CS B551: ELEMENTS OF ARTIFICIAL INTELLIGENCEInstructor: Kris Hauser

http://cs.indiana.edu/~hauserk

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RECAP

http://www.cs.indiana.edu/classes/b551 Brief history and philosophy of AI What is intelligence? Can a machine

act/think intelligently? Turing machine, Chinese room

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AGENDA

Problem Solving using Search Search Algorithms

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EXAMPLE: 8-PUZZLE

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Initial state Goal state

State: Any arrangement of 8 numbered tiles and an empty tile on a 3x3 board

SUCCESSOR FUNCTION: 8-PUZZLE

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SUCC(state) subset of states

The successor function is knowledgeabout the 8-puzzle game, but it does not tell us which outcome to use, nor towhich state of the board to apply it.

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Across history, puzzles and games requiring the exploration of alternatives have been considered a challenge for human intelligence:

Chess originated in Persia and India about 4000 years ago

Checkers appear in 3600-year-old Egyptian paintings

Go originated in China over 3000 years ago

So, it’s not surprising that AI uses games to design and test algorithms

EXPLORING ALTERNATIVES

Problems that seem to require intelligence usually require exploring multiple alternatives

Search: a systematic way of exploring alternatives

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8-QUEENS PROBLEM

State repr. 1 Any non-conflicting

placement of 0-8 queens

State repr. 2 Any placement of 8

queens

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DEFINING A SEARCH PROBLEM

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State space S Successor function: x S SUCC(x) 2S

Initial state s0

Goal test: xS GOAL?(x) =T or F Arc cost

S

STATE GRAPH

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Each state is represented by a distinct node

An arc (or edge) connects a node s to a node s’ if s’ SUCC(s)

The state graph may contain more than one connected component

SOLUTION TO THE SEARCH PROBLEM A solution is a path

connecting the initial node to a goal node (any one)

The cost of a path is the sum of the arc costs along this path

An optimal solution is a solution path of minimum cost

There might be no solution !

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PATHLESS PROBLEMS Sometimes the path

doesn’t matter A solution is any

goal node Arcs represent

potential state transformations

E.g. 8-queens, Simplex for LPs, Map coloring

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REPRESENTATION 1 State: any placement of

0-8 queens Initial state: 0 queens Successor function:

Place queen in empty square

Goal test: Non-conflicting placement

of 8 queens # of states ~ 64x63x…

x57 ~ 3x1014

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REPRESENTATION 2

State: any placement of non-conflicting 0-8 queens in columns starting from left

Initial state: 0 queens Successor function:

A queen placed in leftmost empty column such that it causes no conflicts

Goal test: Any state with 8 queens

# of states = 2057

PATH PLANNING

15What is the state space?

FORMULATION #1

16Cost of one horizontal/vertical step = 1Cost of one diagonal step = 2

OPTIMAL SOLUTION

17This path is the shortest in the discretized state space, but not in the original continuous space

FORMULATION #2

18Cost of one step: length of segment

FORMULATION #2

19Cost of one step: length of segment

Visibility graph

SOLUTION PATH

20The shortest path in this state space is also the shortest in the original continuous space

WHAT IS A STATE?

A state does: Represent all information meaningful to the

problem at a given “instant in time” – past, present, or future

Exist in an abstract, mathematical sense A state DOES NOT:

Necessarily exist in the computer’s memory Tell the computer how it arrived at the state Tell the computer how to choose the next state Need to be a unique representation

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WHAT IS A STATE SPACE?

An abstract mathematical object E.g., the set of all permutations of (1,…,8,empty)

Membership should be trivially testable E.g., S = { s | s is reachable from the start state

through transformations of the successor function } is not easily testable

Arcs should be easily generated

Again: the state space does NOT contain information about which arc to take (or not to take) in a given state

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5-MINUTE QUIZ

Formulate 2x2 Tic-Tac-Toe, where you play both X and O’s actions, as a search problem. The goal state is any state with two in a line. Assume O goes first. Draw entire state graph. For compactness’s

sake, eliminate symmetrical states Indicate initial and goal states on this graph

Suppose one side is allowed to pass. How does the state graph change? Do you need to change anything to the problem definition?

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EXAMPLE: 8-PUZZLE

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Initial state Goal state

State: Any arrangement of 8 numbered tiles and an empty tile on a 3x3 board

15-PUZZLE Introduced (?) in 1878 by Sam Loyd,

who dubbed himself “America’s greatest puzzle-expert”

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15-PUZZLE

Sam Loyd offered $1,000 of his own money to the first person who would solve the following problem:

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4321

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But no one ever won the prize !!27

HOW BIG IS THE STATE SPACE OF THE (N2-1)-PUZZLE?

