1. ARTIFICIAL INTELLIGENCE
CS 512
ARTIFICIAL INTELLIGENCE
LECTURE 03

2. Solving problems by searching
Problem solving
We want:
To automatically solve a problem
We need:
A representation of the problem
Algorithms that use some strategy to solve the problem defined in that representation

3. Problem Representation
State-based Models
State space :
Model task as a graph of all possible states
A state captures all the relevant information about the past in order to act (optimally) in the future
Actions correspond to transitions from one state to another
Solutions are defined as a sequence of steps/actions (i.e., a path in the graph)
State-space graphs

4. Building Goal-Based Agents
What are the key questions needing to be addressed?
What goal does the agent need to achieve?
What knowledge does the agent need?
What actions does the agent need to do?

5. Search Example: Route Finding
Actions: go straight, turn left, turn right
Goal: shortest? fastest? most scenic?

6. Search Example: River Crossing Problem
Goal: All on right side of river
Rules:
Farmer must row the boat
Only room for one other
Without the farmer present:
Dog bites sheep
Sheep eats cabbage
Actions: F>, F<, FC>, FC<, FD>, FD<, FS>, FS<

7. Search Example: 8-Puzzle
Actions: move tiles (e.g., Move2Down)
Goal: reach a certain configuration

8. Search Example: Water Jugs Problem
Given 4-liter and 3-liter pitchers, how do you get exactly 2 liters into the 4-liter pitcher? 4 3

9. Search Example: 8-Queens
Place eight queens on a chess board such that no queen can attack another queen

10. Search Example: Remove 5 Sticks Problem
Remove exactly 5 of the 17 sticks so the resulting figure forms exactly 3 squares

11. Search Space Definitions
State
A description of a possible state of the world
Initial state
the state in which the agent starts the search
Goal test
Conditions the agent is trying to meet
Goal state
Any state which meets the goal condition
Action
Function that maps (transitions) from one state to another

12. Water Jugs Problem
Given 4-liter and 3-liter pitchers, how do you get exactly 2 liters into the 4-liter pitcher? State: (x, y) for # liters in 4-liter and 3-liter pitchers, respectively Actions: empty, fill, pour water between pitchers Initial state: (0, 0) Goal state: (2, *) 4 3

13. Actions / Successor Functions
1. (x, y | x < 4) (4, y) “Fill 4”
2. (x, y | y < 3) (x, 3) “Fill 3”
3. (x, y | x > 0) (0, y) “Empty 4”
4. (x, y | y > 0) (x, 0) “Empty 3”
5. (x, y | x+y ≥ 4 and y > 0) (4, y - (4 - x))
“Pour from 3 to 4 until 4 is full”
6. (x, y | x+y ≥ 3 and x > 0) (x - (3 - y), 3)
“Pour from 4 to 3 until 3 is full”
7. (x, y | x+y ≤ 4 and y > 0) (x+y, 0)
“Pour all water from 3 to 4”

14. State space

15. Example: Map Planning

16. Searching For Solutions
Initial State
e.g. “At Arad”
Successor Function
A set of action state pairs
S(Arad) = {(Arad->Zerind, Zerind), …}
Goal Test
e.g. x = “at Bucharest”
Path Cost
sum of the distances traveled

17. Searching For Solutions
Having formulated some problems…how do we solve them?
Search through a state space
Use a search tree that is generated with an initial state and successor functions that define the state space

18. Search Problem Example (as a tree)

19. Uninformed Search on Trees
Uninformed means we only know:
The goal test
The successors() function
But not which non-goal states are better
For now, also assume state space is a tree
Search process constructs a "search tree"
root is the start state
leaf nodes are:
unexpanded nodes (in the Frontier list)
"dead ends" (nodes that aren't goals and have no successors because no operators were applicable)
goal node is last leaf node found

20. Uninformed Search Strategies
Uninformed Search: strategies that order nodes without using any domain specific information, i.e., don’t use any information stored in a state
BFS: breadth-first search
Queue (FIFO) used for the Frontier
remove from front, add to back
DFS: depth-first search
Stack (LIFO) used for the Frontier
remove from front, add to front
CLICK EACH SUB

21. Depth First Search ( DFS )
Expand deepest unexpanded node

22. Depth First Search ( DFS )

23. Depth First Search ( DFS )

24. Depth First Search ( DFS )

25. Depth First Search (DFS)
Put start state in the agenda
Loop
Get a state from the agenda
If goal, then return
Expand state (put children to the front of agenda)
Avoiding loops
Don’t add a node to the agenda if it’s already in the agenda
Don’t expand a node (or add it to the agenda) if it has already been expanded.

