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ARTIFICIAL

INTELLIGENCE

LECTURE # 07

Artificial Intelligence 2012 Lecture 07 Delivered By Zahid Iqbal 1

Review of Last Lecture


Recap

• Review of Last Lecture

• Searching

• State Space Search

• Depth First Search

• Breath First Search

• Iterative Deepening Search

• Bi-Directional Search


Today’s agenda • Heuristic search

• Hill climbing

• Beam search

• Best first search

• Best first search algorithm

• Greedy approach

• Optimal searches

• Branch and bound

• A* search

• Adversarial search

• Max Min procedure

• Alpha beta pouring


Informed Search Methods

Heuristic = “to find”, “to discover”

• Blind search: •Search the solution space in an uninformed manner •They simply march through the search space until a solution is found.

•Costly with respect time, space or both •Informed search.

•Search the solution space in informed manner -> HEURISTIC SEARCH •Don’t try to explore all possible search path, rather focus on the path which get you closer to the goal state using some kind of a “ GUIDE”. •A heuristic search makes use of available information in making the

search more efficient.

Artificial Intelligence 2012 Lecture 07 Delivered By Zahid

Iqbal 5

HEURISTIC SEARCH

Heuristics:

Rules for choosing the branches in a state space that are most likely to lead to an acceptable problem solution.

Used when:

• Computational costs are high E.g. Chess

But, Heuristics are fallible. They can lead a search algorithm to a suboptimal solution or fail to find a solution at all.


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Informed Search Methods

Heuristic = “to find”, “to discover”

•A heuristic is a rule or method that almost always improves the

decision process.

•Generally can’t be sure that you are near the goal, but heuristic is a good guess for the purpose. •E.g. mouse searching for cheese having fresh smell. •Mouse use “fresh smell” as a guide to reach the goal state.

•For example, if in a shop with several checkouts, it is usually best to go

to the one with the shortest queue. This holds true in most cases but

further information could influence this - (1) if you saw that there was a

checkout with only one person in that queue but that the person currently at

the checkout had three trolleys full of shopping and (2) that at the fast-

checkout all the people had only one item, you may choose to go to the fast-

checkout instead. Also, (3) you don't go to a checkout that doesn't have a

cashier - it may have the shortest queue but you'll never get anywhere.

•Heuristic search can be use for

•General purpose ( if you encounter a problem, remember your past weather you solved such type of problem or not, •Specific purpose ( add some form the problem domain) •E.g. car mechanic example, doctor patient example.


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Heuristic search • Straight line distance of each city form goal city..

S G

C

F D E

A B 5

4

6

3 2

3 3


Heuristic search: road map The no show the distance b/w two cities

S G

C

F D E

A B 2

4 4

3 3

2

3 1

3


Simple heuristic search procedure:

Hill Climbing • Named so because it resembles an eager, but blind

mountain climber, you move in such direction in which

your height increased.

• Your have a altimeter which tells you what will be your

height if you move L, R, U, S.

• Used a heuristic function which measure your height, or

measure remaining distance or find delta.

• And you select such operator which reduce this delta.

• Go uphill along the steepest possible path until you can

go no further.

• As it keeps no history, the algorithm cannot recover from

failures of its strategy


Hill Climbing

• Simplest way to implement heuristic search

• Expand current state in the search and evaluate its

children

• The best child is selected for further expansion; neither its

siblings nor parents are retained


Hill Climbing:

If the Node is better in terms of assigned heuristics, only then you proceed to that Node

Algorithm: 1. Start with current-state (cs) = initial state

2. Until cs = goal-state or there is no change in the cs do:

(a) Get the successor of cs and use the EVALUATION FUNCTION to assign a score to each successor

(b) If one of the successor has a better score than cs then set the new state to be the successor with the best score.


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IIA1: Hill-climbing (cont)

This simple policy has three well-known drawbacks: 1. Local Maxima: a local maximum as opposed to global maximum. 2. Plateaus: An area of the search space where evaluation function is flat, thus requiring random walk. 3. Ridge: Where there are steep slopes and the search direction is not towards the top but towards the side.

(a)

(b)

(c)

Figure 5.9 Local maxima, Plateaus and

ridge situation for Hill

Climbing


IIA1: Hill-climbing (cont)

• All these problems can be mapped to situations in our solution space searching. If we are at a state and the heuristics of all the available options take us to a lower value, we might be at local maxima. Similarly, if all the available heuristics take us to no improvement we might be at a plateau. Same is the case with ridge as we can encounter such states in our search tree.

