1. CSNB234 ARTIFICIAL INTELLIGENCE
Chapter 5
State Space Search
Instructor: Alicia Tang Y. C.

2. State Space Representation
A state space can be considered as consisting of a collection of points, each point corresponding to a state of a problem. So there will be as many number of points in the state space as the number of different possible states of the problem. Thus, a state space is the collection of all possible states, a problem can have. States can be represented as nodes of a tree. The searching of a goal node is by traversing the tree’s nodes using some well-established algorithms.

3. Defining A Search Problem
State Space
is described by
an initial state space
a set of possible actions (operators)
goal state
Path
is any sequence of actions that lead from a state to another state and finally reached (hopefully) the goal node

4. Defining a Search Problem
Other considerations are:
Path cost: it is only relevant if there exists more than one path leading to the goal state.
& certainly that we want the shortest path
Goal test: it is applicable to single state problem, I.e. only one goal is found in the state space problem

5. Relationship between Initial states, actions and goal(s)
Control
Strategy

6. Why ‘Search’?
Search is an important (and powerful ) aspect in AI problem solving
Search will
will help to explore alternatives in tree
will find sequence of steps in a planning situation/case
All goal-driven activities (such as one used in B. C. reasoning) occur in a state space, and we call the problem a state space problem

7. Classical Search Domains
8-Puzzle
Water Jug
Blocks World
Travelling Salesman
Maze
Chess
Tower of Hanoi
etc.

8. An Example of Search Preparation2 4 8 1 8 7 7 3 2 6
Initial
State
Goal
State 1 5 6 3 4 5
Initial State: The location of each of the 8-puzzle in one of the
nine squares
Path cost: Each step costs 1 point, total path cost will be
equal to the number of steps taken
Goal test: State matches the goal configuration (given board pattern)
Operators: blank moves [1] Left, [2] Right, [3] Up & [4] Down.

9. Blind optimal partial Good enough optimal S e a r c complete h A p o s
Search Process
Test
Solution
complete
Comparisons
stop
when all
checked
All possible
solutions are
checked
Blind
optimal
partial
Check only
some
alternatives
Best
among
alternatives
Comparisons
Stop when
solution is
good
enough
Good
enough
Heuristics
Only
promising
solutions
Generate
improved
solutions
Stop when no
improvement
is possible
Optimisation
optimal

10. Methods of Search (I) Popular blind search methods are Depth-first
Breadth-first
Bi-directional
Iterative deepening
Depth-limited
Uniform cost

11. Methods of Search (II) Heuristic Search Optimal Searching
Hill-climbing
Best-first
Generate & test
Induction
Greedy search
Optimal Searching
A* search

12. Methods of search at the first glance
Depth-first
explores search tree branch by branch
Breadth-first
examines search tree row by row
Hill-climbing
it is depth first but most promising child node is examined first
Best-first
expands the most promising partial path of all those so far discovered A*
Best-first and Branch and bound algorithm
To be
calculated
Decision required

13. .… Blind…. The entire search tree is examined in an orderly manner.
It can be classified as exhaustive or partial.
Two well-known partial methods are
breadth-first
depth-first

14. DEPTH-FIRST SEARCH (DFS)
A ~ begins at the root node and
works downward to successively
deeper level.
This process continues until a solution is found or backtracking
is forced by reaching a dead end.
Blind ....

15. DEPTH-FIRST SEARCH Delete FIRSTNODE from start of QUEUE
Basic steps used
Delete FIRSTNODE from start of QUEUE
Take children of FIRSTNODE
Add to the front of QUEUE
Put result in NEWQUEUE

17. BREADTH-FIRST SEARCH (BFS) A ~ examines all the nodes in a
search tree, beginning with the
root node.
The nodes in each level are
examined completely before moving
to the next level.

18. BREADTH-FIRST SEARCH Take children of FIRSTNODE
The algorithm
Take children of FIRSTNODE
Delete FIRSTNODE from start of QUEUE
Append list of children to the end of QUEUE
Put result in NEWQUEUE

20. Heuristics Search
Heuristics search is designed to reduce the amount of search for a solution.
It is based on rule-of-thumb principle.
When a problem is presented, such as a goal state is given, this approach tries to reduce the size of the search tree by pruning non promising or inferior nodes.
It can normally speed up the search process in obtaining a good enough solution.