8-puzzle ?? states

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HOW BIG IS THE STATE SPACE OF THE (N2-1)-PUZZLE?

8-puzzle 9! = 362,880 states 15-puzzle 16! ~ 2.09 x 1013 states 24-puzzle 25! ~ 1025 states

But only half of these states are reachable from any given state(but you may not know that in advance)

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PERMUTATION INVERSIONS Wlg, let the goal be:

A tile j appears after a tile i if either j appears on the same row as i to the right of i, or on another row below the row of i.

For every i = 1, 2, ..., 15, let ni be the number of tiles j < i that appear after tile i (permutation inversions)

N = n2 + n3 + + n15 + row number of empty tile

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4321 n2 = 0 n3 = 0 n4 = 0n5 = 0 n6 = 0 n7 = 1n8 = 1 n9 = 1 n10 = 4n11 = 0n12 = 0n13 = 0n14 = 0n15 = 0

N = 7 + 4

Proposition: (N mod 2) is invariant under any legal move of the empty tile

Proof: Any horizontal move of the empty tile leaves N

unchanged A vertical move of the empty tile changes N by

an even increment ( 1 1 1 1)

311215

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s =

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s’ = N(s’) = N(s) + 3 + 1

Proposition: (N mod 2) is invariant under any legal move of the empty tile

For a goal state g to be reachable from a state s, a necessary condition is that N(g) and N(s) have the same parity

It can be shown that this is also a sufficient condition

The state graph consists of two connected components of equal size

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SEARCHING THE STATE SPACE

It is often not feasible (or too expensive) to build a complete representation of the state graph

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8-, 15-, 24-PUZZLES

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8-puzzle 362,880 states

15-puzzle 2.09 x 1013 states

24-puzzle 1025 states

100 millions states/sec

0.036 sec

~ 55 hours

> 109 years

INTRACTABILITY

Constructing the full state graph is intractable for most interesting problems

n-puzzle: (n+1)! states k-queens:

kk states

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Tractability of search hinges on the ability to explore only a tiny portion

of the state graph!

SEARCHING36

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SEARCHING THE STATE SPACE

Search tree

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SEARCHING THE STATE SPACE

Search tree

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SEARCHING THE STATE SPACE

Search tree

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SEARCHING THE STATE SPACE

Search tree

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SEARCHING THE STATE SPACE

Search tree

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SEARCHING THE STATE SPACE

Search tree

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SEARCH NODES AND STATES

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If states are allowed to be revisited,the search tree may be infinite even

when the state space is finite

If states are allowed to be revisited,the search tree may be infinite even

when the state space is finite

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DATA STRUCTURE OF A NODE

PARENT-NODE

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Depth of a node N = length of path from root to N

(depth of the root = 0)

BOOKKEEPING

5Path-Cost

5Depth

RightAction

Expanded yes...

CHILDREN

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NODE EXPANSION

The expansion of a node N of the search tree consists of: Evaluating the successor

function on STATE(N) Generating a child of N for

each state returned by the function

node generation node expansion

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FRINGE OF SEARCH TREE

The fringe is the set of all search nodes that haven’t been expanded yet

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Is it identical to the set of leaves?

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SEARCH STRATEGY

The fringe is the set of all search nodes that haven’t been expanded yet

The fringe is implemented as a priority queue FRINGE INSERT(node,FRINGE) REMOVE(FRINGE)

The ordering of the nodes in FRINGE defines the search strategy

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SEARCH ALGORITHM #1

SEARCH#11. If GOAL?(initial-state) then return initial-

state2. INSERT(initial-node,FRINGE)3. Repeat:4. If empty(FRINGE) then return failure5. N REMOVE(FRINGE)6. s STATE(N)7. For every state s’ in

SUCCESSORS(s)8. Create a new node N’ as a child

of N9. If GOAL?(s’) then return path or

goal state10. INSERT(N’,FRINGE)

Expansion of N

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PERFORMANCE MEASURES

CompletenessA search algorithm is complete if it finds a solution whenever one exists[What about the case when no solution exists?]

OptimalityA search algorithm is optimal if it returns a minimum-cost path whenever a solution exists

ComplexityIt measures the time and amount of memory required by the algorithm

TOPICS OF NEXT 3-4 CLASSES

Blind (uninformed) Search Little or no knowledge about how to search

Heuristic (informed) Search How to use extra knowledge about the problem

Local Search With knowledge about goal distribution

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RECAP

General problem solving framework State space Successor function Goal test => State graph

Search is a methodical way of exploring alternatives

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HOMEWORK

Register! Readings: R&N Ch. 3.4-3.5 HW1

On OnCourse Writing and programming Due date: 9/6

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