• In each of the previous cases (local maxima, plateaus & ridge), the algorithm reaches a point at which no progress is being made.

• A solution is to do a random-restart hill-climbing - where random initial states are generated, running each until it halts or makes no discernible progress. The best result is then chosen.


Beam search

• Is like hill climbing, but

• Out of n possible choice at any level, beam search follows

only the best k of them.

• K is the parameter to which we set.

• K will be the best node among all the exploring nodes.


Best First Search

• A combination of depth first (DFS) and breadth first search (BFS).

• DFS is good because a solution can be found without computing all nodes and BFS is good because it does not get trapped in dead ends.

• The Best First Search (BestFS) allows us to switch between paths thus gaining the benefits of both approaches.


BestFS1: Greedy Search

• This is one of the simplest BestFS strategy.

• Heuristic function: h(n) - prediction of path cost left to the goal.

• Greedy Search: “To minimize the estimated cost to reach the

goal”.

• The node whose state is judged to be closest to the goal state

is always expanded first.

• Two route-finding methods (1) Straight line distance; (2) minimum

Manhattan Distance - movements constrained to horizontal and

vertical directions.


BestFS1: Greedy Search (cont)

Figure 5.3 Map of Romania with road distances in km, and straight-line distances to Bucharest.

hSLD(n) = straight-line distance between n and the goal location.



Figure 5.4 Stages in a greedy search for Bucharest, using the straight-line distance to Bucharest

as the

heuristics function hSLD. Nodes are labelled with their h-values.



• Noticed that the solution for A S F B is not optimum. It is 32

miles longer than the optimal path A S R P B.

• The strategy prefers to take the biggest bite possible out of the

remaining cost to reach the goal, without worrying whether this is

the best in the long run - hence the name ‘greedy search’.

• Greed is one of the 7 deadly sins, but it turns out that GS perform

quite well though not always optimal.

• GS is susceptible to false start. Consider the case of going from

Iasi to Fagaras. Neamt will be consider before Vaului even though

it is a dead end.



• GS resembles DFS in the way it prefers to follow a single path all

the way to the goal, but will back up when it hits a dead end.

• Thus suffering the same defects as DFS - not optimal and is

incomplete.

• The worst-case time complexity for GS is O(bm), where m is the

max depth of the search space.

• With good heuristic function, the space and time complexity

can be reduced substantially.


• Algorithm: At any time, expand the most promising node

according to the heuristic

If the heuristic is always 0, best-first search is the same as

breadth-first search

Best-first search is a greedy method.

Greedy methods maximize short-term advantage without

worrying about long-term consequences.

Best First Search Algorithm


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Best First Search Method Function best_First_Search; Begin open := [Start]; closed:=[]; while open != [] do begin

remove the leftmost state from open, call it X; If X = goal then return the path from Start to X

Else begin Generate children of X;

For each child of X do If the child is not on open or closed then begin assign the child a heuristic value; add the child to open end; End;

Put X on closed; Re-order states on open by heuristic merit (best leftmost)

End; Return FAIL End.


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F:6

B:5

D:4 E:2

G:0

Solution

C:3

A:10

Best First Search Demonstration


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1. Open [A:10] : closed []

2. Evaluate A:10; open [C:3,B:5]; closed [A:10]

3. Evaluate C:3; open [B:5,F:6]; closed [C:3,A:10]

4. Evaluate B:5; open [E:2,D:4,F:6]; closed [C:3,B:5,A:10].

5. Evaluate E:2; open [G:0,D:4,F:6];

closed [E:2,C:3,B:5,A:10]

6. Evaluate G:0; the solution / goal is reached



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1. Open [A5] : closed []

2. Evaluate A5; open [B4,C4,D6]; closed [A5]

3. Evaluate B4; open [C4,E5,F5,D6]; closed [B4,A5]

4. Evaluate C4; open [H3,G4,E5,F5,D6]; closed [C4,B4,A5].

5. Evaluate H3; open [O2,P3,G4,E5,F5,D6]; closed [H3,C4,B4,A5]

6. Evaluate O2; open [P3,G4,E5,F5,D6]; closed [O2,H3,C4,B4,A5]

7. Evaluate P3; THE SOLUTION IS FOUND!



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•If the evaluation function is good best first search may

significantly cut the amount of search requested

otherwise.

•If the evaluation function is heavy / very expensive the

benefits may be overweighed by the cost of assigning a

score

Restrictions on Evaluation Function


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Optimal searches

• We have covered informed and uninformed

searches.

• Bothe have some advantages and

disadvantages.

