Search

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Search
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Problem State Space

• What is Problem State Space?
• What is valid move within the space?
• Rule of the Game
• Characteristic of the State Space
• How to find the Solution Path in the State Space?
• Search Methods

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What is Problem State Space?

• 8-Puzzel

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2
3
3
2
1
8
4
4
8
5
6
5
6
7
7
3
2
3
8
2
2
3
8
1
1
4
4
4
1
8
6
7
7
5
6
7
5
5
6
a
c
b
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State Space of 8-Puzzle(1)

• List Representation
• (2 0 3 1 8 4 7 6 5) Initial State
• (1 2 3 8 0 4 7 6 5) Goal State
• (0 2 3 1 8 4 7 6 5) Intermediate States
• (2 8 3 1 0 4 7 6 5)
• ..

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State Space of 8-Puzzle(2)

• Position Function
• Pos(1, 1) 2
• Pos(1, 2) 8
• .
• Or Predicate?
• Pos(1, 1, 2) True
• Empty(3, 2) True

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Valid Moves of 8-Puzzle

• List
• (2 0 3 1 8 4 7 6 5) ? (0 2 3 1 8 4 7 6 5)
• (2 0 3 1 8 4 7 6 5) ? (2 8 3 1 0 4 7 6 5)
• (2 0 3 1 8 4 7 6 5) ? (2 3 0 1 8 4 7 6 5)
• Predicate
• (empty 1 2) ? (pos 1 1 x) ?
• (empty 1 1) ? (pos 1 2 x)

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Water Jug Problem

• 2 water jugs 4 liter, 3 liter
• Need to get exactly 2 liters of water
• State space?
• (0, 0) initial (2, 0) Goal
• Valid move?
• (4,0) ?(1, 3) ? (0, 1)?(4, 1)? (2,3)? (2,0 )
• (0,3) ? (3, 0) ? (3, 3) ? (4, 2) ? (0, 2)

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Examples

• Missionaries Cannibals Problem
• Blocks world
• Tower of Hamoi

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Missionaries Cannibals Problem

• 3?? ???? 3?? ????(cannibal)? ? ? ??? ???.
• ??? ??? ?? ??? ???, ??? 1??? 1? ??? ? ?? 2?? ???
  ???
• ???? 1?? ?? ?????......
• ?? ???? ?? ????? ???(??? ????) ???? ???? ?? ????.
  ???? ??? ??? 6? ?? ?? ??? ??.

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The Missionaries and Cannibals
Domain (defun goal-test (state) "The goal is to
have no missionaries or cannibals left on the
first side." ( 0 (m1 state) (c1
state))) (defun successors (state) "Return a
list of (action . state) pairs. An action is a
triple of the form (delta-m delta-c delta-b),
where a positive delta means to move from side
1 to side 2 negative is the opposite. For
example, the action (1 0 1) means move one
missionary and 1 boat from side 1 to side 2."
(let ((pairs nil)) (loop for action in '((1
0 1) (0 1 1) (2 0 1) (0 2 1) (1 1 1)
(-1 0 -1) (0 -1 -1) (-2 0 -1) (0 -2 -1) (-1 -1
-1)) do (let ((new-state (take-the-boat state
action))) (when (and new-state (not
(cannibals-can-eat? new-state))) (push (cons
action new-state) pairs)))) pairs))

• (defun main ()
• (print-actions (solve))
• 'done)
• (defun solve ()
• (labels ((iter (state path)
• (if (goal-test state)
• (reverse path)
• (let ((ss (remove-cyclic (successors state)
  path)))
• (and ss (some '(lambda (s)
• (iter (cdr s) (cons s path)))
• (suffle ss)))))))
• (iter (make-cannibal-state) (list (cons
  'Begin (make-cannibal-state))))))
• (defun remove-cyclic (succs path)
• (remove-if '(lambda (s) (member s path test
  '(lambda (a b)
• (equal (cdr a) (cdr b))))) succs))
• (defun print-actions (path)

