/half eye |
| 1060 | status, but it should never be overestimated. |
| 1061 | @end quotation |
| 1062 | @item @code{void set_eyevalue(struct eyevalue *e, int a, int b, int c, int d)} |
| 1063 | @findex set_eyevalue |
| 1064 | @quotation set parameters into the @code{struct eyevalue} as follows |
| 1065 | (@pxref{Eye Local Game Values}): |
| 1066 | @example |
| 1067 | struct eyevalue @{ /* number of eyes if: */ |
| 1068 | unsigned char a; /* attacker plays first twice */ |
| 1069 | unsigned char b; /* attacker plays first */ |
| 1070 | unsigned char c; /* defender plays first */ |
| 1071 | unsigned char d; /* defender plays first twice */ |
| 1072 | @}; |
| 1073 | @end example |
| 1074 | @end quotation |
| 1075 | @item @code{int min_eye_threat(struct eyevalue *e)} |
| 1076 | @findex min_eye_threat |
| 1077 | @quotation |
| 1078 | Number of eyes if attacker plays first twice (the threat of the first |
| 1079 | move by attacker). |
| 1080 | @end quotation |
| 1081 | @item @code{int min_eyes(struct eyevalue *e)} |
| 1082 | @findex min_eyes |
| 1083 | @quotation |
| 1084 | Number of eyes if attacker plays first followed by alternating play. |
| 1085 | @end quotation |
| 1086 | @item @code{int max_eyes(struct eyevalue *e)} |
| 1087 | @findex max_eyes |
| 1088 | @quotation |
| 1089 | Number of eyes if defender plays first followed by alternating play. |
| 1090 | @end quotation |
| 1091 | @item @code{int max_eye_threat(struct eyevalue *e)} |
| 1092 | @findex max_eye_threat |
| 1093 | @quotation |
| 1094 | Number of eyes if defender plays first twice (the threat of the first |
| 1095 | move by defender). |
| 1096 | @end quotation |
| 1097 | @item @code{void add_eyevalues(struct eyevalue *e1, struct eyevalue *e2, struct eyevalue *sum)} |
| 1098 | @findex add_eyevalues |
| 1099 | @quotation |
| 1100 | Add the eyevalues @code{*e1} and @code{*e2}, leaving the result in *sum. It is |
| 1101 | safe to let @code{sum} be the same as @code{e1} or @code{e2}. |
| 1102 | @end quotation |
| 1103 | @item @code{char * eyevalue_to_string(struct eyevalue *e)} |
| 1104 | @findex eyevalue_to_string |
| 1105 | @quotation |
| 1106 | Produces a string containing the eyevalue. @strong{Note}: the result string is |
| 1107 | stored in a statically allocated buffer which will be overwritten the next |
| 1108 | time this function is called. |
| 1109 | @end quotation |
| 1110 | @item @code{void test_eyeshape(int eyesize, int *eye_vertices)} |
| 1111 | @findex test_eyeshape |
| 1112 | /* Test whether the optics code evaluates an eyeshape consistently. */ |
| 1113 | @item @code{int analyze_eyegraph(const char *coded_eyegraph, struct eyevalue *value, char *analyzed_eyegraph)} |
| 1114 | @findex analyze_eyegraph |
| 1115 | @quotation |
| 1116 | Analyze an eye graph to determine the eye value and vital moves. |
| 1117 | The eye graph is given by a string which is encoded with @samp{%} for |
| 1118 | newlines and @samp{O} for spaces. E.g., the eye graph |
| 1119 | |
| 1120 | @example |
| 1121 | ! |
| 1122 | .X |
| 1123 | !... |
| 1124 | @end example |
| 1125 | |
| 1126 | is encoded as @code{OO!%O.X%!...}. (The encoding is needed for the GTP |
| 1127 | interface to this function.) The result is an eye value and a (nonencoded) |
| 1128 | pattern showing the vital moves, using the same notation as eyes.db. In the |
| 1129 | example above we would get the eye value 0112 and the graph (showing ko threat |
| 1130 | moves) |
| 1131 | |
| 1132 | @example |
| 1133 | @ |
| 1134 | .X |
| 1135 | !.*. |
| 1136 | @end example |
| 1137 | |
| 1138 | If the eye graph cannot be realized, 0 is returned, 1 otherwise. |
| 1139 | @end quotation |
| 1140 | @end itemize |