globals [row-clues col-clues globalError] patches-own [on old-on] to setup clear-all-plots ask patches [ ifelse random 1.0 < fraction-on [set on true] [set on false] set old-on on ] setupChickenClues checkClues displayState calculateGlobalError updateErrorPlot end to go locals [old-score new-score accept ] ;ask random-one-of patches [ ;this line if using modifyPatch0 ask random-one-of patches with [ on ] [ ;this line if using modifyPatch1 set old-score calculateStrip pxcor pycor rememberStates modifyPatch1 set new-score calculateStrip pxcor pycor set accept acceptModification new-score old-score if not accept [ revertStates ] ] displayState calculateGlobalError updateErrorPlot if globalError = 0 [stop] end to setupChickenClues ;row-clues read top to bottom ;col-clues read left to right set row-clues [ [2] [1] [4] [5] [3 2] [2 1] [3] ] set col-clues [ [1] [3] [1 3] [5 1] [4] [3] [2] ] end to setupTeaBreakClues ;row-clues read top to bottom ;col-clues read left to right set row-clues [ [10 3 3] [9 3 2] [8 9 1] [7 7 1] [6 2 2] [5 3 3] [4 2 2] [3 1 1 3] [2 5] [1 8 5] [8 3] [8 3] [3] [3] [3 3] [2 4] [1 5] [2] [3 3] [3 8] [3 8 1] [2 8 2] [1 1 3] [2 4] [3 5] [3 6] [3 7] [11 8] [11 9] ;[11 10] ] set col-clues [ [2 10] [2 9] [2 8] [2 7] [5 5 5] [6 4 5] [7 3 4] [2 3] [2 4 8 2] [2 5 7 1] [2 6 6] [3 2 3] [3 3 3] [3 3 3] [3 3 3 3] [3 3 6] [1 8] [2 2 2 2] [3 5 2] [4 4 2] [5 5 2] [6 5 2] [7 4 3] [8 3 2] [9 1 1] ] end to checkClues if (length row-clues != screen-size-y) [ show sentence "screen-size-y should be " length row-clues stop ] if (length col-clues != screen-size-x) [ show sentence "screen size-x should be " length col-clues stop ] end to rememberStates ask neighbors [ set old-on on ] set old-on on end to-report calculateStrip [ x y ] report ( evaluateRow dec y screen-edge-y ) + ( evaluateRow y ) + ( evaluateRow inc y screen-edge-y ) + ( evaluateCol dec x screen-edge-x ) + ( evaluateCol pxcor ) + ( evaluateCol inc x screen-edge-x ) end to revertStates ask neighbors [ set on old-on ] set on old-on end to-report acceptModification [ new old ] locals [ prob ] ifelse new < old [ report true ] [ set prob exp ( ( old - new ) / temperature ) ifelse random 1.0 < prob [ report true ] [ report false ] ] end to calculateGlobalError locals [ x y row-error col-error ] set y ( -1 * screen-edge-y ) set row-error 0 repeat screen-size-y [ set row-error ( row-error + evaluateRow y ) set y ( y + 1 ) ] set x ( -1 * screen-edge-x ) set col-error 0 repeat screen-size-x [ set col-error ( col-error + evaluateCol x ) set x ( x + 1 ) ] set globalError ( row-error + col-error ) end to-report evaluateRow [aRow] locals [ row-elements row-states a-row-clue row-error target ] set row-elements patches with [pycor = aRow] set row-states values-from row-elements [on] set a-row-clue convertStatesToClue row-states set target item ( aRow + screen-edge-y ) row-clues set row-error calculateError a-row-clue target report row-error end to-report evaluateCol [aCol] locals [ col-elements col-states a-col-clue col-error target ] set col-elements patches with [pxcor = aCol] set col-states values-from col-elements [on] set a-col-clue convertStatesToClue col-states set a-col-clue reverse a-col-clue set target item ( aCol + screen-edge-x ) col-clues set col-error calculateError a-col-clue target report col-error end to-report calculateError [ vector1 vector2 ] locals [ error i diff ] ; penalise a difference in vector length set diff abs (length vector1 - length vector2) ; get the two vectors to be the same length by padding the shorter with zeroes while [ (length vector1) < (length vector2) ] [ set vector1 lput 0 vector1 ] while [ (length vector2) < (length vector1) ] [ set vector2 lput 0 vector2 ] ; calculate the distance between the two vectors set error 0 set i 0 repeat length vector1 [ ; set error ( error + abs( ( item i vector1 ) - ( item i vector2 ) ) ) set error error + ( ( item i vector1 ) - ( item i vector2 ) ) * ( ( item i vector1 ) - ( item i vector2 ) ) set i ( i + 1 ) ] set error error + ( diff * wt-diff) report error end to-report convertStatesToClue [ states ] locals [ a-clue i i-max ] ifelse item 0 states [set a-clue [0 1] ] [set a-clue [0] ] set i 1 set i-max length states repeat ( i-max - 1 ) [ if (item i states) and (item ( i - 1 ) states) [ set a-clue replace-item ( -1 + length a-clue ) a-clue (1 + last a-clue) ] if (item i