1. Problem Solving as Search

2. Chapter 3 : Problem Solving by Searching
“In which we see how an agent can find a sequence of actions that achieves its goals when no single action will do.”
Such agents must be able to:
Formulate a goal
Formulate the overall problem
Find a solution

3. Appropriate environment for Searching Agents
Observable??
Deterministic??
Episodic??
Static??
Discrete??
Agents??
Yes
Either

4. Problem Types Deterministic, fully observable  single-state problem
Agent knows exactly which state it will be in
Solution is a sequence
Non-observable  conformant problem
Agent may have no idea where it is
Solution (if any) is a sequence
Nondeterministic and/or partially observable  contingency problem
percepts provide new information about current state
solution is a tree or policy
often interleave search, execution
Unknown state space  exploration problem ( “online” )

5. Problem Types Example: vacuum world Start in #5. Solution??
[Right, Suck]

6. Problem Types Deterministic, fully observable  single-state problem
Agent knows exactly which state it will be in
Solution is a sequence
Non-observable  conformant problem
Agent may have no idea where it is
Solution (if any) is a sequence
Nondeterministic and/or partially observable  contingency problem
percepts provide new information about current state
solution is a tree or policy
often interleave search, execution
Unknown state space  exploration problem ( “online” )

7. Problem Types
Conformant, start in {1,2,3,4,5,6,7,8} Solution??

8. Problem Types Conformant, start in {1,2,3,4,5,6,7,8}
e.g., Right goes to {2,4,6,8}.
[Right, Suck, Left, Suck]

9. Problem Types Deterministic, fully observable  single-state problem
Agent knows exactly which state it will be in
Solution is a sequence
Non-observable  conformant problem
Agent may have no idea where it is
Solution (if any) is a sequence
Nondeterministic and/or partially observable  contingency problem
percepts provide new information about current state
solution is a tree or policy
often interleave search, execution
Unknown state space  exploration problem ( “online” )

10. Problem Types Contingency, start in #5
Murphy’s Law: Suck can dirty a clean carpet
Local Sensing: dirt, location only.
Solution??
[Right, if dirt then Suck]

11. Problem Types Deterministic, fully observable  single-state problem
Agent knows exactly which state it will be in
Solution is a sequence
Non-observable  conformant problem
Agent may have no idea where it is
Solution (if any) is a sequence
Nondeterministic and/or partially observable  contingency problem
percepts provide new information about current state
solution is a tree or policy
often interleave search, execution
Unknown state space  exploration problem
“online” search

12. Problem Types Deterministic, fully observable  single-state problem
Non-observable  conformant problem
Nondeterministic and/or partially observable  contingency problem
Unknown state space  exploration problem

13. Single-State Problem Formulation
A problem is defined by four items:
initial state
successor function (which actually defines all reachable states)
goal test
path cost (additive) e.g., sum of distances, number of actions executed, etc. C(x,a,y) is the step cost, assumed to be  0

14. Consider this problem Five missionaries and five cannibals
Want to cross a river using one canoe.
Canoe can hold up to three people.
Can never be more cannibals than missionaries on either side of the river.
Aim: To get all safely across the river without any missionaries being eaten.
States?? Actions?? Goal test?? Path cost??

15. Problem Representation : Cannibals and Missionaries
Initial State
We can show number of cannibals, missionaries and canoes on each side of the river.
Start state is therefore:
[5,5,1,0,0,0]

16. Problem Representation : Cannibals and Missionaries
Initial State
However, since the system is closed, we only need to represent one side of the river, as we can deduce the other side.
We will represent the starting side of the river, and omit the ending side.
So start state is:
[5,5,1]

17. Problem Representation : Cannibals and Missionaries
Goal State(s)
[0,0,0]
TECHNICALLY also [0,0,1]

18. Problem Representation : Cannibals and Missionaries
Successor Function (9 actions)
Move one missionary.
Move one cannibal.
Move two missionaries.
Move two cannibals.
Move three missionaries.
Move three cannibals.
Move one missionary and one cannibal.
Move one missionary and two cannibals.
Move two missionaries and one cannibal.

19. Problem Representation : Cannibals and Missionaries
Successor Function (9 actions)
Move one missionary.
Move one cannibal.
Move two missionaries.
Move two cannibals.
Move three missionaries.
Move three cannibals.
Move one missionary and one cannibal.
Move one missionary and two cannibals.
Move two missionaries and one cannibal.