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(defstruct (cannibal-state (conc-name nil)
(type list)) "The state says how many
missionaries, cannibals, and boats on each
side. The components m1,c1,b1 stand for the
number of missionaries, cannibals and boats,
respectively, on the first side of the river.
The components m2,c2,b2 are for the other side of
the river." We need to represent both sides
(rather than just one as on p 68) because
we have generalized from 33 people to MC.
Incidently, we also generalized from 1 boat
to B boats. (m1 3) (c1 3) (b1 1) (m2 0) (c2 0)
(b2 0)) (defun take-the-boat (state action)
"Move a certain number of missionaries,
cannibals, and boats (if possible)."
(destructuring-bind (delta-m delta-c delta-b)
action (if (or (and ( delta-b 1) (gt (b1
state) 0)) (and ( delta-b -1) (gt (b2 state)
0))) (let ((new (copy-cannibal-state state)))
(decf (m1 new) delta-m) (incf (m2 new) delta-m)
(decf (c1 new) delta-c) (incf (c2 new) delta-c)
(decf (b1 new) delta-b) (incf (b2 new)
delta-b) (if (and (gt (m1 new) 0) (gt (m2 new)
0) (gt (c1 new) 0) (gt (c2 new) 0))
new nil)) nil))) (defun
cannibals-can-eat? (state) "The cannibals feast
if they outnumber the missionaries on either
side." (or (gt (c1 state) (m1 state) 0) (gt
(c2 state) (m2 state) 0)))
CL-USER 1 gt (main) Action M1 C1 B1 M2 C2
B2 BEGIN 3 3 1 0 0 0 (0 2 1)
3 1 0 0 2 1 (0 -1 -1) 3 2 1
0 1 0 (0 2 1) 3 0 0 0 3 1 (0
-1 -1) 3 1 1 0 2 0 (2 0 1) 1
1 0 2 2 1 (-1 -1 -1) 2 2 1 1 1
0 (2 0 1) 0 2 0 3 1 1 (0 -1 -1)
0 3 1 3 0 0 (0 2 1) 0 1 0 3
2 1 (-1 0 -1) 1 1 1 2 2 0 (1 1
1) 0 0 0 3 3 1 DONE CL-USER 2 gt
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Tower of Hamoi

• ??? ?? A? ???? ?? 3?? ?? C? ??? ???. ? ??? ???
  ??? ??.
• ? ?? ??? ??? ??? ? ??
• ? ?? ?? ??? ??? ? ??
• ??? ?? ???? ? ??? ?? ? ??.
• ??? ??? ????? ??? ? ??? ?? ???? ??? ??.

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3?? ??? ??? ??
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n?? ??? ??

• ??? ???
• n-1?? ??? A?? B? ???
• n?? ??? A?? C? ?? ??,
• n-1?? ??? B?? C? ???.
• ??? ??
• ??? ?? n
• ???? ???? from ? to
• ???? 1? tmp
• ???, ??? prototype

void hanoi_tower(int n, char from, char tmp, char
to)
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void hanoi_tower(int n, char from, char tmp, char
to) ?

• if (n1)
• 1. 1? ??? from?to ?, A?C? ???.
• else ngt1 ??
• 2. from? ? ?? ??? ??? (n-1)?? ??? from?tmp, ?
  A?B? ???.
• 3. 2??? ??? ?? from? 1???? from?to, A?C? ???.
• 4. ????? tmp? ???? to? ???.

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???? ??
include ltstdio.hgt void hanoi_tower(int n, char
from, char tmp, char to) if( n1 )
printf("?? 1? c ?? c?? ???.\n",from,to)
else hanoi_tower(n-1, from, to,
tmp) printf("?? d? c?? c?? ???.\n",n, from,
to) hanoi_tower(n-1, tmp, from, to) int
main(void) hanoi_tower(4, 'A', 'B', 'C')
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Characteristics of State Space

• Decomposable?
• goal -- subgoals
• Blind vs Informed (heuristic) Search
• Revocable vs Irrevocable
• backtracking
• Predictable vs Stochastic
• Optimal vs Plausible Solution

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Search Algorithm

• Guarantee to find a solution ?
• Always terminate ?
• Guarantee the solution to be optimal
• -- admissible
• Complexity - time and space
• How can the complexity be reduced ?
• Characteristics of the space?

