AI and 2048. Part 2: Minimax + alpha beta clipping

Monotone

Smoothness (smoothness, evenness)

Loose tiles

Small change

''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''( - )''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '       -- 'Position -  4  4     'Depth - ,    'Alpha, Beta -         'MaximisingPlayer -      ? Private Function MiniMaxAlpaBeta_Evaluation(Position As Variant, Depth As Long, _ Alpha As Double, Beta As Double, _ MaximisingPlayer As Boolean, _ Optional MainLevel As Boolean = False) As Double Dim MaxEval As Double '  Dim MinEval As Double '  Dim PositionNext As Variant '     Dim PositionTemp As Variant '     Dim Eval As Double '   Dim Way As Long '   -      Dim Row As Long '     Dim Col As Long '     Dim TileNew As Long '      '   (  ,  '     ) If GameOverPosition(Position) Then '    ? '     MiniMaxAlpaBeta_Evaluation = -1000000 + TileMax(Position) '         ElseIf Depth = 0 Then '     MiniMaxAlpaBeta_Evaluation = StaticEvaluation(Position) '  ,    '     () ElseIf MaximisingPlayer Then MaxEval = -1000000 '      For Way = 1 To 4 ' 4   - (, , , ) ChangeCount = 0 ' ,      ',       PositionNext = StepHuman(Position, Way) If ChangeCount > 0 Then '     '      , '    () Eval = MiniMaxAlpaBeta_Evaluation(PositionNext, Depth - 1, _ Alpha, Beta, False) If Eval > MaxEval Then MaxEval = Eval '  '     If Eval > Alpha Then Alpha = Eval '    ,   '   -    If Beta > Alpha Then Exit For End If Next '          MiniMaxAlpaBeta_Evaluation = MaxEval '  ,    '     () Else 'Not MaximisingPlayer MinEval = 1000000 '      For Row = 1 To 4 '     For Col = 1 To 4 '     If Position(Row, Col) = 0 Then '   For TileNew = 2 To 4 Step 2 '    2  4 ',       '    PositionNext = StepComp(Position, Row, Col, TileNew) '     , '    () Eval = MiniMaxAlpaBeta_Evaluation(PositionNext, Depth - 1, _ Alpha, Beta, True) If Eval < MinEval Then MinEval = Eval '  '     If Eval < Beta Then Beta = Eval '    ,   '   -    If Alpha < Beta Then Exit For Next End If Next Next '          MiniMaxAlpaBeta_Evaluation = MinEval End If End Function
      
      

        
        
        
      

    
        
        
        
      
      

        
        
        
      

    
     

