Search in AI

Towers of Hanoi Solve the problem by finding the appropriate

state transitions from a start state to a goal state. A state is a configuration of discs A transition takes us from one state to another via a

legal move

Towers of Hanoi Observations

Moving the smallest disc The smallest disc can always be moved to two

other poles If the smallest disc was moved on the previous

move, it does not make sense to move it again because One choice is to move back to where it just was. The other choice was available on the previous

move Conclusion: There is never a reason to

move the smallest disc twice in a row.

Towers of Hanoi Observations (continued)

Moving a disc other than the smallest The disc holding the smallest disc is not a

legal destination for a larger disc The largest top-level disc can not be

moved, as both possible destinations contain smaller discs

Conclusion: If we aren’t moving the smallest disc, there is only one move choice.

Towers of Hanoi Optimization We will always have a choice of two

destinations for the smallest disc, followed by a “forced” move of the smaller of the two other top-level discs.

We can therefore reduce the possible transitions at any point to two.

The Eight Puzzle

Eight puzzle - one start state and one solution state.

Eight puzzle can also have state optimizations Change the goal-state:From to this: To this:

1 2 3

4 5 6

7 8

1 2 3

4 5

7 8 6

From each state there are 8 possible “corner twists.”

Rotate clockwise Top-left quadrant Top-right quadrant Bottom-left quadrant Bottom-right quadrant

Rotate counter-clockwise Top-left quadrant Top-right quadrant Bottom-left quadrant Bottom-right quadrant

Using state-transition graphs, many of these types of problems become be solved using graph search algorithms.

You probably already know some graph-search algorithms Breadth-first search

Look at all the possible solutions one move deep, before looking at any possible solutions two moves deep and so on.

Depth-first search Try a first move, then another move from

that state. When you run out of moves, backtrack to the next available move.

Breadth-first Search Place the root node in a queue While a solution has not been found

Remove a state from the queue If that state is not a solution

Place all possible transitions from that state into the queue.

Positive qualities: With enough resources, the algorithm will find a solution.Negative qualities: Memory requirements and processing times grow exponentially for every move depth where the branching factor is greater than 1.

Depth-first Search Depth-first-search(root)

For every transition from the root If the state is not a solution

Depth-first-search(transition state)

Positive qualities: Memory requirements do not grow exponentiallyNegative qualities: There is no guarantee that depth-first will find an answer, even if one exists.

Depth-First Search with Iterative Deepening Set the depth limit to 0 While a solution is not found

Perform Depth-First Search to the depth limit

Increase the depth limit by 1

This algorithm has positive qualities of breadth-first, without the exponential memory costs.

Heuristics-based Search Use a set of rules to evaluate the

“quality” of a state. The quality of a state is usually an estimate of how close the state is to the solution.

Sample Heuristics For the 8-puzzle

Compute the sum of the minimum number of moves needed to move each tile from its current position its destination

Manhattan Distance – sum of the minimum number of moves each tile must make to get to its destination position

4 must move a minimum of 1 space1 must move a minimum of 1 space2 must move a minimum of 1 space8 must move a minimum of 1 space7 must move a minimum of 3 spaces6 must move a minimum of 3 spaces3 must move a minimum of 3 spaces5 must move a minimum of 2 spacesManhattan distance = 1 + 1 + 1 +1 + 3 + 3 + 3 + 2 = 15

Possible (naïve) Heuristic for Towers of Hanoi Number of discs in final position (i.e., on

the third pole on top of disc that it is on top of in the solution) – the minimum number of moves to reach a complete solution.

Hill Climbing Make any transition that gets you from your current

state to a better state. Do not store any state other than your current state. The value of the state is typically created by a heuristic

Examples: 8-Puzzle

Make any move that transitions to a state with a lower Manhattan distance.

Lost in Hoan Kiem District: Turn onto a different street if it is larger than the street

that you are on until you find something that you recognize.