State Space Search
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Strategies of SS Search

• Data-Driven vs. Goal Driven
• Forward chaining / Backward chaining
• Symptom ? diagnosis Forward
• eg. Fever, cough ? flu
• Goal/Sub-goal Backward
• eg. Good Husband?
• Rich? Or Handsome Edu

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Forward vs Backward

• When is forward (backward) better?
• Bi-Directional
• Factors deciding Forward/Backward
• - Size of the known states initial, goal
• small to big
• - Branching Factor converging
• - computational simplicity
• eg. Water jug problem

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Weak Methods

• General Purpose Control Strategies
• Simplicity is the KEY of Weak Methods
• Usually
• - Not Optimal
• - No Backtracking
• - No Guarantee to the Solution

• Generate Test
• Hill Climbing
• Best First Search
• Means-Ends Analysis
• Etc.

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Heuristic Function

• Heuristics Knowledge obtained from human
  experience
• Heuristic function h SS ? number
• a preference measure for each position
  of the state space
• What is the heuristic function for 8-puzzle?
• Necessary for Informed Search (cf. Blind)

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Generate Test

• British Museum Algorithm
• Guess a Solution Test if it is right
• DENDRAL molecular structure analysis
• plan-generate-test (constraint)
• AM (Automated Mathematician)
• Number Theory generate a conjecture
• Generate Test is especially useful when
  combined with other strategy

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Hill Climbing

• Greed Method
• Iterative Improvement Algorithm
• Select the Best Looking Candidate
• Use heuristic function
• Stop when no more better looking candidates ?
  Assume it is the Goal!
• No Backtracking Simplicity!!

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Hill Climbing Algorithm

• Hill-climbing Attempt to maximize Eval(X) by
  moving to
• the highest configuration in our moveset.
• 1. Let X initial config
• 2. Let E Eval(X)
• 3. Let N moveset_size(X)
• 4. For ( i 0 iltN i i1)
• Let Ei Eval(move(X,i))
• 5. If all Eis are E, terminate, return X
• 6. Else let i argmaxi Ei
• 7. X move(X,i)
• 8. E Ei
• 9. Goto 3

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Hill Climbing Issues

• Low Memory Requirement
• -- No backtracking
• Moveset (next move) design is critical
• Evaluation Function design is critical
• If the number of move is too big
• -- Inefficient
• Too small?
• -- Easily stuck (plateau)

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Problems of Hill Climbing

• Local Maxima (?????)
• Plateau(??)
• Ridge(????)

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Avoiding Problems in H.C.

• Many Starting Points
• Random Jump Mutation
• (Genetic Algorithm)
• Simulated Annealing
• Stochastic Method

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Simulated Annealing Algorithm

• 1. Let X initial config
• 2. Let E Eval(X)
• 3. Let i random move from the moveset
• 4. Let Ei Eval(move(X,i))
• 5. If E lt Ei then
• X move(X,i)
• E Ei
• Else with some probability, accept the move
  even though
• things get worse
• X move(X,i)
• E Ei
• 6. Goto 3 unless bored.

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Simulated Annealing

• If Ei gt E then definitely accept the change.
• If Ei lt E then accept the change with
  probability
• exp (-(E - Ei)/Ti) (called
  the Boltzman distribution)
• where Ti is a temperature
  parameter that
• gradually decreases.
• High temp accept all moves (Random Walk)
• Low temp Stochastic Hill-Climbing
• When enough iterations have passed without
  improvement,
• terminate.