states) and not (item ( i - 1 ) states) [ set a-clue lput 1 a-clue ] set i ( i + 1 ) ] if a-clue != [0] [ set a-clue remove 0 a-clue ] report a-clue end ;================== ; UTILITY FUNCTIONS ;================== to-report inc [ x edge ] ifelse x < edge [ report x + 1 ] [ report -1 * edge ] end to-report dec [ x edge ] ifelse x > (-1 * edge) [ report x - 1 ] [ report edge ] end ;======================== ; PERTURBATION STRATEGIES ;======================== to modifyPatch0 set on not on end to modifyPatch1 locals [ wt-total p-kill p-breed ran ] set wt-total (wt-kill + wt-breed + wt-move) set ran random wt-total if ran < wt-kill [ killPatch ] if (ran >= wt-kill) and (ran < (wt-kill + wt-breed) ) [ breedPatch ] if ran >= (wt-kill + wt-breed) [ movePatch ] end to killPatch set on false end to breedPatch locals [ vacant ] set vacant neighbors with [not on] if count vacant > 0 [ set on-of random-one-of vacant on ] end to movePatch locals [ vacant ] set vacant neighbors with [not on] if count vacant > 0 [ set on-of random-one-of vacant on set on false ] end ;================== ; PLOTTING ROUTINES ;================== to displayState ask patches [ ifelse on [set pcolor black] [set pcolor white] ] end to updateErrorPlot set-current-plot "global error" plot globalError end @#$#@#$#@ GRAPHICS-WINDOW 321 10 426 115 3 3 15.0 1 10 0 0 CC-WINDOW 19 429 271 681 Command Center BUTTON 13 18 80 51 setup setup NIL 1 T OBSERVER SLIDER 14 114 186 147 fraction-on fraction-on 0 1.0 0.9 0.1 1 NIL BUTTON 13 65 81 98 go go T 1 T OBSERVER SLIDER 14 161 187 194 temperature temperature 0.1 5 0.6 0.1 1 NIL PLOT 304 236 710 538 global error NIL NIL 0.0 10.0 0.0 100.0 true false PENS "default" 1.0 0 -65536 true MONITOR 304 183 390 232 globalError globalError 3 1 SLIDER 15 231 187 264 wt-kill wt-kill 0 10 5 1 1 NIL SLIDER 15 264 187 297 wt-breed wt-breed 0 10 5 1 1 NIL SLIDER 15 295 187 328 wt-move wt-move 0 10 5 1 1 NIL SLIDER 16 364 188 397 wt-diff wt-diff 0 50 0 1 1 NIL @#$#@#$#@ WHAT IS IT? ----------- This program solves nonograms (sometimes) using simulated annealing. A nonogram is a logic problem, invented in Japan and popularised in the UK. It consists of a rectangular grid with one set of clues for each row and column of the grid. Solving the clues determines which cells in the puzzle are to be shaded to produce a picture. Clues are of the form ‘x1,x2, ... ,xn’, which translates as x1 shaded cells followed by at least one blank cell, followed by x2 shaded squares followed by at least one blank cell etc. For example, the following nonogram (it's a chicken, honest) has clues as shown |· # # # · · · 3 |# # · # · · · 2,1 |· # # # · # # 3,2 |· · # # # # # 5 |· · # # # # · 4 |· · · · # · · 1 |· · · # # · · 2 |1 3 1 5 4 3 2 | 3 1 If you want to solve anything other than the 'chicken' nonogram, you'll need to download the file and tinker with the input routines. HOW IT WORKS ------------ Although there is a deterministic (and fast) algorithm - see "Related Models" below - I decided to try a simulated annealing approach: 1) The 'energy' of the grid is calculated (an objective function comparing the clues that would be created by the current pattern against the desired target clues - when this reaches zero the nonagram is solved), E1. 2) An agent is selected at random and allowed to move, breed or die. 3) The energy of the grid is calculated again, E2. 4) If E2<=E1, then the change in (2) is accepted. If E2>E1, the change in (2) has a probability of acceptance of exp(-(E2-E1)/T), where T is a 'temperature' that is gradually reduced as the annealing proceeds. HOW TO USE IT ------------- 1) Initialise the nonagram by selecting a value for fraction-on (the fraction of patches that are originally occupied) and pressing 'setup'. 2) Set temperature (somewhere around the top end of the scale is good), and set the relative weights of agent death, breeding and moving (the bigger the weight, the more likely that action will be chosen by an agent). Leave wt-diff at zero for the time being (see 'Things to Try'). 