Monotonicity Admissible Search: That finds the shortest path to the Goal Monotonicity: local admissibility is called MONOTONICITY This property ensures consistently minimal path to each state they encounter in the search.
It takes accumulative effect into consideration (for a distance problem) Along any path from the root, the cost never decreases (If this is true then the Heuristic is Monotonic in nature.) f(n) = g(n) + h(n) = 3 + 4 = 7 f (n) = g' (n ) + h' (n ) = 4 + 2 = 6 417 n 366 390 526 n
Non-Monotonic Monotonic If f(n`) < f(n) (Non - Monotonic) Then f(n`) = max (f(n), g(n`) + h(n`)) We take the cost of parent Node. Pathmax. This when only heuristic cost is taken Another representation = h(n) - h(n`) <= cost (n, n`)
Informedness. For two A heuristic h 1 and h 2 if h 1 (n) <= h 2 (n) for all states n in the search space, heuristic h 2 is said to be more informed than h 1. Both h 1 and h 2 can give OPTIMAL path but h 2 examines very few states in the process.
Monotonic Heuristics are Admissible States = S 1, S 2,, S g S 1 = Start S g = goal h(s 1 ) - h(s 2 ) <= cost (s 1, s 2 ) h(s 2 ) - h(s 3 ) <= cost (s 2, s 3 ) h (g-1) - h (g) <= cost (S g-1, S g ) ADD h (s1) - h (g) <= cost (S 1, S g )
h (n) = 0 Uninformed search Example Breadth - First search A * is more informed then Breadth - first search
Adversary Search (Games) AIM: The aim is to move in such a way as to stop the opponent from making a good / winning move. Game playing can use Tree - Search. The tree or game - tree alternates between two players.
Things to Remember: 1. Every move is vital 2. The opponent could win at the next move or subsequent moves. 3. Keep track of the safest moves 4. The opponent is well - informed 5. How the opponent is likely to response to your moves.
Two move win Player 1 = P 1 P1 moves B C A D Player 2 = P 2 P2 moves wins Safest move for P 1 is always A Safest move for P 2 is always A E F G H I J P1 P2 P1 P1 P2 P2 C D (if allowed 1 st move)
Minimax Procedure for Games Assumption: MIN: MAX: Opponent has same knowledge of state space and make a consistent effort to WIN. Label for the opponent trying to minimize other player s (MAX) score. Player trying to win (maximise advantage) Both MAX and MIN are equally informed
Rules 1. Label level s MAX and MIN 2. Assign values to leaf nodes: 0 if MIN wins 1 if MAX wins 3. Propagate values up the graph. If parent is MAX, assign it Max-value of its children If parent is MIN, assign it min-value of its children
Minimaxing to fixed to play depth (Complex games) Strategy: n - move look ahead - Suppose you start in the middle of the game. - One cannot assign WIN/LOOSE values at that stage - In this case some heuristics evaluation is applied - values are then projected back to supply indicate WINNING/LOOSING trend.
Summary Assign heuristic values to leaves of n-level graph Propagate value to root node This value indicates best state that can be reached in n moves from this start - state or node MAXIMIZE for MAX Parents MINIMIZE for MIN parents
Example: TIC - TAC - TOE X O X O X M(n) = Total of my possible O winning lines O(n) = Trial of Opponents winning lines E(n) = M(n) - O(n)
Horizon Effect Heuristics applied with limited ahead may lead to a bad situation and the person may leave the game. Same depth in search can be added to partially reduce this affect.
Alpha - Beta Procedures Minimax procedure pursues all branches in the space. Same of them could have been ignored or pruned. To improve efficiency pruning is applied to two person games
Simple Idea if A > 5 or B < 0 If A >5 and B <0 If the first condition A > 5 succeeds then B < 0 may not be evaluated. If the first condition failed then evaluating B < 0 is unnecessary.
a MAX MIN b = 0.4 c -0.2 e MAX d = 0.6-0.2 MIN f = -0.5 g = -0.2 - MAX can score maximum of -0.2 when moves a - c - e. MAX has a batter option to move to b
- MAX node neglects values <= a (atleast it can score) at MIN nodes below it. A MAX - MIN node neglects values >= b (almost it can score) at MAX nodes below it B =10 C MIN G=0 H C node can score ATMOST 0 nothing above 0 (beta) A node can score ATLEAST 10 nothing less than 10 (alpha)
Complexity Reduction Complexity Cost: Can be estimated roughly through measuring the size of open and closed list. (A) Beam Search: In beam search only the n most pronishing state are best for future consideration Bound applied to the open list. - The procedure may miss the solution by pruning it too early.
(B) More Informed ness - Apply more informed heuristics to reduce the complexity. - This may increase the computational cost in computing the heuristic