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Using Search in Using Search in Problem SolvingProblem Solving

Part IIPart II

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Basic ConceptsBasic ConceptsBasic concepts:

• Initial state

• Goal/Target state

• Intermediate states

• Path from the initial to the target state• Operators/rules to get from one state to another• All states - search space

We search for a path / sequence of operators to go from the initial state to the goal state

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Search methodsSearch methods

Basic (uninformed):Basic (uninformed): breadth-firstbreadth-first depth-firstdepth-first

Heuristic (informed): Heuristic (informed): hill climbing, hill climbing, best-first, best-first, A*A*

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Heuristic SearchHeuristic Search

Heuristic search is used to reduce the search space.

Basic idea: explore only promising states/paths.

We need an evaluation function to estimate each state/path.

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Hill climbingHill climbing

Basic idea: always head towards a state which is better than the current one.

There is no exhaustive search, so no node list is maintained.

If a solution is found, it is found for a very short time and with minimum memory requirements. However it is not guaranteed that a solution will be found - the local maxima problem.

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Hill Climbing - AlgorithmHill Climbing - Algorithm Start with current-state = initial-state. Until current-state = goal-state OR there is no

change in current-state do:

Get the successors of the current state and use the evaluation function to assign a score to each successor.

If one of the successors has a better score than the current-state then set the new current-state to be the successor with the best score.

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Hill ClimbingHill Climbing

Node list is not maintainedNode list is not maintained No problems with loops as we move No problems with loops as we move

to a better nodeto a better node If a solution is found, it is found for a If a solution is found, it is found for a

very short time with minimal very short time with minimal memory requirementsmemory requirements

Finding a solution is not guaranteed – Finding a solution is not guaranteed – the the local maxima problemlocal maxima problem

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Best First SearchBest First Search The node with the best score is chosen to

be expanded. Works in breadth-first manner, keeps a

data structure (called agenda, based on priority queues) of all successors and their scores.

If a node that has been chosen does not lead to a solution, the next "best" node is chosen, so eventually the solution is found

Always finds a solution, not guaranteed to be the optimal one.

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Best First Search Best First Search AlgorithmAlgorithm

1. Start with agenda = [initial-state].

2. While agenda is not empty do

A. Pick the best node on agenda.

B. If it is the goal node then return with success. Otherwise find its

successors.

C. Assign the successor nodes a score using the evaluation function and add the scored nodes to the agenda

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Comparison with hill-Comparison with hill-climbingclimbing

Similarities: best-first always chooses the best node

Difference: best-first search keeps an agenda as in breadth-first search, and in case of a dead end it will backtrack, choosing the next-best node.

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Beam SearchBeam Search

Best-first methodBest-first method

Expands the best few paths at each Expands the best few paths at each levellevel

Has the memory advantages of the Has the memory advantages of the depth-first searchdepth-first search

Not exhaustive and may not find the Not exhaustive and may not find the best solution. Finding a solution is best solution. Finding a solution is not guaranteednot guaranteed

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The A* AlgorithmThe A* AlgorithmAn evaluation function that accounts for - the cost of the paths - the score of the nodes

F(Node) = g(Node) + h(Node)

g(Node) - the costs from the initial state to the current node

h(Node) - future costs, i.e. node score

Disadvantage of A* is the memory requirement - the algorithm keeps records for the entire tree.

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The A* AlgorithmThe A* Algorithm

A* always finds the best

solution, provided that

h(Node) does not

overestimate the future

costs.

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Example

107

8

8

4

4

6

10

92

4

A

B: 9D : 10

F : 6

H : 5

C: 8E: 7

K: 5

2

G: 0

J: 4 Start: A

Goal: G

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Search and Problem-Search and Problem-solvingsolving

Generate and TestGenerate and Test Means-end analysis in planningMeans-end analysis in planning

work out the path backwardswork out the path backwards use rules with preconditions and effectsuse rules with preconditions and effects the target goal is the effect of some operator/rulethe target goal is the effect of some operator/rule set the conditions as subgoals to be achievedset the conditions as subgoals to be achieved

Minimax algorithm in games,Minimax algorithm in games, Alpha-Beta pruningAlpha-Beta pruning

Constraint satisfactionConstraint satisfaction

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Planning TechniquesPlanning Techniques

Means-Ends analysisInitial state, Goal state, intermediate states.Operators to change the states.

Each operator:Preconditions

Effects

Go backwards : from the goal to the initial state.

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Basic AlgorithmBasic AlgorithmPut the goal state in a list of subgoals

While the list is not empty, doTake a subgoal.

Is there an operation that has as an effect this goal state?

If no - the problem is not solvable.If yes, put the operation in the plan.

Are the preconditions fulfilled?If yes - we have constructed the planIf no - Put the preconditions in the list

of subgoals

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More complex problemsMore complex problems

Goal state - a set of sub-states

Each sub-state is a goal.

To achieve the target state we have to achieve each goal in the Target state.

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Extensions of the basic Extensions of the basic methodmethod

Planning with goal protectionPlanning with goal protection Nonlinear planningNonlinear planning Hierarchical planningHierarchical planning Reactive planningReactive planning

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Game Playing SystemsGame Playing Systems

Special feature of games: there is an opponent that tries to destroy our plans.

The search tree (game tree) has to reflect the possible moves of both players

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Player1

Player2

Player1

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Minimax procedureMinimax procedure

Scoring the final nodes: Winning : 10Draw : 0Loosing : -10

Each player is trying to maximize their score and minimize the score of the other player.

The levels in the tree for player P1 are maximizing if it is P1's move, and minimizing if it is P2's move.

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Minimax algorithmMinimax algorithm • Build the search tree.

• Assign scores to the leaves• Assign scores to intermediate nodes:

If node is a leaf of the search tree - return score of that nodeElse:

If it is a maximizing node return maximum of scores of the successor nodes

Else if it is a minimizing node - return minimum of scores of the successor nodes

After that, choose nodes with highest score

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Alpha-Beta PruningAlpha-Beta Pruning

At maximizing level the score is at least the score of any successor.

At minimizing level the score is at most the score of any successor.

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A:10

B:10 C: 0

E:10 F:10 G: 0 H:10

I:10 J: 0 K:10 L:-10 M:-10 N:0 O:10 P:-10

Maximizing score

Minimizing score

P1's move

P2's move

P1's minimax tree

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Constraint satisfaction Constraint satisfaction searchsearch

State description: a set of variables.A set of constraints on the assigned values of variables.

Each variable may be assigned a value within its domain, and the value must not violate the constraints.

Initial state - the variables do not have valuesGoal state: Variables have values and the constraints are obeyed.

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Constraint satisfaction Constraint satisfaction searchsearch

Different types of constraints: Different types of constraints:

unaryunary - on one variable only - on one variable only

binarybinary - on two variables, etc. - on two variables, etc.

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ExampleExample

WEEK LOAN WEEK + LOAN + WEEK

LOAN WEEK ----------------- -----------------

DEBT MONTH

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Example - continuedExample - continued

The problem is to assign a digit to each letter, with different letters being different digits, so that when the letters are replaced by the digits, the addition is done correctly.

Initially no letters have values.

The goal state would be an assignment of values so that the addition is correct.

Operators: assign a digit to a letter, so that no two different digits have same values, and the addition rules are not violated.