1. Instructor: Kris Hauser http://cs.indiana.edu/~hauserk
Search
Instructor: Kris Hauser

2. Recap http://www.cs.indiana.edu/classes/b351
Brief history and philosophy of AI
What is intelligence? Can a machine act/think intelligently?
Turing machine, Chinese room

3. Agenda
Problem Solving using Search
Search Algorithms

4. Exploring Alternatives
Problems that seem to require intelligence usually require exploring multiple alternatives
Search: a systematic way of exploring alternatives

5. 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

6. Example: 8-Puzzle 8 2 1 2 3 3 4 7 4 5 6 5 1 6 7 8 Initial state
Goal state
State: Any arrangement of 8 numbered tiles and an empty tile on a 3x3 board

7. Successor Function: 8-Puzzle2 7
SUCC(state)  subset of states
The successor function is knowledge about the 8-puzzle game, but it does
not tell us which outcome to use, nor to
which state of the board to apply it. 3 4 5 1 6 8 2 8 2 7 8 2 7 3 4 7 3 4 6 3 4 5 1 6 5 1 5 1 6

8. Defining a Search Problem
State space S
Successor function: x  S  SUCC(x)  2S
Initial state s0
Goal test:
xS  GOAL?(x) =T or F
Arc cost

9. 8-Queens Problem State repr. 1 State repr. 2
Any non-conflicting placement of 0-8 queens
State repr. 2
Any placement of 8 queens

10. State Graph 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

11. 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 ! I G

12. 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 G I

13. 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

14. 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

15. Path Planning
What is the state space?

16. Formulation #1 Cost of one horizontal/vertical step = 1
Cost of one diagonal step = 2

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

18. Formulation #2
Cost of one step: length of segment

19. Formulation #2
Cost of one step: length of segment
Visibility graph

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

21. What is a State? A state does: A state DOES NOT:
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

22. 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

23. Example: 8-Puzzle 8 2 1 2 3 3 4 7 4 5 6 5 1 6 7 8 Initial state
Goal state
State: Any arrangement of 8 numbered tiles and an empty tile on a 3x3 board

24. 15-Puzzle
Introduced (?) in 1878 by Sam Loyd, who dubbed himself “America’s greatest puzzle- expert”

25. 15-Puzzle
Sam Loyd offered $1,000 of his own money to the first person who would solve the following problem: 12 15 11 14 10 13 9 5 6 7 8 4 3 2 1 ? 12 14 11 15 10 13 9 5 6 7 8 4 3 2 1

26. But no one ever won the prize !!

27. How big is the state space of the (n2-1)-puzzle?
8-puzzle  ?? states

28. 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)

29. 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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4
n2 = 0 n3 = 0 n4 = 0
n5 = 0 n6 = 0 n7 = 1
n8 = 1 n9 = 1 n10 = 4
n11 = 0 n12 = 0 n13 = 0
n14 = 0 n15 = 0 5 10 7 8
 N = 7 + 4 9 6 11 12 13 14 15

30. 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) 12 15 11 14 10 13 9 5 6 7 8 4 3 2 1 12 15 11 14 10 13 9 5 6 7 8 4 3 2 1
s’ =
s =
N(s’) = N(s)

31. 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

32. Searching the State Space
It is often not feasible (or too expensive) to build a complete representation of the state graph

33. 8-, 15-, 24-Puzzles 8-puzzle  362,880 states
15-puzzle  2.09 x 1013 states
24-puzzle  1025 states
0.036 sec
~ 55 hours
> 109 years
100 millions states/sec

34. Intractability
Constructing the full state graph is intractable for most interesting problems
n-puzzle: (n+1)! states
k-queens: kk states
Tractability of search hinges on the ability to explore only a tiny portion of the state graph!

35. Searching

36. Searching the State Space
Search tree

37. Searching the State Space
Search tree

38. Searching the State Space
Search tree

39. Searching the State Space
Search tree

40. Searching the State Space
Search tree

41. Searching the State Space
Search tree

42. Search Nodes and States8 3 5 2 4 7 1 6 8 3 5 2 4 7 1 6 8 3 5 2 4 7 1 6
If states are allowed to be revisited, the search tree may be infinite even
when the state space is finite

43. Data Structure of a Node
PARENT-NODE 8 3 5 2 4 7 1 6
STATE
BOOKKEEPING 5
Path-Cost
Depth
Right
Action
Expanded
yes
...
CHILDREN
Depth of a node N = length of path from root to N
(depth of the root = 0)

44. Node expansion 1 2 3 4 5 6 7 8
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 N 1 3 5 6 8 4 7 2

45. Fringe of Search Tree
The fringe is the set of all search nodes that haven’t been expanded yet 8 2 3 4 7 5 1 6 8 2 7 3 4 5 1 6 8 2 8 2 8 3 5 2 4 7 1 6 3 4 7 3 4 7 5 1 6 5 1 6

46. Is it identical to the set of leaves?

47. 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

48. Search Algorithm #1
SEARCH#1 1. If GOAL?(initial-state) then return initial-state 2. INSERT(initial-node,FRINGE) 3. Repeat: 4. If empty(FRINGE) then return failure 5. 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 N 9. If GOAL?(s’) then return path or goal state 10. INSERT(N’,FRINGE)
Expansion of N

49. Performance Measures
Completeness A search algorithm is complete if it finds a solution whenever one exists [What about the case when no solution exists?]
Optimality A search algorithm is optimal if it returns a minimum-cost path whenever a solution exists
Complexity It measures the time and amount of memory required by the algorithm

50. 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

51. Recap General problem solving framework
State space
Successor function
Goal test
=> State graph
Search is a methodical way of exploring alternatives

52. Homework Readings: R&N Ch. 3.4-3.5 HW1 On OnCourse
Writing and programming
Due date: 9/12