 function AI(grid) { this.grid = grid; } //   () AI.prototype.eval = function() { var emptyCells = this.grid.availableCells().length; var smoothWeight = 0.1, //monoWeight = 0.0, //islandWeight = 0.0, mono2Weight = 1.0, emptyWeight = 2.7, maxWeight = 1.0; return this.grid.smoothness() * smoothWeight //+ this.grid.monotonicity() * monoWeight //- this.grid.islands() * islandWeight + this.grid.monotonicity2() * mono2Weight + Math.log(emptyCells) * emptyWeight + this.grid.maxValue() * maxWeight; }; // alpha-beta depth first search AI.prototype.search = function(depth, alpha, beta, positions, cutoffs) { var bestScore; var bestMove = -1; var result; // the maxing player if (this.grid.playerTurn) { bestScore = alpha; for (var direction in [0, 1, 2, 3]) { var newGrid = this.grid.clone(); if (newGrid.move(direction).moved) { positions++; if (newGrid.isWin()) { return { move: direction, score: 10000, positions: positions, cutoffs: cutoffs }; } var newAI = new AI(newGrid); if (depth == 0) { result = { move: direction, score: newAI.eval() }; } else { result = newAI.search(depth-1, bestScore, beta, positions, cutoffs); if (result.score > 9900) { // win result.score--; // to slightly penalize higher depth from win } positions = result.positions; cutoffs = result.cutoffs; } if (result.score > bestScore) { bestScore = result.score; bestMove = direction; } if (bestScore > beta) { cutoffs++ return { move: bestMove, score: beta, positions: positions, cutoffs: cutoffs }; } } } } else { // computer's turn, we'll do heavy pruning to keep the branching factor low bestScore = beta; // try a 2 and 4 in each cell and measure how annoying it is // with metrics from eval var candidates = []; var cells = this.grid.availableCells(); var scores = { 2: [], 4: [] }; for (var value in scores) { for (var i in cells) { scores[value].push(null); var cell = cells[i]; var tile = new Tile(cell, parseInt(value, 10)); this.grid.insertTile(tile); scores[value][i] = -this.grid.smoothness() + this.grid.islands(); this.grid.removeTile(cell); } } // now just pick out the most annoying moves var maxScore = Math.max(Math.max.apply(null, scores[2]), Math.max.apply(null, scores[4])); for (var value in scores) { // 2 and 4 for (var i=0; i<scores[value].length; i++) { if (scores[value][i] == maxScore) { candidates.push( { position: cells[i], value: parseInt(value, 10) } ); } } } // search on each candidate for (var i=0; i<candidates.length; i++) { var position = candidates[i].position; var value = candidates[i].value; var newGrid = this.grid.clone(); var tile = new Tile(position, value); newGrid.insertTile(tile); newGrid.playerTurn = true; positions++; newAI = new AI(newGrid); result = newAI.search(depth, alpha, bestScore, positions, cutoffs); positions = result.positions; cutoffs = result.cutoffs; if (result.score < bestScore) { bestScore = result.score; } if (bestScore < alpha) { cutoffs++; return { move: null, score: alpha, positions: positions, cutoffs: cutoffs }; } } } return { move: bestMove, score: bestScore, positions: positions, cutoffs: cutoffs }; } // performs a search and returns the best move AI.prototype.getBest = function() { return this.iterativeDeep(); } // performs iterative deepening over the alpha-beta search AI.prototype.iterativeDeep = function() { var start = (new Date()).getTime(); var depth = 0; var best; do { var newBest = this.search(depth, -10000, 10000, 0 ,0); if (newBest.move == -1) { break; } else { best = newBest; } depth++; } while ( (new Date()).getTime() - start < minSearchTime); return best } AI.prototype.translate = function(move) { return { 0: 'up', 1: 'right', 2: 'down', 3: 'left' }[move]; }
      
      

        
        
        
      

    
        
        
        
      
      

        
        
        
      

    
     

 ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''  '''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' '     'Position -  4  4     Private Function StaticEvaluation(Position As Variant) As Double Dim Smoothness As Double ' Dim Monotonicity As Double ' Dim EmptyCount As Double '  Dim MaxValue As Long '  '   Const SmoothWeight = 0.1 Const MonoWeight = 1 Const EmptyWeight = 2.7 Const MaxWeight = 1 Dim k As Long '   Dim i As Long '  Dim j As Long '  Dim x As Long '  Dim y As Long '  ' Dim Value As Double '       '         Dim TargetValue As Double Smoothness = 0 '    For i = 1 To 4 '     For j = 1 To 4 '     If Position(i, j) > 0 Then '   Value = Log(Position(i, j)) / Log(2) If i < 4 Then '       For x = i + 1 To 4 '    If Position(x, j) > 0 Then '    '    TargetValue = Log(Position(x, j)) / Log(2) ',   Smoothness = Abs(Value - TargetValue) '       Exit For End If Next End If If j < 4 Then '       For y = j + 1 To 4 '    If Position(i, y) > 0 Then '    '    TargetValue = Log(Position(i, y)) / Log(2) ',   Smoothness = Abs(Value - TargetValue) '        Exit For End If Next End If End If Next Next ' Dim arrTotals(1 To 4) As Double '     Dim Current As Long '   Dim Next_ As Long '      Dim CurrentValue As Double '      Dim NextValue As Double '        Monotonicity = 0 '    '      For k = 1 To 4 arrTotals(k) = 0 Next ' -  - For x = 1 To 4 '   Current = 1 '      '   (     )  Next_ = Current + 1 Do While Next_ <= 4 '       '      '(       ) Do While Next_ <= 4 '       If Position(x, Next_) = 0 Then '     Next_ = Next_ + 1 '   Else ' -    Exit Do ' ,  ,   End If Loop '         If Next_ > 4 Then Next_ = 4 '          If Position(x, Current) > 0 Then CurrentValue = Log(Position(x, Current)) / Log(2) Else CurrentValue = 0 End If ' MsgBox "Position[" & x & ", " & Next_ & "]=" & Position(x, Next_) If Position(x, Next_) > 0 Then NextValue = Log(Position(x, Next_)) / Log(2) Else NextValue = 0 End If If CurrentValue > NextValue Then '   ? arrTotals(Up) = arrTotals(Up) + NextValue - CurrentValue ElseIf NextValue > CurrentValue Then '   ? arrTotals(Down) = arrTotals(Down) + CurrentValue - NextValue End If Current = Next_ '       Next_ = Next_ + 1 '    Loop Next '      -  - Monotonicity = IIf(arrTotals(Up) >= arrTotals(Down), _ Monotonicity + arrTotals(Up), _ Monotonicity + arrTotals(Down)) ' -  - For y = 1 To 4 '   Current = 1 '      '   (     )  Next_ = Current + 1 Do While Next_ <= 4 '       '      '(       ) Do While Next_ <= 4 '       If Position(Next_, y) = 0 Then '     Next_ = Next_ + 1 '   Else ' -    Exit Do ' ,  ,   End If Loop '         If Next_ > 4 Then Next_ = 4 '          If Position(Current, y) > 0 Then CurrentValue = Log(Position(Current, y)) / Log(2) Else CurrentValue = 0 End If If Position(Next_, y) > 0 Then NextValue = Log(Position(Next_, y)) / Log(2) Else NextValue = 0 End If If CurrentValue > NextValue Then '   ? arrTotals(Left) = arrTotals(Left) + NextValue - CurrentValue ElseIf NextValue > CurrentValue Then '   ? arrTotals(Right) = arrTotals(Right) + CurrentValue - NextValue End If Current = Next_ '       Next_ = Next_ + 1 '    Loop Next '      -  - Monotonicity = IIf(arrTotals(Left) >= arrTotals(Right), _ Monotonicity + arrTotals(Left), _ Monotonicity + arrTotals(Right)) '     EmptyCount = 0 '      MaxValue = 0 '    For i = 1 To 4 '     For j = 1 To 4 '     If Position(i, j) = 0 Then '  ... '...     EmptyCount = EmptyCount + 1 '     ... ElseIf Position(i, j) > MaxValue Then MaxValue = Position(i, j) '...    End If Next Next '   StaticEvaluation = Smoothness * SmoothWeight + _ Monotonicity * MonoWeight + _ Log_Base_Arg(EmptyCount) * EmptyWeight + _ MaxValue * MaxWeight End Function
      
      

        
        
        
      

    
        
        
        
      
      

        
        
        
      

    
     