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Means Ends Analysis(1)

• Bi-Directional Method
• GPS(General Problem Solver) Newell
• Operator, Pre-condition, Result Table
• Choose the operator that reduce the difference
  most
• Recursively reduce the first and rest parts with
  other operators

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Means Ends Analysis(2)

• GPS(I, G, O) Iinitial, G goal, O op table
• select o from O which reduce the
  difference
• GPS(I, pre-condition(o), O)
• print(o)
• GPS(result(o), G, O)
• end

• Going to Central Park, N.Y
• GPS(Konkuk, C.Park, O-Table)
• Operation Table
• op pre
  result
• -------------------------------------------
  -----
• AirPlane Inchon J.F.K
• AirBus ????? Inchon
• Taxi Konkuk
  Shinchon
• Subway J.F.K
  C.Park

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Implementing Search(1)

• Blind Search
• -- Search does not depend on the nature of
  the positional value
• -- Systematic Search Method
• Breadth-First Search
• Depth-First Search
• Iterative Deepening Search
• Lowest Cost First Search
• Heuristic Search

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Implementing Search(2)

• Blind Search
• Heuristic Search
• Hill Climbing
• Best First Search
• A Algorithm

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Blind Search

• Abstract Data Type - Tree
• (defun make-TREE(label value children)
• (list tree label value children))
• (defun TREE-label (tree) (second tree))
• (defun TREE-value(tree) (third tree))
• (defun TREE-children(tree)(fourth tree))
• (defun TREE-print(tree)
• (princ(TREE-label tree)))

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Depth First Search

• (defun dfs(nodes goalp next)
• (cond ((null nodes) nil)
• ((funcall goalp (first nodes)) (first
  nodes))
• (t (dfs (append (funcall next (first
  nodes))
• (rest
  nodes))
• goalp
• next))))

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Function DFS

• gt (setq tree
• (make-TREE a 6
• (list (make-TREE b 3
• (list(make-TREE d 5 nil)
• (make-TREE e 4
  nil))))))
• //? ???? ??? ??? ???
• gt (dfs (list tree)
• (lambda(x) (TREE-print x)
• (eq g
  (TREE-label x)))
• TREE-children)

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Depth-First Search Example

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Breadth First Search

• (defun bfs(nodes goalp next)
• (cond ((null nodes) nil)
• ((funcall goalp (first nodes)) (first
  nodes))
• (t (bfs (append (rest nodes)
• (funcall next
  (first nodes)))
• goalp
• next))))

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Breadth-First Search Example
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Comparison of BFS and DFS

• BFS always terminate if goal exist
• cf. DFS on infinite tree
• BFS Guarantee shortest path to the goal
  (admissible)
• eg. Multiple Goals
• Space requirement
• BFS - Exponential
• DFS - Linear
• Which is better ? BFS or DFS ?

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Iterative Deepening Search

• Compromise of BFS and DFS
• Branch and Bound
• Set the Limit then DFS
• If Fail, extend the limit (one step)
• Save on Storage, guarantee shortest path

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Function ids

• (defun ids (start goalp next depth)
• (or (dfs-fd start goalp next 0 depth)
• (ids start goalp next (1 depth))))
• (defun dfs-fd(node goalp next depth max)
• (cond((funcall goalp node) node)
• (( depth max) nil)
• (t (some (lambda (n)
• (dfs-fd n goalp next
  (1 depth) max))
• (funcall next node)))))

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Heuristic Search

• Blind search assumes no information about the
  domain Too expensive
• Heuristic Information can be helpful
• Heuristic Function
• f(n) g(n) h(n)
• f total cost
• g cost from start to node n (real
  value)
• h cost from node n to goal
• (usually unknown)

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Lowest Cost First Search

• Borderline between Blind and Heuristic Search
• Cost value is assigned to each arc
• Moveset (candidate list) is a priority queue
  ordered by path cost
• Eg. Dijkstras algorithm
• When Arc cost is unit value, BFS

