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CPSC 322 Introduction to Artificial Intelligence. October 25, 2004. Things. Assignment 3 was posted Friday night Midterm exam 2 is one week from today Bring me your first midterms before the second midterm if you want to be retested on problem 3 Jessica Hodgins talk, Thursday,

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things
Things...

Assignment 3 was posted Friday night

Midterm exam 2 is one week from

today

Bring me your first midterms before

the second midterm if you want to be

retested on problem 3

Jessica Hodgins talk, Thursday,

October 28, 1-2pm, MacLeod 214

game search
Game search

(also known as adversarial search) has

these components:

move (or board) generator

static board evaluation function (this is

the heuristic part - it doesn’t generate

moves or look ahead - it’s static)

minimax algorithm to alternately propagate

minima and maxima upward from “bottom”

minimax algorithm
Minimax algorithm

Start with the following:

a) there are two players, MAX and MIN

b) it’s MAX’s turn to move

c) MAX has a static board evaluation function that returns

bigger values if a board is favorable to MAX

d) the evaluation function gets better as the game gets

closer to a goal state (else why bother to generate

the game space?)

e) MAX believes that MIN’s evaluation function is no

better than MAX’s (if that’s not true, then MAX

should at least avoid betting money on this game)

minimax algorithm1
Minimax algorithm
  • Generate the game tree to as many levels (plies) that time and space

constraints allow. The top level is called MAX (as in it’s now MAX’s

turn to move), the next level is called MIN, the next level is MAX, and

so on.

  • Apply the evaluation function to all the terminal (leaf) states/boards

to get “goodness” values

  • Use those terminal board values to determine the values to be

assigned to the immediate parents:

a) if the parent is at a MIN level, then the value is

the minimum of the values of its children

b) if the parent is at a MAX level, then the value is

the maximum of the values of its children

  • Keep propagating values upward as in step 3
  • When the values reach the top of the game tree, MAX chooses the

move indicated by the highest value

slide6
1997

Deep Blue Garry Kasparov

vital statistics: vital statistics:

• 200,000,000 moves per second • 3 moves per second

• 480 custom chess-playing chips • meat

puzzles games and ai
Puzzles, games, and AI

Puzzles and two-player board games have long

served as a laboratory for experiments with

heuristic search

Solving puzzles and playing games suggests

intelligence on the part of a human player, but

humans don’t do this stuff the same way

that search procedures do

Electronic Arts probably won’t be hiring you

solely on your mastery of minimax search

(but it couldn’t hurt)

puzzles games and ai1
Puzzles, games, and AI

Still, heuristic search is a simple but useful

tool, and the minimax game playing approach

is clearly very powerful

You probably don’t fully understand these

search techniques until you’ve had to

implement them...especially game search

Let’s look at just one more game...

slide9
Oska

• two players

• four pieces each

slide10
Oska

• two players

• four pieces each

slide11
Oska

• two players

• four pieces each

• piece movement:

slide12
Oska

• two players

• four pieces each

• piece movement:

• piece may be moved one

space forward on the diagonal

slide13
Oska

• two players

• four pieces each

• piece movement:

• piece may be moved one

space forward on the diagonal

slide14
Oska

• two players

• four pieces each

• piece movement:

• piece may be moved one

space forward on the diagonal

slide15
Oska

• two players

• four pieces each

• piece movement:

• piece may be moved one

space forward on the diagonal

slide16
Oska

• two players

• four pieces each

• piece movement:

• piece may be moved one

space forward on the diagonal

• piece may jump forward on

the diagonal over an

opponent’s piece to an

empty space, thus capturing

the opponent’s piece

slide17
Oska

• two players

• four pieces each

• piece movement:

• piece may be moved one

space forward on the diagonal

• piece may jump forward on

the diagonal over an

opponent’s piece to an

empty space, thus capturing

the opponent’s piece

slide18
Oska

• two players

• four pieces each

• piece movement:

• piece may be moved one

space forward on the diagonal

• piece may jump forward on

the diagonal over an

opponent’s piece to an

empty space, thus capturing

the opponent’s piece

slide19
Oska

• two players

• four pieces each

• piece movement:

• piece may be moved one

space forward on the diagonal

• piece may jump forward on

the diagonal over an

opponent’s piece to an

empty space, thus capturing

the opponent’s piece

slide20
Oska

• two players

• four pieces each

• piece movement:

• piece may be moved one

space forward on the diagonal

• piece may jump forward on

the diagonal over an

opponent’s piece to an

empty space, thus capturing

the opponent’s piece

slide21
Oska

• two players

• four pieces each

• piece movement:

• piece may be moved one

space forward on the diagonal

• piece may jump forward on

the diagonal over an

opponent’s piece to an

empty space, thus capturing

the opponent’s piece

• you don’t have to capture if you don’t want to

slide22
Oska

• two players

• four pieces each

• piece movement:

• piece may be moved one

space forward on the diagonal

• piece may jump forward on

the diagonal over an

opponent’s piece to an

empty space, thus capturing

the opponent’s piece

• you don’t have to capture if you don’t want to

slide23
Oska

• two players

• four pieces each

• piece movement:

• piece may be moved one

space forward on the diagonal

• piece may jump forward on

the diagonal over an

opponent’s piece to an

empty space, thus capturing

the opponent’s piece

• you don’t have to capture if you don’t want to

slide24
Oska

• two players

• four pieces each

• piece movement:

• piece may be moved one

space forward on the diagonal

• piece may jump forward on

the diagonal over an

opponent’s piece to an

empty space, thus capturing

the opponent’s piece

• you don’t have to capture if you don’t want to

• multiple jumps not permitted

slide25
Oska

• two players

• four pieces each

• piece movement:

• piece may be moved one

space forward on the diagonal

• piece may jump forward on

the diagonal over an

opponent’s piece to an

empty space, thus capturing

the opponent’s piece

• you don’t have to capture if you don’t want to

• multiple jumps not permitted

slide26
Oska

• two players

• four pieces each

• piece movement:

• piece may be moved one

space forward on the diagonal

• piece may jump forward on

the diagonal over an

opponent’s piece to an

empty space, thus capturing

the opponent’s piece

• you don’t have to capture if you don’t want to

• multiple jumps not permitted

slide27
Oska

• two players

• four pieces each

• piece movement:

• piece may be moved one

space forward on the diagonal

• piece may jump forward on

the diagonal over an

opponent’s piece to an

empty space, thus capturing

the opponent’s piece

• you don’t have to capture if you don’t want to

• multiple jumps not permitted

slide28
Oska

• two players

• four pieces each

• piece movement:

• piece may be moved one

space forward on the diagonal

• piece may jump forward on

the diagonal over an

opponent’s piece to an

empty space, thus capturing

the opponent’s piece

• you don’t have to capture if you don’t want to

• multiple jumps not permitted

slide29
Oska

• two players

• four pieces each

• piece movement:

• piece may be moved one

space forward on the diagonal

• piece may jump forward on

the diagonal over an

opponent’s piece to an

empty space, thus capturing

the opponent’s piece

• you don’t have to capture if you don’t want to

• multiple jumps not permitted

slide30
Oska

• two players

• four pieces each

• piece movement:

• piece may be moved one

space forward on the diagonal

• piece may jump forward on

the diagonal over an

opponent’s piece to an

empty space, thus capturing

the opponent’s piece

• you don’t have to capture if you don’t want to

• multiple jumps not permitted

• if a player can’t make a legal move on a turn,

the player loses turn and opponent moves again

slide31
Oska

• how to win

slide32
Oska

• how to win

• capture all your

opponent’s pieces

slide33
Oska

• how to win

• capture all your

opponent’s pieces

slide34
Oska

• how to win

• capture all your

opponent’s pieces

slide35
Oska

• how to win

• capture all your

opponent’s pieces

slide36
Oska

• how to win black wins

• capture all your

opponent’s pieces

slide37
Oska

• how to win

• capture all your

opponent’s pieces

slide38
Oska

• how to win

• capture all your

opponent’s pieces

• move all your remaining

pieces to your opponent’s

starting row

slide39
Oska

• how to win

• capture all your

opponent’s pieces

• move all your remaining

pieces to your opponent’s

starting row

slide40
Oska

• how to win

• capture all your

opponent’s pieces

• move all your remaining

pieces to your opponent’s

starting row

slide41
Oska

• how to win

• capture all your

opponent’s pieces

• move all your remaining

pieces to your opponent’s

starting row

slide42
Oska

• how to win

• capture all your

opponent’s pieces

• move all your remaining

pieces to your opponent’s

starting row

slide43
Oska

• how to win white wins

• capture all your

opponent’s pieces

• move all your remaining

pieces to your opponent’s

starting row

what does this have to do with you consider this predicate
What does this have to do with you?Consider this predicate:

oska([[w w w w][0 0 0][0 0][0 0 0][b b b b]],

w,

3,

[[w w w 0][0 0 w][0 0][0 0 0][b b b b]]).

This predicate is true if

• the first argument is a legal board in an Oska game

• the second argument indicates the colour of the pieces

being “moved” by the oska predicate

• the third argument indicates how deep (number of

moves/plies/levels) the game search goes

• the fourth argument is the best move that the oska

predicate can find using its move generator, board

evaluation function, and minimax algorithm

welcome to your term project
Welcome to your term project

oska([[w w w w][0 0 0][0 0][0 0 0][b b b b]],

w,

3,

[[w w w 0][0 0 w][0 0][0 0 0][b b b b]]).

• due 6:00am Monday, November 29, 2004

• you can work individually or in pairs

but I need to know who the pairs no later than this

Sunday, October 31

• more details will be posted this week, including

documentation requirements

one other thing
One other thing....

oska([[w w w w][0 0 0][0 0][0 0 0][b b b b]],

w,

3,

[[w w w 0][0 0 w][0 0][0 0 0][b b b b]]).

oska([[w w w w w][0 0 0 0][0 0 0][0 0]

[0 0 0][0 0 0 0][b b b b b]],

w,

3,

[[w w w w 0][0 0 0 w][0 0 0][0 0]

[0 0 0][0 0 0 0][b b b b b]]).

could look like this instead:

your program vs deep blue
Your program vs. Deep Blue

game search game search

move generator move generator

static board evaluator static board evaluator

minimax algorithm minimax algorithm

pruning (maybe) pruning

one processor 480 custom-fabricated

chess-playing chips

CILOG algorithms and representations

hardwired on chips

grit and determination selective deepening by

reallocating chips on the fly

giant libraries of openings,

endgames, and championship

matches

heuristic depth first search algorithm
Heuristic depth-first search algorithm

Given a set of start nodes, a set of goal nodes,

and a graph (i.e., the nodes and arcs):

apply heuristic h(n) to start nodes

make a “list” of the start nodes - let’s call it the “frontier”

sort the frontier by h(n) values

repeat

if no nodes on the frontier then terminate with failure

choose one node from the front of the frontier and remove it

if the chosen node matches the goal node

then terminate with success

else get next nodes (neighbors) and h(n) values

and sort those nodes by h(n) values

and put those sorted nodes on the front of the frontier

end repeat

heuristic depth first search tiles out of place
Heuristic depth-first search - tiles out of place

2 8 3 1 2 3

1 _ 4 goal: 8 _ 4

7 6 5 7 6 5

2 8 3 2 _ 3 2 8 3

_ 1 4 1 8 4 1 4 _

7 6 5 7 6 5 7 6 5

_ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3

2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5

7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _

8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3

2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5

7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6

8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3

2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5

7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6

heuristic depth first search tiles out of place1
Heuristic depth-first search - tiles out of place

2 8 3 1 2 3

1 _ 4 3 goal: 8 _ 4

7 6 5 7 6 5

2 8 3 2 _ 3 2 8 3

_ 1 4 1 8 4 1 4 _

7 6 5 7 6 5 7 6 5

_ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3

2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5

7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _

8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3

2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5

7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6

8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3

2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5

7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6

heuristic depth first search tiles out of place2
Heuristic depth-first search - tiles out of place