• One common lacks in both:

• Immediately stop when find a solution

• Don’t look for an optimal solution

• Optimal search:

• Find all the possible solution using either blind/heuristic

search.

• Choose the most optimal amongst them.


Branch and bound • Some how related to heuristic search.

• Cut of the unnecessary search.

S

G B F

D

E A

9 9


.

• .

Branch and bound The no show the distance b/w two cities

S G

C

F E

A B 2

4 4

3 3

2

3 1

3

D


A* • A* (pronounced "A star") is a best-first, graph search algorithm that finds the least-cost path from a given initial node to one goal node (out of one or more possible goals).

• Improve version of branch and bound.

• Uses a distance-plus-cost heuristic function (usually denoted f(x)) to determine the order in which the search visits nodes in the tree.

• Initially called algorithm A. Since using this algorithm yields optimal behavior for a given heuristic, it has been called A*.


Distance-Plus-Cost Heuristics function

• The distance-plus-cost heuristic is a sum of two functions:

• The path-cost function (usually denoted g(x)): cost from the starting

node to the current node.

• A heuristic estimate of the distance to the goal (usually denoted

h(x)).

• Total Cost: f(x) = g(x) + h(x)


.

• .

A* Search Algorithm The no show the distance b/w two cities

And the color no shows the under estimated

distance between that particular node and the goal

node.

S G

C

F E

A B 2

4 4

3 3

2

3 1

3

D

8

2 3

5

0

2 5

6 Artificial Intelligence 2012 Lecture 07 Delivered By Zahid Iqbal 36

A* Search Algorithm

• At every step, take node n from the front of the queue

• Enqueue all its successors n" with priorities:

• f(n") = g(n") + h(n")

= cost of getting to n" + estimated cost from n" to goal

• Terminate when a goal state is popped from the queue.


A* Search Method Function A*; Begin open := [Start]; closed:=[]; while open != [] do begin

remove the leftmost state from open, call it X; If X = goal then return the path from Start to X

Else begin Generate children of X;

For each child (m) of X do case the child (m) is not on open or closed then begin assign the child (m) a heuristic value; { g(m) = g(n) + C(n,m) f(m) = g(m) + h(m) } add the child to open end;


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the child (m) belongs to open or closed then begin g(m) = min{g(m), g(n) + C(n,m)} f(m) = g(m) + h(m) End; if f(m) has decreased and m belongs to closed then move m to open End

Put X on closed; Re-order states on open by heuristic merit (best leftmost)

End; Return FAIL End.


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Examples of Heuristics Artificial Intelligence 2012 Lecture 07 Delivered By Zahid

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Last Solution


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Adversarial Search

Adversary:= opponent

We have experience in search where we assume that we are the only

intelligent being and we have explicit control over the “world”.

Your are the only agent in complete search space so you are

responsible what you have done in achieving your goal

Adversarial Search

Lets consider what happens when we relax those assumptions. Problem solving agent is not alone any more

Multiagent, conflict You have no more control over the search space, what to done next. You will be given just one chance to apply a legal operator to move. Two searches will be in progress at the same time, each try to beat the other search.


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Two Player Games Max always moves first.

Min is the opponent.

We have

•An initial state.

•A set of operators.

•A terminal test (which tells us when the game is over).

•A utility function (evaluation function).

The utility function is like the heuristic function we have seen in the past, except

it evaluates a node in terms of how good it is for each player. Positive values

indicate states advantageous for Max, negative values indicate states

advantageous for Min.

Max Vs Min


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X O

X

X O

X

X

O

X X O

O O X

X

X O X

O

O O

X X X

O

O

X

... ... ...

-1 O 1

Utility

Terminal

States

Max

Min

Max ... ... ... ... ... ...


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Search with an opponent

• Trivial approximation: generating the tree for all moves

• Terminal moves are tagged with a utility value, for example: “+1” or “-1” depending on if the winner is MAX or MIN.

• The goal is to find a path to a winning state.

• Even if a depth-first search would minimize memory space, in complex games this kind of search cannot be carried out.

• Even a simple game like tic-tac-toe is too complex to draw the entire game tree.


3 12 8

A11 A12 A13

2 4 6

A21 A22 A23

14 5 2

A31 A32 A33

max

A simple abstract game.

Max makes a move, then Min replies.

A3 A2 A1

An action by one player is called a ply, two ply (a action and a counter

action) is called a move.

min


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• Generate the game tree down to the terminal nodes.

• Apply the utility function to the terminal nodes to gets its value.

• For a S set of sibling nodes, pass up to the parent…

• the lowest value in S if the siblings are (moves by max)

• the largest value in S if the siblings are moves by min)

• Recursively do the above, until the backed-up values reach the

initial state.