20. Problem Representation : Cannibals and Missionaries
Successor Function
To be a little more mathematical/computer like, we want to represent this in a true successor function format…
S(state)  state
Move one cannibal across the river.
S([x,y,1])  [x-1,y,0]
S([x,y,0])  [x+1,y,1]
[Note, this is a slight simplification.
We also require, 0  x, y, [x+1 or x-1]  5 ]

21. Problem Representation : Cannibals and Missionaries
Successor Function
S([x,y,0])  [x+1,y,1] //1 missionary
S([x,y,1])  [x-1,y,0] …
S([x,y,0])  [x+2,y,1] //2 missionaries
S([x,y,1])  [x-2,y,0]
S([x,y,0])  [x+1,y+1,1] // 1 of each
S([x,y,1])  [x-1,y-1,0]

22. Problem Representation : Cannibals and Missionaries
Path Cost
One unit per trip across the river.

23. Why do we look at “toy” problems.
Real world is absurdly complex state space must be abstracted for problem solving
(Abstract) state = set of real states (Abstract) action = complex combination of real actions e.g., “Arad  Zerind” represents a complex set of possible route, detours, rest stops, etc. For guaranteed realizability, any real state “in Arad” must get to some real state “in Zerind”
(Abstract) solution = set of real paths that are solutions in the real world each abstract action should e “easier” than the original problem!

24. Implementation: General Tree Search

25. Tree Search Example

26. Tree Search Example

27. Uninformed Search Strategies
Uninformed strategies use only information available in the problem definition
Breadth-first search
Uniform-cost search
Depth-first search
Depth-limited search
Iterative deepening search

28. Uninformed Search Strategies
Uninformed strategies use only information available in the problem definition
Breadth-first search
Uniform-cost search
Depth-first search
Depth-limited search
Iterative deepening search

29. Breadth-First Search
Expanding shallowest unexpanded node Implementation fringe is a FIFO queue, i.e., new successors go at end

30. Breadth-First Search
Expanding shallowest unexpanded node Implementation fringe is a FIFO queue, i.e., new successors go at end

31. Breadth-First Search
Expanding shallowest unexpanded node Implementation fringe is a FIFO queue, i.e., new successors go at end

32. Breadth-First Search
Expanding shallowest unexpanded node Implementation fringe is a FIFO queue, i.e., new successors go at end

33. Depth-First Search
Expand deepest unexpanded node Implementation: fringe = LIFO queue, I.e., put successors at front

34. Depth-First Search
Expand deepest unexpanded node Implementation: fringe = LIFO queue, I.e., put successors at front

35. Depth-First Search
Expand deepest unexpanded node Implementation: fringe = LIFO queue, I.e., put successors at front

36. Depth-First Search
Expand deepest unexpanded node Implementation: fringe = LIFO queue, I.e., put successors at front

37. Depth-First Search
Expand deepest unexpanded node Implementation: fringe = LIFO queue, I.e., put successors at front

38. Depth-First Search
Expand deepest unexpanded node Implementation: fringe = LIFO queue, I.e., put successors at front

39. Depth-First Search
Expand deepest unexpanded node Implementation: fringe = LIFO queue, I.e., put successors at front

40. Depth-First Search
Expand deepest unexpanded node Implementation: fringe = LIFO queue, I.e., put successors at front

41. Depth-First Search
Expand deepest unexpanded node Implementation: fringe = LIFO queue, I.e., put successors at front

42. Depth-First Search
Expand deepest unexpanded node Implementation: fringe = LIFO queue, I.e., put successors at front

43. Depth-First Search
Expand deepest unexpanded node Implementation: fringe = LIFO queue, I.e., put successors at front

44. Depth-First Search
Expand deepest unexpanded node Implementation: fringe = LIFO queue, I.e., put successors at front

45. Search Strategies
A strategy is defined by picking the order of node expansion
Strategies are evaluated along the following dimensions: completeness – does it always find a solution if one exists? time complexity – number of nodes generated/expanded space complexity – maximum number of nodes in memory optimality – does it always find a least-cost solution

46. Search Strategies
Time and space complexity are measured in terms of b – maximum branching factor of the search tree d – depth of the least-cost solution m – maximum depth of the state space (may be infinite)