26. Depth-First Search (DFS)
generalSearch(problem, stack)
S start
# of nodes tested: 0, expanded: 0
expnd. node
Frontier
{S} 5 2 4 A B C
CLICK DESC: 1. interior nodes, 2. leaf nodes, 3. arcs 9 4 6 2 D E 6 G
goal 1 F 7 H

27. Depth-First Search (DFS)
generalSearch(problem, stack)
S start
# of nodes tested: 1, expanded: 1
expnd. node
Frontier
{S}
S not goal
{A,B,C} 5 2 4 A B C
CLICK ONCE 9 4 6 2 D E 6 G
goal 1 F 7 H

28. Depth-First Search (DFS)
generalSearch(problem, stack)
S start
# of nodes tested: 2, expanded: 2
expnd. node
Frontier
{S} S
{A,B,C}
A not goal
{D,E,B,C} 5 2 4 A B C
CLICK ONCE 9 4 6 2 D E 6 G
goal 1 F 7 H

29. Depth-First Search (DFS)
generalSearch(problem, stack)
S start
# of nodes tested: 3, expanded: 3
expnd. node
Frontier
{S} S
{A,B,C} A
{D,E,B,C}
D not goal
{H,E,B,C} 5 2 4 A B C
CLICK ONCE 9 4 6 2 D E 6 G
goal 1 F 7 H

30. Depth-First Search (DFS)
generalSearch(problem, stack)
S start
# of nodes tested: 4, expanded: 4
expnd. node
Frontier
{S} S
{A,B,C} A
{D,E,B,C} D
{H,E,B,C}
H not goal
{E,B,C} 5 2 4 A B C
CLICK ONCE 9 4 6 2 D E 6 G
goal 1 F 7 H

31. Depth-First Search (DFS)
generalSearch(problem, stack)
S start
# of nodes tested: 5, expanded: 5
expnd. node
Frontier
{S} S
{A,B,C} A
{D,E,B,C} D
{H,E,B,C} H
{E,B,C}
E not goal
{G,B,C} 5 2 4 A B C
CLICK ONCE 9 4 6 2 D E 6 G
goal 1 F 7 H

32. Depth-First Search (DFS)
generalSearch(problem, stack)
S start
# of nodes tested: 6, expanded: 5
expnd. node
Frontier
{S} S
{A,B,C} A
{D,E,B,C} D
{H,E,B,C} H
{E,B,C} E
{G,B,C}
G goal
{B,C} no expand 5 2 4 A B C
CLICK ONCE 9 4 6 2 D E 6 G
goal 1 F 7 H

33. Depth-First Search (DFS)
generalSearch(problem, stack)
S start
# of nodes tested: 6, expanded: 5
expnd. node
Frontier
{S} S
{A,B,C} A
{D,E,B,C} D
{H,E,B,C} H
{E,B,C} E
{G,B,C} G
{B,C} 5 2 4 A B C
CLICK ONCE 9 4 6 2 D E 6 G
goal 1 F 7
path: S,A,E,G cost: 15 H

34. Depth First Search (DFS)
Try this
Depth First Search (DFS)
Find a path from node S to the goal nod I. Use DFS method.

35. Try this Find a path from node A to the goal nod B. Use DFS method. AZ O S F C P R T L M D

36. Breadth First Search (BFS)
Expand shallowest unexpanded node

37. Breadth First Search (BFS)

38. Breadth First Search (BFS)

39. Breadth First Search (BFS)
Put start state in the agenda
Loop
Get a state from the agenda
If goal, then return
Expand state (put children to the back of agenda)
Avoiding loops
Don’t add a node to the agenda if it’s already in the agenda
Don’t expand a node (or add it to the agenda) if it has already been expanded.