• The value of the initial state is the minimum score for Max.

The Minimax Algorithm


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The minimax algorithm


3 12 8

A11 A12 A13

3

2 4 6

A21 A22 A23

2

14 5 2

A31 A32 A33

2

3

A3 A2 A1

In this game Max’s best move is A1, because he is guaranteed a score of at least

3.

• Minimax-value(n): utility for MAX of being in state n, assuming both players are playing optimally =

– Utility(n), if n is a terminal state

– maxs ∈ Successors(n) Minimax-value(s), if n is a MAX state

– mins ∈ Successors(n) Minimax-value(s), if n is a MIN state


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• The algorithm first recurses down to the tree bottom-left

nodes

• and uses the Utility function on them to discover that their

values are 3, 12 and 8. 52



• Then it takes the minimum of these values, 3, and returns

it as the backed-up value of node B.

• Similar process for the other nodes.

53

B

A


Although the Minimax algorithm is optimal, there is a

problem…

The time complexity is O(bm) where b is the effective branching factor (legal moves at

each point) and m is the depth of the terminal states (maximum depth of the tree).

(Note space complexity is only linear in b and m, because we can do depth first

search).

One possible solution is to do depth limited Minimax search.

• Search the game tree as deep as you can in the given time.

• Evaluate the fringe (border) nodes with the utility function.

• Back up the values to the root.

• Choose best move, repeat.

We would like

to do Minimax

on this full

game tree...

… but we don’t

have time, so

we will explore

it to some

manageable

depth.

cutoff


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The minimax algorithm: problems

• For real games the time cost of minimax is totally

impractical, but this algorithm serves as the basis:

• for the mathematical analysis of games and

• for more practical algorithms

• Problem with minimax search:

• The number of game states it has to examine is exponential in

the number of moves.

• Unfortunately, the exponent can’t be eliminated, but it can

be cut in half.

55


Adversarial search game playing

we will represent the game as a tree.

We will focus on Board games.

G

C

F E

A

B

D

Original Board

situation

New Board

situation

New Board

situation


Alpha-beta pruning

• It is possible to compute the correct minimax decision

without looking at every node in the game tree.

• Alpha-beta pruning allows to eliminate large parts of the

tree from consideration, without influencing the final

decision.

57


Alpha-Beta Pruning

3 12 8

A11 A12 A13

3

2

A21 A22 A23

2

14 5 2

A31 A32 A33

2

3

A3 A2 A1

If you have an idea that is surely bad, don't take the

time to see how truly awful it is.


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Alpha-Beta Example

[-∞, +∞]

[-∞,+∞]

Range of possible values

Do DF-search until first leaf


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Alpha-Beta Example (continued)

[-∞,3]

[-∞,+∞]


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[-∞,3]

[-∞,+∞]


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[3,+∞]

[3,3]


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[-∞,2]

[3,+∞]

[3,3]

This node is worse

for MAX


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[-∞,2]

[3,14]

[3,3] [-∞,14]

,


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[−∞,2]

[3,5]

[3,3] [-∞,5]

,


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[2,2] [−∞,2]

[3,3]

[3,3]


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[2,2] [-∞,2]

[3,3]

[3,3]


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Alpha-beta pruning

• Alpha-beta pruning gets its name from two parameters.

• They describe bounds on the values that appear anywhere

along the path under consideration:

• α = the value of the best (i.e., highest value) choice found so far

along the path for MAX

• β = the value of the best (i.e., lowest value) choice found so far

along the path for MIN

68


Alpha-beta pruning

• Alpha-beta search updates the values of α and β as it

goes along.

• It prunes the remaining branches at a node (i.e.,

terminates the recursive call)

• as soon as the value of the current node is known to be worse

than the current α or β value for MAX or MIN, respectively.

69


The alpha-beta search algorithm


Alpha-Beta Example

Do DF-search until first leaf


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Alpha-Beta Example


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Final comments about alpha-beta pruning

• Pruning does not affect final results.

• Entire subtrees can be pruned, not just leaves.

• Good move ordering improves effectiveness of

pruning.

• With perfect ordering, time complexity is

O(bm/2).

• Effective branching factor of sqrt(b)

• Consequence: alpha-beta pruning can look twice as

deep as minimax in the same amount of time.

77


References

• Artificial Intelligence, A modern approach by Russell:

(Chapter 4 & 6)

• Artificial Intelligence: Structures and Strategies for

Complex Problem Solving by George F Luger:

(Chapter 4)


End of Lecture


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