2 8 3 1 2 3

1 _ 4 3 goal: 8 _ 4

7 6 5 7 6 5

2 8 3 2 _ 3 2 8 3

_ 1 4 4 1 8 4 4 1 4 _ 5

7 6 5 7 6 5 7 6 5

_ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3

2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5

7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _

8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3

2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5

7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6

8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3

2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5

7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6

heuristic depth first search tiles out of place3
Heuristic depth-first search - tiles out of place

2 8 3 1 2 3

1 _ 4 3 goal: 8 _ 4

7 6 5 7 6 5

2 8 3 2 _ 3 2 8 3

_ 1 44 1 8 4 4 1 4 _ 5

7 6 5 7 6 5 7 6 5

_ 8 32 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3

2 1 4 4 7 1 4 5 1 8 4 1 8 4 1 4 3 1 4 5

7 6 5_ 6 5 7 6 5 7 6 5 7 6 5 7 6 _

8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3

2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5

7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6

8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3

2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5

7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6

heuristic depth first search tiles out of place4
Heuristic depth-first search - tiles out of place

2 8 3 1 2 3

1 _ 4 3 goal: 8 _ 4

7 6 5 7 6 5

2 8 3 2 _ 3 2 8 3

_ 1 44 1 8 4 4 1 4 _ 5

7 6 5 7 6 5 7 6 5

_ 8 32 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3

2 1 4 4 7 1 4 5 1 8 4 1 8 4 1 4 3 1 4 5

7 6 5_ 6 5 7 6 5 7 6 5 7 6 5 7 6 _

8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3

2 1 4 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5

7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6

8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3

2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5

7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6

heuristic depth first search tiles out of place5
Heuristic depth-first search - tiles out of place

2 8 3 1 2 3

1 _ 4 3 goal: 8 _ 4

7 6 5 7 6 5

2 8 3 2 _ 3 2 8 3

_ 1 44 1 8 4 4 1 4 _ 5

7 6 5 7 6 5 7 6 5

_ 8 32 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3

2 1 4 4 7 1 4 5 1 8 4 1 8 4 1 4 3 1 4 5

7 6 5_ 6 5 7 6 5 7 6 5 7 6 5 7 6 _

8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3

2 1 4 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5

7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6

8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3

2 1 4 5 2 _ 4 3 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5

7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6

heuristic depth first search tiles out of place6
Heuristic depth-first search - tiles out of place

2 8 3 1 2 3

1 _ 4 3 goal: 8 _ 4

7 6 5 7 6 5

2 8 3 2 _ 3 2 8 3

_ 1 44 1 8 4 4 1 4 _ 5

7 6 5 7 6 5 7 6 5

_ 8 32 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3

2 1 4 4 7 1 4 5 1 8 4 1 8 4 1 4 3 1 4 5

7 6 5_ 6 5 7 6 5 7 6 5 7 6 5 7 6 _

8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3

2 1 4 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5

7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6

8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3

2 1 4 52 _ 4 3 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5

7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6

heuristic depth first search tiles out of place7
Heuristic depth-first search - tiles out of place

2 8 3 1 2 3

1 _ 4 3 goal: 8 _ 4

7 6 5 7 6 5

2 8 3 2 _ 3 2 8 3

_ 1 44 1 8 4 4 1 4 _ 5

7 6 5 7 6 5 7 6 5

_ 8 32 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3

2 1 4 4 7 1 4 5 1 8 4 1 8 4 1 4 3 1 4 5

7 6 5_ 6 5 7 6 5 7 6 5 7 6 5 7 6 _

8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3

2 1 4 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5

7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6

8 3 _8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3

2 1 452 _ 4 3 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5

7 6 57 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6

heuristic depth first search tiles out of place8
Heuristic depth-first search - tiles out of place