3) Press go. You'll see the developing solution in the grid and a graph of the global error as the annealing progresses. When this graph hits zero, the solution is found and the run automatically stops. THINGS TO NOTICE ---------------- As the global error reduces and you start to see the emergence of some vague pattern, start reducing the temperature. Somewhere round the 0.4 - 0.6 mark, you'll get a solution. If the temperature is much hotter, the probability of moving away from the solution is too high. Once you have reduced the temperature, you might find that the global error gets stuck at some low but non-zero number (i.e. it's fallen into a local minimum). If this happens just briefly boil the chicken again and return to a low simmer. THINGS TO TRY ------------- 1) Different objective functions (I). Use a Euclidean distance in the objective function rather than a city-block distance. Any effect? 2) Different objective functions (II). On a large nonogram with large contiguous blocks, it might be a good idea to penalise differences in number of clues (see to-report calculateError). Does this have an effect? 3) Large nonagrams. I have included the clues for a 25 x 29 nonagram called 'Tea Break' (I got it off the Web somewhere). To use it: (i) In 'to setup', replace setupChickenClues by setupTeaBreakClues (ii) change the world size (by manual edit of the interface) to screen-edge-x = screen-edge-y = 3 The solution is below. I've not been able to converge to it. This rather suggests that simulated annealing is *not* the best solution to solving nonagrams (also see 'Related models' below). |###########·····######### |###########······######## |····###···········####### |····###············###### |····###··············#### |·····##···············### |····#·#·########·······## |····##··########········# |····###·########········· |····###·###·············· |····###··##·············· |········#·#####·········· |········##·####·········· |········###·###·········· |········###·············· |········###·············· |········########··###···· |········########··###···· |#·······########·#####··· |##···············#####··· |###·············#·#·###·· |####············##···##·· |#####··········###···###· |######·········##·····##· |#######········#######·#· |########······#########·# |#########·····###·····##· |##########····###·····### NETLOGO FEATURES ---------------- I used patches rather than turtles as my agents because of the one-to-one correspondence between patches and cells of the grid - I didn't need to worry about how to evaluate objective functions if multiple turtles decided to squat on the same patch. RELATED MODELS -------------- Simulated annealing might be fun but it's criminally slow. It's actually possible to work out an extremely fast algorithm that relies solely on the same sort of logical processes you'd use if you were solving one long-hand. For example, you'd probably start with that 5-row in the middle. The extreme positions for the block are either all the way left and all the way right, that is: # # # # # · · and · · # # # # # which means that the three central blocks must be occupied thus: · · # # # · · You can then use this as a constraint on the column clues, working out what cells can and cannot be occupied, and continue in this fashion till the nonogram is solved. You can find a Java program that uses this approach at http://www.morleysoft.freeserve.co.uk/computing/java/griddler.html. It might be fast, but it's not as much fun as boiling the chicken and cooling it down. @#$#@#$#@ default true 0 Polygon -7566196 true true 150 5 40 250 150 205 260 250 arrow true 0 Polygon -7566196 true true 150 0 0 150 105 150 105 293 195 293 195 150 300 150 box true 0 Polygon -7566196 true true 45 255 255 255 255 45 45 45 spacecraft true 0 Polygon -7566196 true true 150 0 180 135 255 255 225 240 150 180 75 240 45 255 120 135 thin-arrow true 0 Polygon -7566196 true true 150 0 0 150 120 150 120 293 180 293 180 150 300 150 turtle true 0 Polygon -7566196 true true 138 75 162 75 165 105 225 105 225 142 195 135 195 187 225 195 225 225 195 217 195 202 105 202 105 217 75 225 75 195 105 187 105 135 75 142 75 105 135 105 person false 0 Circle -7566196 true true 155 20 63 Rectangle -7566196 true true 158 79 217 164 Polygon -7566196 true true 158 81 110 129 131 143 158 109 165 110 Polygon -7566196 true true 216 83 267 123 248 143 215 107 Polygon -7566196 true true 167 163 145 234 183 234 183 163 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279 75 276 22 100 67 31 0 bird2 false 0 Polygon -7566196 true true 2 4 33 4 298 270 298 298 272 298 155 184 117 289 61 295 61 105 0 43 @#$#@#$#@ NetLogo 1.2 @#$#@#$#@ @#$#@#$#@ @#$#@#$#@