21. Benefits of Heuristics
It has inherent flexibility
It is better when an optimal/best solution is too costly to generate
even though it can only produce “good enough” answer
Simpler for decision maker to understand (especially the managers)
because they think in the same way as “heuristics” does

22. A heuristics search in general
Problem: Tic-Tac-Toe x x x o o x x o o o o o o o x o x x x x x x x x x x x x x x x

23. Consider the Tic-Tac-Toe game
Blind search is surely not practical…
Each nine first moves has eight possible responses,
which in turn have seven continuing moves, and so on.
A simple analysis tells that the number of states to be considered in an exhaustive search is 9 * 8 * 7 *… or 9!
However, symmetry reduction in second level leads to only 12 * 7! (much smaller than above), further reduction in search space will further reduce the number or search required

24. HILL-CLIMBING SEARCH Delete FIRSTNODE from start of QUEUE
Main steps used
Delete FIRSTNODE from start of QUEUE
Take children of FIRSTNODE
Order children with most promising first
Place them at the front of QUEUE
Put result in NEWQUEUE

25. HILL-CLIMBING SEARCH
A complete function:

26. Hill-Climbing
Hill-climbing combines depth-first search with a method for ordering the alternatives by measuring the probability of success at each decision point. In other word it tries to reach a goal by choosing those nodes which are predicted to be nearest to the goal. Quality measurements turn depth-first search into hill-climbing.

27. Hill-Climbing
It is about fin ding the shortest h(n)
Earlier slide says that it is quite similar to depth-first blind search.
However, paths are not selected arbitrarily, but in relationship to their proximity to the desired goal (i.e. how close to it)
let’s take a look at this tree:
notice that there are numbers attached nodes
those are potential numbers of defects in a product

28. Apply hill-climbing search to find ‘1’ defect node
START A I
III II B 13 C D 8 11 4 E 3 F G 5 H 7 2 I 1 J

29. Each production process I, II and III can continue for several stages.
If this is done by depth-first, it goes to I then to II and then to III .
In Hill-climbing method, nodes B, C and D are compared, and it starts in branch I (lowest, 8 here).
Since one defect was not found, backtracking is exercised to branch III. (branch III is done before II, why)?
The search goes to D and H; backtracking is then done and D, G, J path leads to the desired solution.
Path A--C--F was not visited
therefore much time and cost are saved
This is exactly the aims of ‘heuristics’!

30. Hill-climbing
Few more words on hill-climbing method: If we imagine the goal as the top of hill, h(n) as the difference in heights between n and the top, the hill-climbing procedure corresponds to climbing the hill by always going upwards.

31. Problems with hill-climbing:
Foothill problem.
Go up to the top of a hill and not advancing further. - misses the overall maximum
Plateau problem.
Problem with multidimensional problem space, i.e. no hill to climb (is flat). Gradient equals to zero. - not informative
Ridges.
Where there are steep slopes and the search direction is not towards the way up (but side)

32. Some solutions to hill-climbing problems are:
Random restart hill-climbing
where random initial states are generated
Simulated annealing
allow for bad moves as well, where the probability of such a move decreases exponentially with its badness

33. BEST-FIRST SEARCH Based on some heuristics evaluation function
More flexible than hill-climbing
An evaluation function is used to assign a score to each candidate node
Next move is made by selecting the best value node
It expands the best partial path, for that It could lead to “shortsighted” situation

34. BEST-FIRST SEARCH
Best-first search has been used in such applications as games and web crawlers
In a web crawler, each web page is treated as a node, and all the hyperlinks on the page are treated as unvisited successor nodes in the search space.
A crawler that uses best-first search generally uses an evaluation function that assigns priority to links based on how closely the contents of their parent page resemble the search query

35. BEST-FIRST SEARCH Delete FIRSTNODE from start of QUEUE
Take children of FIRSTNODE
Append children to QUEUE
Order result with most promising first
Put ordered result in NEWQUEUE

36. Apply best-first search to find ‘3’ defect in a production
[B D C]
[E I D C]
[I D C]
[D C]
[H G C]
[G C]
[J C]
[A]
[B D C]
[D C E I]
[H G C E I]
[G C E I]
[J C E I] A B 13 C D 8 11 E I 14 16 23 F G 5 H 7
Traversing order (i.e. the search path) is
A B D H G J 3 J
If hill climbing is used, the solution path will be
A B E I D H G J (it is longer, less ‘intelligent’)

37. Optimal Search Will produce optimal (best) answer
Based on some optimisation function
Mathematical functions are used
for improvements
for “optimisation”

38. BRANCH AND BOUND ALGORITHM
One way to find optimal paths with less work is to use branch-and-bound.
It always keeps track of all partial paths contending for further consideration.
The shortest one is extended one level, creating as many new partial paths.
Next, these new paths are considered, along with the remaining old ones, again, the shortest is extended.
The process is repeated until the goal is reached along some path.