2 8 3 1 2 3

1 _ 4 3 goal: 8 _ 4

7 6 5 7 6 5

2 8 3 2 _ 3 2 8 3

_ 1 44 1 8 4 4 1 4 _ 5

7 6 5 7 6 5 7 6 5

_ 8 32 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3

2 1 4 47 1 4 5 1 8 4 1 8 4 1 4 3 1 4 5

7 6 5_ 6 5 7 6 5 7 6 5 7 6 5 7 6 _

8 _ 32 8 3 1 2 3 2 3 4 2 _ 8 2 8 3

2 1 4 4 7 1 4 6 _ 8 4 1 8 _ 1 4 3 1 4 5

7 6 56 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6

8 3 _8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3

2 1 452 _ 4 3 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5

7 6 57 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6

heuristic depth first search tiles out of place9
Heuristic depth-first search - tiles out of place

2 8 3 1 2 3

1 _ 4 3 goal: 8 _ 4

7 6 5 7 6 5

2 8 3 2 _ 3 2 8 3

_ 1 44 1 8 4 4 1 4 _ 5

7 6 5 7 6 5 7 6 5

_ 8 32 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3

2 1 4 47 1 4 5 1 8 4 1 8 4 1 4 3 1 4 5

7 6 5_ 6 5 7 6 5 7 6 5 7 6 5 7 6 _

8 _ 32 8 3 1 2 3 2 3 4 2 _ 8 2 8 3

2 1 4 4 7 1 4 6 _ 8 4 1 8 _ 1 4 3 1 4 5

7 6 56 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6

8 3 _8 1 32 8 32 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3

2 1 452 _ 4 3 7 _ 4 5 7 1 4 7 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5

7 6 57 6 56 1 56 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6

heuristic depth first search tiles out of place10
Heuristic depth-first search - tiles out of place

2 8 3 1 2 3

1 _ 4 3 goal: 8 _ 4

7 6 5 7 6 5

2 8 3 2 _ 3 2 8 3

_ 1 44 1 8 4 4 1 4 _ 5

7 6 5 7 6 5 7 6 5

_ 8 32 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3

2 1 4 47 1 4 5 1 8 4 1 8 4 1 4 3 1 4 5

7 6 5_ 6 5 7 6 5 7 6 5 7 6 5 7 6 _

8 _ 32 8 3 1 2 3 2 3 4 2 _ 8 2 8 3

2 1 4 4 7 1 4 6 _ 8 4 1 8 _ 1 4 3 1 4 5

7 6 56 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6

8 3 _8 1 32 8 32 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3

2 1 452 _ 4 3 7 _ 4 5 7 1 4 7 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5

7 6 57 6 56 1 56 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6

heuristic depth first search tiles out of place11
Heuristic depth-first search - tiles out of place

2 8 3 1 2 3

1 _ 4 3 goal: 8 _ 4

7 6 5 7 6 5

2 8 3 2 _ 3 2 8 3

_ 1 44 1 8 4 4 1 4 _ 5

7 6 5 7 6 5 7 6 5

_ 8 32 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3

2 1 4 47 1 4 5 1 8 4 1 8 4 1 4 3 1 4 5

7 6 5_ 6 5 7 6 5 7 6 5 7 6 5 7 6 _

8 _ 32 8 3 1 2 3 2 3 4 2 _ 8 2 8 3

2 1 4 4 7 1 4 6 _ 8 4 1 8 _ 1 4 3 1 4 5

7 6 56 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6

8 3 _8 1 32 8 32 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3

2 1 452 _ 4 3 7 _ 4 5 7 1 4 7 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5

7 6 57 6 56 1 56 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6

heuristic depth first search tiles out of place12
Heuristic depth-first search - tiles out of place

2 8 3 1 2 3

1 _ 4 3 goal: 8 _ 4

7 6 5 7 6 5

2 8 32 _ 3 2 8 3

_ 1 441 8 44 1 4 _ 5

7 6 57 6 5 7 6 5

_ 8 32 8 3_ 2 32 3 _ 2 8 _ 2 8 3

2 1 4 47 1 4 5 1 8 4 4 1 8 4 5 1 4 3 1 4 5

7 6 5_ 6 57 6 57 6 5 7 6 5 7 6 _

8 _ 32 8 3 1 2 3 2 3 4 2 _ 8 2 8 3

2 1 4 4 7 1 4 6 _ 8 4 1 8 _ 1 4 3 1 4 5

7 6 56 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6

8 3 _8 1 32 8 32 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3

2 1 452 _ 4 3 7 _ 4 5 7 1 4 7 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5