40. Breadth-First Search (BFS)
generalSearch(problem, queue) 5 2 9 6 4 1 7
S start A E D F B G
goal C H
# of nodes tested: 0, expanded: 0
expnd. node
Frontier list
{S}
CLIKC ANIM: 1. generalSearch, 2. graph, 3. table, 4. # expanded

41. Breadth-First Search (BFS)
generalSearch(problem, queue)
S start
# of nodes tested: 1, expanded: 1
expnd. node
Frontier list
{S}
S not goal
{A,B,C} 5 2 4 A B C
CLICK ONCE 9 4 6 2 D E 6 G
goal 1 F 7 H

42. Breadth-First Search (BFS)
generalSearch(problem, queue)
S start
# of nodes tested: 2, expanded: 2
expnd. node
Frontier list
{S} S
{A,B,C}
A not goal
{B,C,D,E} 5 2 4 A B C
CLICK ONCE 9 4 6 2 D E 6 G
goal 1 F 7 H

43. Breadth-First Search (BFS)
generalSearch(problem, queue)
S start
# of nodes tested: 3, expanded: 3
expnd. node
Frontier list
{S} S
{A,B,C} A
{B,C,D,E}
B not goal
{C,D,E,G} 5 2 4 A B C
CLICK ONCE 9 4 6 2 D E 6 G
goal 1 F 7 H

44. Breadth-First Search (BFS)
generalSearch(problem, queue)
S start
# of nodes tested: 4, expanded: 4
expnd. node
Frontier list
{S} S
{A,B,C} A
{B,C,D,E} B
{C,D,E,G}
C not goal
{D,E,G,F} 5 2 4 A B C
CLICK ONCE 9 4 6 2 D E 6 G
goal 1 F 7 H

45. Breadth-First Search (BFS)
generalSearch(problem, queue)
S start
# of nodes tested: 5, expanded: 5
expnd. node
Frontier list
{S} S
{A,B,C} A
{B,C,D,E} B
{C,D,E,G} C
{D,E,G,F}
D not goal
{E,G,F,H} 5 2 4 A B C
CLICK ONCE 9 4 6 2 D E 6 G
goal 1 F 7 H

46. Breadth-First Search (BFS)
generalSearch(problem, queue)
S start
# of nodes tested: 6, expanded: 6
expnd. node
Frontier list
{S} S
{A,B,C} A
{B,C,D,E} B
{C,D,E,G} C
{D,E,G,F} D
{E,G,F,H}
E not goal
{G,F,H,G} 5 2 4 A B C
CLICK ONCE 9 4 6 2 D E 6 G
goal 1 F 7 H

47. Breadth-First Search (BFS)
generalSearch(problem, queue)
S start
# of nodes tested: 7, expanded: 6
expnd. node
Frontier list
{S} S
{A,B,C} A
{B,C,D,E} B
{C,D,E,G} C
{D,E,G,F} D
{E,G,F,H} E
{G,F,H,G}
G goal
{F,H,G} no expand 5 2 4 A B C
CLICK ONCE 9 4 6 2 D E 6 G
goal 1 F 7 H

48. Breadth-First Search (BFS)
generalSearch(problem, queue)
S start
# of nodes tested: 7, expanded: 6
expnd. node
Frontier list
{S} S
{A,B,C} A
{B,C,D,E} B
{C,D,E,G} C
{D,E,G,F} D
{E,G,F,H} E
{G,F,H,G} G
{F,H,G} 5 2 4 A B C
CLICK ONCE 9 4 6 2 D E 6 G
goal 1 F 7
path: S,B,G cost: 8 H

49. Depth First Search (DFS)
Try this
Depth First Search (DFS)
Find a path from node S to the goal nod I. Use BFS method.

50. Try this Find a path from node A to the goal nod B. Use BFS method. AZ O S F C P R T L M D