39. Branch-and-Bound Keys to remember:
To turn likely to certain, you have to extend all partial paths until they are as long or longer than the complete path. The reason is that the last step in reaching the goal may be long enough to make the supposed solution longer than one or more partial paths.
It might be that only a tiny step would extend one of the partial paths to the solution node.
To be sure that this is not so, instead of terminating when a path is found, you terminate when the shortest partial path is longer than the shortest complete path

40. An Example for B-N-B S D A E F B 15 G 15 The length of the complete
path from S to G,
S-D-E-F-G is 15.
Similarly, the length of the
partial path S-D-A-B also 15
and any additional movement along a branch will make it longer than 15.
Accordingly, there is no need
to try S-D-A-B any further.
Because it will be longer
than the complete path
already known.
Only other paths emerging from S and from S-D-E have to be considered, as they may provide a shorter path. S D A E F B 15 G 15

41. To conduct a branch-and-bound search:
Form a one-element queue consisting of zero length path (only root node)
Until the first path in the queue terminated at the goal node or the queue is empty,
remove the first path from the queue; create new paths by extending the first path to all the neighbours of the terminal node
reject all paths with loops
add the reaming new paths, if any, to the queue
sort the entire queue by path length with least-cost paths in front
If the goal is found, announce success; otherwise announce failure.

42. Consider this search tree:

43. Solution Search path to goal node: [S(0)] [B(1) A(4)]
[E(3) A(4) F(4)]
[A(4) F(4) I(6) H(7)]
[F(4) C(5) I(6) D(6) H(7)]
[G(5) C(5) I(6) D(6) J(6) H(7)] 
Goal node reached and stopped.
Use your tie breaker solution

44. A* Search All A* algorithms are admissible
A search algorithm is admissible if it always produces an optimal solution
The A* search is branch-and-bound, with an estimate of remaining cost combined with dynamic programming principle
Theorem of A* algorithm
For each node n, let h*(n) denote the cost of an optimal path from n to a goal node
The algorithm uses a heuristic function h that satisfies h(n) <= h*(n) for all n in the state space, is admissible

45. A* SEARCH Delete FIRSTNODE from start of QUEUE
Append children of FIRSTNODE
Order resulting list according to
cost-so-far + underestimate of remaining cost
Put result in NEWQUEUE

46. A* SEARCH

47. To conduct a A* search:
Form a one-element queue consisting of zero length path (only root node)
Until the first path in the queue terminated at the goal node or the queue is empty,
remove the first path from the queue; create new paths by extending the first path to all the neighbours of the terminal node
reject all paths with loops
if two or more paths reach a common node, delete all those paths except the one that reaches the common node with the minimum cost
sort the entire queue by SUM of the path length and a lower-bound estimate of the cost remaining, with least-cost paths in front
If the goal is found, announce success; otherwise announce failure.

48. Function definition in A*
Consider the evaluation function
f(n) = g(n) + h(n)
where n is any state in the search
g(n) is the cost from the start state
h(n) is the heuristic estimate of the cost
going from n state to goal node

49. Using A* in 8-puzzle problem
Figure 1 shows a start state and the first set of moves 2 8 3
START 1 6 4 7 5 2 8 3 2 8 3 2 8 3
First level of
The tree search 1 4 1 6 4 1 6 4 7 6 5 7 5 7 5 A2 A3 A1
Which one will the A* takes?
& Why? On what basis it is chosen?