7 6 57 6 56 1 56 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6

heuristic depth first search tiles out of place13
Heuristic depth-first search - tiles out of place

2 8 3 1 2 3

1 _ 4 3 goal: 8 _ 4

7 6 5 7 6 5

2 8 32 _ 3 2 8 3

_ 1 441 8 44 1 4 _ 5

7 6 57 6 5 7 6 5

_ 8 32 8 3_ 2 32 3 _ 2 8 _ 2 8 3

2 1 4 47 1 4 5 1 8 4 4 1 8 4 5 1 4 3 1 4 5

7 6 5_ 6 57 6 57 6 5 7 6 5 7 6 _

8 _ 32 8 31 2 3 2 3 4 2 _ 8 2 8 3

2 1 4 4 7 1 4 6 _ 8 4 2 1 8 _ 1 4 3 1 4 5

7 6 56 _ 57 6 5 7 6 5 7 6 5 7 _ 6

8 3 _8 1 32 8 32 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3

2 1 452 _ 4 3 7 _ 4 5 7 1 4 7 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5

7 6 57 6 56 1 56 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6

heuristic depth first search tiles out of place14
Heuristic depth-first search - tiles out of place

2 8 3 1 2 3

1 _ 4 3 goal: 8 _ 4

7 6 5 7 6 5

2 8 32 _ 3 2 8 3

_ 1 441 8 44 1 4 _ 5

7 6 57 6 5 7 6 5

_ 8 32 8 3_ 2 32 3 _ 2 8 _ 2 8 3

2 1 4 47 1 4 5 1 8 4 4 1 8 4 5 1 4 3 1 4 5

7 6 5_ 6 57 6 57 6 5 7 6 5 7 6 _

8 _ 32 8 31 2 3 2 3 4 2 _ 8 2 8 3

2 1 4 4 7 1 4 6 _ 8 4 2 1 8 _ 1 4 3 1 4 5

7 6 56 _ 57 6 5 7 6 5 7 6 5 7 _ 6

8 3 _8 1 32 8 32 8 31 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3

2 1 452 _ 4 3 7 _ 4 5 7 1 4 7 8 _ 4 0 7 8 4 3 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5

7 6 57 6 56 1 56 5 _7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6

heuristic depth first search tiles out of place15
Heuristic depth-first search - tiles out of place

2 8 3 1 2 3

1 _ 4 3 goal: 8 _ 4

7 6 5 7 6 5

2 8 32 _ 3 2 8 3

_ 1 441 8 44 1 4 _ 5

7 6 57 6 5 7 6 5

_ 8 32 8 3_ 2 32 3 _ 2 8 _ 2 8 3

2 1 4 47 1 4 5 1 8 4 4 1 8 4 5 1 4 3 1 4 5

7 6 5_ 6 57 6 57 6 5 7 6 5 7 6 _

8 _ 32 8 31 2 3 2 3 4 2 _ 8 2 8 3

2 1 4 4 7 1 4 6 _ 8 4 2 1 8 _ 1 4 3 1 4 5

7 6 56 _ 57 6 5 7 6 5 7 6 5 7 _ 6

8 3 _8 1 32 8 32 8 31 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3

2 1 452 _ 4 3 7 _ 4 5 7 1 4 7 8 _ 4 0 7 8 4 3 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5

7 6 57 6 56 1 56 5 _7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6

comparing search strategies
Comparing search strategies

strategy selection halts? space optimal

path

depth-first last node added no linear no

breadth-first first node added yes exponential yes*

lowest-cost minimal g(n) yes exponential yes

first

best-first globally minimal no exponential no

h(n)

heuristic locally minimal no linear no

depth-first h(n)

* assuming all arcs have the same cost

what s missing
What’s missing?

A heuristic search strategy that is guaranteed to

find the optimal (lowest-cost) path from start node

to goal node

What do the strategies that find optimal paths

have in common?

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