 function Grid(size) { this.size = size; this.startTiles = 2; this.cells = []; this.build(); this.playerTurn = true; } // pre-allocate these objects (for speed) Grid.prototype.indexes = []; for (var x=0; x<4; x++) { Grid.prototype.indexes.push([]); for (var y=0; y<4; y++) { Grid.prototype.indexes[x].push( {x:x, y:y} ); } } // Build a grid of the specified size Grid.prototype.build = function () { for (var x = 0; x < this.size; x++) { var row = this.cells[x] = []; for (var y = 0; y < this.size; y++) { row.push(null); } } }; // Find the first available random position Grid.prototype.randomAvailableCell = function () { var cells = this.availableCells(); if (cells.length) { return cells[Math.floor(Math.random() * cells.length)]; } }; Grid.prototype.availableCells = function () { var cells = []; var self = this; this.eachCell(function (x, y, tile) { if (!tile) { //cells.push(self.indexes[x][y]); cells.push( {x:x, y:y} ); } }); return cells; }; // Call callback for every cell Grid.prototype.eachCell = function (callback) { for (var x = 0; x < this.size; x++) { for (var y = 0; y < this.size; y++) { callback(x, y, this.cells[x][y]); } } }; // Check if there are any cells available Grid.prototype.cellsAvailable = function () { return !!this.availableCells().length; }; // Check if the specified cell is taken Grid.prototype.cellAvailable = function (cell) { return !this.cellOccupied(cell); }; Grid.prototype.cellOccupied = function (cell) { return !!this.cellContent(cell); }; Grid.prototype.cellContent = function (cell) { if (this.withinBounds(cell)) { return this.cells[cell.x][cell.y]; } else { return null; } }; // Inserts a tile at its position Grid.prototype.insertTile = function (tile) { this.cells[tile.x][tile.y] = tile; }; Grid.prototype.removeTile = function (tile) { this.cells[tile.x][tile.y] = null; }; Grid.prototype.withinBounds = function (position) { return position.x >= 0 && position.x < this.size && position.y >= 0 && position.y < this.size; }; Grid.prototype.clone = function() { newGrid = new Grid(this.size); newGrid.playerTurn = this.playerTurn; for (var x = 0; x < this.size; x++) { for (var y = 0; y < this.size; y++) { if (this.cells[x][y]) { newGrid.insertTile(this.cells[x][y].clone()); } } } return newGrid; }; // Set up the initial tiles to start the game with Grid.prototype.addStartTiles = function () { for (var i=0; i<this.startTiles; i++) { this.addRandomTile(); } }; // Adds a tile in a random position Grid.prototype.addRandomTile = function () { if (this.cellsAvailable()) { var value = Math.random() < 0.9 ? 2 : 4; //var value = Math.random() < 0.9 ? 256 : 512; var tile = new Tile(this.randomAvailableCell(), value); this.insertTile(tile); } }; // Save all tile positions and remove merger info Grid.prototype.prepareTiles = function () { this.eachCell(function (x, y, tile) { if (tile) { tile.mergedFrom = null; tile.savePosition(); } }); }; // Move a tile and its representation Grid.prototype.moveTile = function (tile, cell) { this.cells[tile.x][tile.y] = null; this.cells[cell.x][cell.y] = tile; tile.updatePosition(cell); }; Grid.prototype.vectors = { 0: { x: 0, y: -1 }, // up 1: { x: 1, y: 0 }, // right 2: { x: 0, y: 1 }, // down 3: { x: -1, y: 0 } // left } // Get the vector representing the chosen direction Grid.prototype.getVector = function (direction) { // Vectors representing tile movement return this.vectors[direction]; }; // Move tiles on the grid in the specified direction // returns true if move was successful Grid.prototype.move = function (direction) { // 0: up, 1: right, 2:down, 3: left var self = this; var cell, tile; var vector = this.getVector(direction); var traversals = this.buildTraversals(vector); var moved = false; var score = 0; var won = false; // Save the current tile positions and remove merger information this.prepareTiles(); // Traverse the grid in the right direction and move tiles traversals.x.forEach(function (x) { traversals.y.forEach(function (y) { cell = self.indexes[x][y]; tile = self.cellContent(cell); if (tile) { //if (debug) { //console.log('tile @', x, y); //} var positions = self.findFarthestPosition(cell, vector); var next = self.cellContent(positions.next); // Only one merger per row traversal? if (next && next.value === tile.value && !next.mergedFrom) { var merged = new Tile(positions.next, tile.value * 2); merged.mergedFrom = [tile, next]; self.insertTile(merged); self.removeTile(tile); // Converge