The document discusses problem solving through search. It defines intelligent agents, search problems, and search graphs. Search problems are formulated using states, operators, start states, and goal states. Several search algorithms are introduced, including depth-first search and breadth-first search. Examples of search problems discussed include finding a route from Arad to Bucharest in Romania, the vacuum world problem, the 8-queens problem, and the 8-puzzle problem. The document outlines how to represent these problems as state spaces and formulates them in terms of states, actions, initial states, and goal tests. It also introduces tree search algorithms and strategies for searching state spaces, such as uninformed blind search and informed heuristic search.

1. 1
Solving problems by searching
171, Class 2
Chapter 3

2. 2
Overview
 Intelligent agents: problem solving as search
 Search consists of
 state space
 operators
 start state
 goal states
 The search graph
 A Search Tree is an efficient way to represent the search
process
 There are a variety of search algorithms, including
 Depth-First Search
 Breadth-First Search
 Others which use heuristic knowledge (in future lectures)

3. 3
Example: Romania
 On holiday in Romania; currently in Arad.
 Flight leaves tomorrow from Bucharest
 Formulate goal:
 be in Bucharest
 Formulate problem:
 states: various cities
 actions: drive between cities
 Find solution:
 sequence of cities, e.g., Arad, Sibiu, Fagaras, Bucharest

4. 4
Example: Romania

5. 5
Problem types
 Static / Dynamic
Previous problem was static: no attention to changes in environment
 Observable / Partially Observable / Unobservable
Previous problem was observable: it knew its initial state.
 Deterministic / Stochastic
Previous problem was deterministic: no new percepts
were necessary, we can predict the future perfectly
 Discrete / continuous
Previous problem was discrete: we can enumerate all possibilities

6. 6
Example: vacuum world
 Observable, start in #5.
Solution?

7. 7
Example: vacuum world
 Observable, start in #5.
Solution? [Right, Suck]
 Unobservable, start in
{1,2,3,4,5,6,7,8} e.g.,
Solution?

8. 8
Example: vacuum world
 Unobservable, start in
{1,2,3,4,5,6,7,8} e.g.,
Solution?
[Right,Suck,Left,Suck]

9. 9

10. 11
State-Space Problem Formulation
A problem is defined by four items:
initial state e.g., "at Arad“
actions or successor function S(x) = set of action–state pairs
 e.g., S(Arad) = {<Arad  Zerind, Zerind>, … }
goal test,
e.g., x = "at Bucharest”, Checkmate(x)
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
A solution is a sequence of actions leading from the initial state to a
goal state

11. 13
Vacuum world state space graph
 states? discrete: dirt and robot location
 initial state? any
 actions? Left, Right, Suck
 goal test? no dirt at all locations
 path cost? 1 per action

12. 14
Example: 8-queen problem

13. 15
Example: 8-Queens
 states? -any arrangement of n<=8 queens
-or arrangements of n<=8 queens in leftmost n
columns, 1 per column, such that no queen
attacks any other.
 initial state? no queens on the board
 actions? -add queen to any empty square
-or add queen to leftmost empty square such
that it is not attacked by other queens.
 goal test? 8 queens on the board, none attacked.
 path cost? 1 per move

14. 16
Example: robotic assembly
 states?: real-valued coordinates of robot joint
angles parts of the object to be assembled
 initial state?: rest configuration
 actions?: continuous motions of robot joints
 goal test?: complete assembly
 path cost?: time to execute

15. 17
Robot block world
 Given a set of blocks in a certain configuration,
 Move the blocks into a goal configuration.
 Example :
 (c,b,a)  (b,c,a)
A
B
C
A
C
B
Move (x,y)

16. 18
Operator Description

17. 19
Example: The 8-puzzle
 states?
 initial state?
 actions?
 goal test?
 path cost?
Try yourselves

18. 20
Example: The 8-puzzle
 states? locations of tiles
 initial state? given
 actions? move blank left, right, up, down
 goal test? goal state (given)
 path cost? 1 per move
[Note: optimal solution of n-Puzzle family is NP-hard]

19. 22
The “8-Puzzle” Problem
1 2 3
4
8
6
7 5
1 2 3
4 5 6
7 8
Goal State
Start State
1 2 3
4
8
6
7
5

20. 23
State space of the 8 puzzle problem

21. 24
The Traveling Salesperson Problem
 Find the shortest tour that visits all cities
without visiting any city twice and return
to starting point.
 State: sequence of cities visited
 S0 = A
C
D
A
E
F
B

22. 25
The Traveling Salesperson Problem
 Find the shortest tour that visits all cities
without visiting any city twice and return
to starting point.
 State: sequence of cities visited
 S0 = A
}
,
,
{ d
c
a }
,
,
|
)
,
,
,
{( d
c
a
X
x
d
c
a 
 SG = a complete tour
C
D
A
E
F
B

23. 26
The state-space graph
 Graphs:
 nodes, arcs, directed arcs, paths
 Search graphs:
 States are nodes
 operators are directed arcs
 solution is a path from start to goal
 Problem formulation:
 Give an abstract description of states, operators, initial state
and goal state.
 Problem solving:
 Generate a part of the search space that contains a solution

24. 27
Tree search algorithms
 Basic idea:
 Exploration of state space graph by generating
successors of already-explored states (a.k.a. expanding
states).
 Every states is evaluated: is it a goal state?

25. 28
Tree search example

26. 29
Tree search example

27. 30
Tree search example

28. 31
Implementation: states vs. nodes
 A state is a (representation of) a physical configuration
 A node is a data structure constituting part of a search tree
contains info such as: state, parent node, action, path cost
g(x), depth
 The Expand function creates new nodes, filling in the
various fields and using the SuccessorFn of the problem
to create the corresponding states.

29. 32
Search strategies
 A search 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
 space complexity: maximum number of nodes in memory
 optimality: does it always find a least-cost solution?
 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 ∞)

30. 33
Searching the search space
 Uninformed Blind search
 Breadth-first
 uniform first
 depth-first
 Iterative deepening depth-first
 Bidirectional
 Branch and Bound
 Informed Heuristic search (next class)
 Greedy search, hill climbing, Heuristics

31. 34
Next time
 Search Strategies
Questions?