50. What is the GOAL we wanted to achieve?
i.e. what is the question?
To reach this GOAL state 1 2 3 8 4 7 6 5
Many levels of search and many solutions are possible
to reach the goal state

51. Using A* in 8-puzzle problem
Figure 2 : Three heuristics applied to states in the 8-puzzle 2 8 3 5 6
Recall the goal state 1 6 4 7 5 1 2 3 8 4 2 8 3 3 4 7 6 5 1 4 7 6 5 2 8 3 5 6 1 6 4
Tiles
out of
place
Sum of
distances
out of place
2 * the no.
of direct tiles
reversals
These are potential
Criteria for the A* formula 7 5

52. Using A* in 8-puzzle problem
Figure 3 : The heuristic f applied to states in the 8-puzzle 2 8 3
g(n) = 0 1 6 4 7 5 2 8 3 2 8 3 2 8 3
g(n)=1 1 6 4 1 4 1 6 4 7 5 7 6 5 7 5 6
f(n) 4 6
f(n) = g(n) + h(n)
where g(n) actual distance from n (cost from start state)
h(n) the no. of tiles out of place 1 2 3 8 4
GOAL 7 6 5

53. List 2(two) limitations of heuristics search methods
Exercise #1
List 4(four) criteria that are normally used to evaluate search methods
Exercise #2
List 2(two) limitations of heuristics search methods

54. Answer to Exercise #1 Completeness Space complexity Time complexity
will the solution be eventually found? if there is one at all?
Space complexity
how much memory will it need or is necessary?
Time complexity
how long will it take to complete the search?
Optimality
will the search method find the highest quality of solution path when there are several.

55. Supplementary Slides

56. Properties of Search Algorithms
Breadth-first Depth-first A* Greedy
Complete Yes No Yes No
Space Exponential Linear Exponential Linear
Time Exponential Exponential Exponential Exponential
Optimal Yes No Yes No

57. Bi-directional Search
Blind Search..
It searches both forward and backward in the same state space (run simultaneously).
It stops when the two moves meet in the middle.
Start
Goal
A schematic view of ~ that when a branch from the start node meets a
Branch of the goal node, it stops

58. Iterative Deepening (Korf 1987)
As depth-first search gets quickly into a deep level in the search space. Depth-first search can get lost deep in the search tree/graph. It may also stuck in infinite path that does not lead to a goal.
A compromise is to use a depth bound on depth-first :
the depth bound forces failure on a search path once it gets below certain level
Blind Search..

59. Iterative Deepening (Korf 1987)
Blind Search..
Iterative Deepening (Korf 1987)
The hard part of Depth-limit search is to determine what limit is “good” limit.
Iterative deepening tries all possible depth limits and pick up the best one!
Thus:
It is optimal and complete.
It uses modest memory.

60. Blind Search..
Depth-Limited search
Similar to DFS, except that it avoids the pitfalls of DFS by imposing a cut off on the maximum depth of a given path
Drawback:
if a chosen limit is too small, the scheme is not “complete”
optimality = no

61. Blind Search..
Uniform Cost Search
We learnt that BFS finds the shallowest path leading to the goal state
uniform cost search modifies BFS by expanding only the lowest cost node (n)
the cost of a path should remain low but not decreasing hence the term “uniform”
Note that if all step costs are equal, this is identical to BFS

62. Comparison of 6 Blind search methods
Breadth Depth Depth Bi-directional Iterative Uniform
first first limited cost
Time bd bm bl bd/ bd bd
Space bd bm bl bd/ bd bd
Complete yes no yes, l d yes yes yes
Optimal yes no no yes yes yes

63. Heuristics …
Induction
Induction means to generalise from a smaller version of the same problem
Two essential features
problem must be modelled in terms of the associated data
induced result must be tested against real examples

64. Generate & Test
Heuristics …
The basic idea is to generate possible solutions and devise a test to determine if the solutions are indeed good, i.e. acceptable.
Steps:
add a specification criterion
try to open a path that satisfies the specification
determine whether the path is plausible, prune it if it is not plausible
move to the next path
check whether all specifications have been mentioned. If not, add the next specification and reiterate the steps by returning to first step

65. Heuristics …
Greedy Search
There is a heuristic function, h(n), which serves to hold values of prediction of path-cost left to the goal
Greedy search is to minimise the estimated cost to reach the goal
Therefore, the node (state) closest to the goal will be expanded first

66. Heuristics …
Greedy Search
It resembles DFS in the way that it follows a single path all the way to the goal, if there is no dead end
Thus, it is incomplete and not optimal
With good quality heuristic function, however, the space an time complexity can be reduced