the two tiles' positions tile.updatePosition(positions.next); // Update the score score += merged.value; // The mighty 2048 tile if (merged.value === 2048) { won = true; } } else { //if (debug) { //console.log(cell); //console.log(tile); //} self.moveTile(tile, positions.farthest); } if (!self.positionsEqual(cell, tile)) { self.playerTurn = false; //console.log('setting player turn to ', self.playerTurn); moved = true; // The tile moved from its original cell! } } }); }); //console.log('returning, playerturn is', self.playerTurn); //if (!moved) { //console.log('cell', cell); //console.log('tile', tile); //console.log('direction', direction); //console.log(this.toString()); //} return {moved: moved, score: score, won: won}; }; Grid.prototype.computerMove = function() { this.addRandomTile(); this.playerTurn = true; } // Build a list of positions to traverse in the right order Grid.prototype.buildTraversals = function (vector) { var traversals = { x: [], y: [] }; for (var pos = 0; pos < this.size; pos++) { traversals.x.push(pos); traversals.y.push(pos); } // Always traverse from the farthest cell in the chosen direction if (vector.x === 1) traversals.x = traversals.x.reverse(); if (vector.y === 1) traversals.y = traversals.y.reverse(); return traversals; }; Grid.prototype.findFarthestPosition = function (cell, vector) { var previous; // Progress towards the vector direction until an obstacle is found do { previous = cell; cell = { x: previous.x + vector.x, y: previous.y + vector.y }; } while (this.withinBounds(cell) && this.cellAvailable(cell)); return { farthest: previous, next: cell // Used to check if a merge is required }; }; Grid.prototype.movesAvailable = function () { return this.cellsAvailable() || this.tileMatchesAvailable(); }; // Check for available matches between tiles (more expensive check) // returns the number of matches Grid.prototype.tileMatchesAvailable = function () { var self = this; //var matches = 0; var tile; for (var x = 0; x < this.size; x++) { for (var y = 0; y < this.size; y++) { tile = this.cellContent({ x: x, y: y }); if (tile) { for (var direction = 0; direction < 4; direction++) { var vector = self.getVector(direction); var cell = { x: x + vector.x, y: y + vector.y }; var other = self.cellContent(cell); if (other && other.value === tile.value) { return true; //matches++; // These two tiles can be merged } } } } } //console.log(matches); return false; //matches; }; Grid.prototype.positionsEqual = function (first, second) { return first.x === second.x && first.y === second.y; }; Grid.prototype.toString = function() { string = ''; for (var i=0; i<4; i++) { for (var j=0; j<4; j++) { if (this.cells[j][i]) { string += this.cells[j][i].value + ' '; } else { string += '_ '; } } string += '\n'; } return string; } // counts the number of isolated groups. Grid.prototype.islands = function() { var self = this; var mark = function(x, y, value) { if (x >= 0 && x <= 3 && y >= 0 && y <= 3 && self.cells[x][y] && self.cells[x][y].value == value && !self.cells[x][y].marked ) { self.cells[x][y].marked = true; for (direction in [0,1,2,3]) { var vector = self.getVector(direction); mark(x + vector.x, y + vector.y, value); } } } var islands = 0; for (var x=0; x<4; x++) { for (var y=0; y<4; y++) { if (this.cells[x][y]) { this.cells[x][y].marked = false } } } for (var x=0; x<4; x++) { for (var y=0; y<4; y++) { if (this.cells[x][y] && !this.cells[x][y].marked) { islands++; mark(x, y , this.cells[x][y].value); } } } return islands; } // measures how smooth the grid is (as if the values of the pieces // were interpreted as elevations). Sums of the pairwise difference // between neighboring tiles (in log space, so it represents the // number of merges that need to happen before they can merge). // Note that the pieces can be distant Grid.prototype.smoothness = function() { var smoothness = 0; for (var x=0; x<4; x++) { for (var y=0; y<4; y++) { if ( this.cellOccupied( this.indexes[x][y] )) { var value = Math.log(this.cellContent( this.indexes[x][y] ).value) / Math.log(2); for (var direction=1; direction<=2; direction++) { var vector = this.getVector(direction); var targetCell = this.findFarthestPosition(this.indexes[x][y], vector).next; if (this.cellOccupied(targetCell)) { var target = this.cellContent(targetCell); var targetValue = Math.log(target.value) / Math.log(2); smoothness -= Math.abs(value - targetValue); } } } } } return smoothness; } Grid.prototype.monotonicity = function() { var self = this; var marked = []; var queued = []; var highestValue = 0; var highestCell = {x:0, y:0}; for (var x=0; x<4; x++) { marked.push([]); queued.push([]); for (var y=0; y<4; y++) { marked[x].push(false); queued[x].push(false); if (this.cells[x][y] && this.cells[x][y].value > highestValue) { highestValue = this.cells[x][y].value; highestCell.x = x; highestCell.y = y; } } } increases = 0; cellQueue = [highestCell]; queued[highestCell.x][highestCell.y] = true; markList = [highestCell]; markAfter = 1; // only mark after all queued moves are done, as if searching in parallel var markAndScore = function(cell) { markList.push(cell); var value; if (self.cellOccupied(cell)) { value = Math.log(self.cellContent(cell).value) / Math.log(2); } else { value = 0; } for (direction in [0,1,2,3]) { var vector = self.getVector(direction); var target = { x: cell.x + vector.x, y: cell.y+vector.y } if (self.withinBounds(target) && !marked[target.x][target.y]) { if ( self.cellOccupied(target) ) { targetValue = Math.log(self.cellContent(target).value ) / Math.log(2); if ( targetValue > value ) { //console.log(cell, value, target, targetValue); increases += targetValue - value; } } if (!queued[target.x][target.y]) { cellQueue.push(target); queued[target.x][target.y] = true; } } } if (markAfter == 0) { while (markList.length > 0) { var cel = markList.pop(); marked[cel.x][cel.y] = true; } markAfter = cellQueue.length; } } while (cellQueue.length > 0) { markAfter--; markAndScore(cellQueue.shift()) } return -increases; } // measures how monotonic the grid is. This means the values of the tiles are strictly increasing // or decreasing in both the left/right and up/down directions Grid.prototype.monotonicity2 = function() { // scores for all four directions var totals = [0, 0, 0, 0]; // up/down direction for (var x=0; x<4; x++) { var current = 0; var next = current+1; while ( next<4 ) { while ( next<4 && !this.cellOccupied( this.indexes[x][next] )) { next++; } if (next>=4) { next--; } var currentValue = this.cellOccupied({x:x, y:current}) ? Math.log(this.cellContent( this.indexes[x][current] ).value) / Math.log(2) : 0; var nextValue = this.cellOccupied({x:x, y:next}) ? Math.log(this.cellContent( this.indexes[x][next] ).value) / Math.log(2) : 0; if (currentValue > nextValue) { totals[0] += nextValue - currentValue; } else if (nextValue > currentValue) { totals[1] += currentValue - nextValue; } current = next; next++; } } // left/right direction for (var y=0; y<4; y++) { var current = 0; var next = current+1; while ( next<4 ) { while ( next<4 && !this.cellOccupied( this.indexes[next][y] )) { next++; } if (next>=4) { next--; } var currentValue = this.cellOccupied({x:current, y:y}) ? Math.log(this.cellContent( this.indexes[current][y] ).value) / Math.log(2) : 0; var nextValue = this.cellOccupied({x:next, y:y}) ? Math.log(this.cellContent( this.indexes[next][y] ).value) / Math.log(2) : 0; if (currentValue > nextValue) { totals[2] += nextValue - currentValue; } else if (nextValue > currentValue) { totals[3] += currentValue - nextValue; } current = next; next++; } } return Math.max(totals[0], totals[1]) + Math.max(totals[2], totals[3]); } Grid.prototype.maxValue = function() { var max = 0; for (var x=0; x<4; x++) { for (var y=0; y<4; y++) { if (this.cellOccupied(this.indexes[x][y])) { var value = this.cellContent(this.indexes[x][y]).value; if (value > max) { max = value; } } } } return Math.log(max) / Math.log(2); } // WIP. trying to favor top-heavy distributions (force consolidation of higher value tiles) /* Grid.prototype.valueSum = function() { var valueCount = []; for (var i=0; i<11; i++) { valueCount.push(0); } for (var x=0; x<4; x++) { for (var y=0; y<4; y++) { if (this.cellOccupied(this.indexes[x][y])) { valueCount[Math.log(this.cellContent(this.indexes[x][y]).value) / Math.log(2)]++; } } } var sum = 0; for (var i=1; i<11; i++) { sum += valueCount[i] * Math.pow(2, i) + i; } return sum; } */ // check for win Grid.prototype.isWin = function() { var self = this; for (var x=0; x<4; x++) { for (var y=0; y<4; y++) { if (self.cellOccupied(this.indexes[x][y])) { if (self.cellContent(this.indexes[x][y]).value == 2048) { return true; } } } } return false; } //Grid.prototype.zobristTable = {} //for //Grid.prototype.hash = function() { //}