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Best First Search

• Moveset (candidate list) is a priority queue
  ordered by heuristic cost h(n)
• Does not guarantee the Shortest Path
• Does not always terminate even though there is a
  solution
• Performance depends on the h function

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Best First Search - Program

• (defun best(nodes goalp next comparep)
• (cond ((null nodes) nil)
• ((funcall goalp (first nodes)) (first
  nodes))
• (t (best (sort (append
• (funcall next
  (first nodes))
• (rest nodes))
• comparep)
• goalp
• next
• comparep))))

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A Algorithm

• Best First Search only considers h
• Lowest Cost First only considers g
• A algorithm takes f (note. f g h)
• A is optimal if h never overestimates the cost
  (optimistic)
• - h is admissible
• See the extra slide about A algorithm

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And/Or Graph

• Divide a problem into sub-problems and solve
  them individually
• divide and conquer
• And/Or Graph
• Node sub-problems
• Links And Link, Or Link
• And connect parent and sub-problems
• Or represents alternatives

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Searching And/Or Graph(1)

• Objective of Search
• To show whether start node is Solvable?
• or Unsolvable ?
• Definition of Solvable
• Terminal node is solvable
• A non-terminal OR node is solvable if at least
  one of its successor is solvable
• A non-terminal AND node is solvable iff all of
  its successors are solvable

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Searching And/Or Graph(2)

• Definition of UnSolvable
• Non-Terminal node with no successor is unsolvable
• A non-terminal OR node is unsolvable iff all of
  its successors are unsolvable
• A non-terminal AND node is unsolvable if at least
  one of its successors is unsolvable
• Search terminate when start node is labeled
  either solvable or unsolvable

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Island Driven Search

• Idea Find a set of islands between s, g
• s ? i1 ? i2 ? . ? im-1 ? g
• Problem is to identify the islands
• Optimal solution not guaranteed
• Hierarchical abstraction can be used

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Game Tree Search

• Game Tree Special case of AND/OR Graph
• Objective of Game Tree Search
• To find good first move
• Static Evaluation Funtion e
• Measure the worth of a tip node
• if p is win for MAX, then e(p) infinity
• if p is win for MIN, then e(p) - infinity

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MiniMax Procedure

• Select the maximum worth alternative
• Under Assumption of that the opponent do his best
• Back-Up Value(BUV)
• 1. At tip node p, BUV is e(p)
• 2. At max node, BUV is max BUV of children
• 3. At min node, BUV is min BUV of children

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MiniMax for Tic-Tac-Toe

• Two player, X and O, X play first
• Static Evaluation function
• If p is not a winning position
• e(p) (complete rows, columns, or diagonal that
  are still open for X) - (complete rows,
  columns, or diagonal that are still open for O)
• If p is a win for X, e(p) infinity
• if p is a win for O, e(p) -infinity

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1
-1
X
X
1
X
O
O
-2
6 - 5
X
X
X
O
X
X
O
5 - 4
6 - 4
O
X
5 - 5
6 - 5
O
X
5 - 5
O
O
O
O
X
X
O
X
X
X
O
4 - 5
5 - 6
5 - 5
5 - 6
5 - 5
4 - 6
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Alpha-Beta Procedure

• Improvement of MiniMax Procedure
• combining search procedure evaluation

max
S
A
B
C
1
-2
-1
0
1
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Production System

• Model of Search and Human Problem Solving
• Heavily used in AI (Expert) system
• Production System components
• Production rules (or production)
• also called Long-term memory
• Working Memory
• also called Short-term memory
• Recognize-act cycle
• also called control structure, engine

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Production Rules

• Production
• condition part and action part
• premise and conclusion
• LHS and RHS
• Condition Part - pattern matching
• Action Part - problem solving step
• Single chunk of knowledge
• Good for representing Judgmental knowledge

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Control of Production System

• Data Driven vs Goal Driven
• Forward vs Backward Search
• Bidirectional Search