Steepest Ascent Hill Climbing Look at all possible state transitions

before selecting the one with the highest value.

Problems with Hill-climbing The heuristic may not be a good measure,

or may not work in this instance It is easy to get to a “local maximum”

You may not be climbing the highest hill in the search space

There is no mechanism for backtracking If you make a wrong turn you may not be

able to recover There is no mechanism for detecting cycles

Problems with Hill Climbing (continued) Plateau Problem

All transition states might have the same value as the current state.

Best-first Search Maintain a list of nodes (the open list) that

we can return to if we take a wrong path Maintain a list of nodes that we no longer

need to consider (the closed list) Best-first search

While a solution has not been found Remove the node with the best heuristic value

from the open list and place it on the closed list For each child of that node, if the node is not

already on the closed list, place it on the open list.

Beam-Search Expand the best k nodes on the open

list Similar to breadth-first, but each level

only has k nodes explored.

Completeness A search algorithm is said to be

complete if it is guaranteed to find a solution if a solution exists.

Optimal A search algorithm is said to be optimal

if it is guaranteed to find the lowest-cost (best) solution.

Admissible Heuristic An admissible heuristic is one that

provides an estimate that is always less than or equal to the actual cost.

Admissible heuristics allow us to know that we have found the best answer when the heuristic estimates from all nodes on the open list exceed the value of the current node.

Monotonicity A search algorithm is said to be

monotonic if it is guaranteed to produce an optimal path to every intermediate node it generates.

Branch and Bound Create a priority queue sorted on lowest cost Insert root node into priority queue While a solution is not found or the queue

contains nodes with a cost less then the solution found Remove a node from the priority queue If it is the best solution found so far, record it Add all the children to the priority queue

Return the solution

Principal of Optimality Optimal paths are constructed from

optimal subpaths If we find a shorter alternate path to a

node we can discard the longer path

Estimating the cost to a goal Let g(x) be the cost from the start state to

node x Let h(x) be the cost from x to the goal state The total cost from start to finish, through

node x is: f(x) = g(x) + h(x) We do not know h(x). So we create an

estimate that we call h*(x) f*(x) = g*(x) + h*(x) is our estimate of f(x)

A* Algorithm Create a priority queue sorted on f*(x) Add the start node to the queue While a solution has not been found

Remove a node from the queue Compute f*(x) for all the nodes children

(that have not already been encountered) Insert the node’s children into the queue

Assignment Use the C++ code posted on the

website to solve the 8-puzzle

Two-Player Games Different search techniques are required

to find solutions to two-player, move-based games.

Minimax Trees Alternate between choosing the

maximum and minimum values of the child nodes.

The “Value of the Game” can be computed using a bottom-up computation of the value of each parent node

Depth of Game-Trees Ideally, all the way to the bottom

We can “solve” such games Use some fixed depth and then apply a

“static evaluation function” that estimates the value of the game at that node.

Use heuristics to determine a depth that is not necessarily uniform e.g., search to a quiescent state

Minimax with alpha-beta pruning If terminal node, return the value of the

terminal node If non-terminal

For each child Score = alpha-beta(child) If (Score > alpha) alpha

Two types of pruning Alpha cut-off

Happens during min-level analysis The max player will not allow this to

happen, so there is no point looking any further

Beta cut-off Happens during max-level analysis The min player will not allow this to happen,

so there is no point looking any further

Homework Assignment Use the letters in your full name to assign values for the

16 terminal nodes in a binary game tree (lower-case only). If you don’t have 16 letters in your name, just repeat your

name j o h n q u e s m i t h j o h n

The first player wants to maximize the alphabetic value of the selected letter

The second player wants to minimize the alphabetic value of the selected letter

Draw the minimax tree and annotate each node with the value of the subtree

Show which branches of the tree would be pruned through alpha-beta search Specify whether alpha or beta cutoff was used for each

pruning