Problems problem spaces and search

Mr. Amey D. S. Kerkar,
Asst. Professor ,
Computer Engineering Department,
Don Bosco College of Engineering,
Fatorda-Goa.

Terminologies:
 State – Configuration of the game at any Point
1. State Space
 - Description of all possible states reachable from initial
state.
 Forms a graph.
 Nodes = States
 Arcs= actions
2. Initial State
3. Goal State
4. Path Cost

5. Production Rules:
 Rules/actions/operators applied to the current state. Results into
the next state.
rule: XY
Where, LHS represents current state in problem space
RHS is the resultant state.
6. Production System
- AI system developed for the solution of any problem.
Components of Production System:
1. Production Rules
2. Knowledge Base(KB)
3. Control Strategy- Order in which rule is applied compared to
Database/Knowledge base rules
4. Rule applier- checks current state with LHS of rules in KB and
finds appropriate rule to apply.

Example 1: 8-Puzzle Problem
7 4 2
1 5
6 3 8
Initial State
1 2 3
8 4
7 6 5
Initial State
Rules /Operators/Actions: 1. Move blank tile UP
2. Move blank tile DOWN
3. Move blank tile LEFT
4. Move blank tile RIGHT
Control Strategy= Search Technique.
Types of Search Techniques:
1. Uninformed/ Blind Search
2. Informed Search

Problem Formulation for 8 puzzle:
- Data Structure: 3 x 3 vector used for board
- Blank Tile is represented with ‘B’
- Numbered tiles are represented with corresponding
tile number
- Ex: 7 4 2
1 5
6 3 8
Is represented as:
7 4 2
1 𝑩 5
6 3 8

7 4 2
1 5
6 3 8
7 4 2
1 5
6 3 8
7 4
1 5 2
6 3 8
7 4 2
1 8 8
6 3
7 2
1 4 5
6 3 8
7 4 2
1 5
6 3 8
7 4 2
1 3 5
6 8
7 4 2
1 5
6 3 8
1 2 3
8 4
7 6 5

Search Process contains:
1. Expanding a node
2. Generating a node
3. Comparing
4. Tracking the path and path cost
Terminologies: 1.Generated node
2. Explored node

Example 2: TIC-TAC-TOE Problem
Initial state
X X X
X X X
X X X
X
X
X
X
X
X
5 POSSIBLE GOAL STATES FOR PLAYER 1
Player 1 will mark X
Player 2 will mark O

Problem Formulation for TIC-TAC-TOE:
- Data Structure: 3 x 3 vector used for board
- Blank Cell is represented with ‘0’
- Cell with ‘X’ marking represented with ‘1’
- Cell with ‘O’ marking represented with ‘2’
- Ex: Initial state 0 0 0
0 𝟎 0
0 0 0
0 0 X
X 0
X 0
- One of the
Goal State :
2 2 1
0 𝟏 2
1 2 0

Problem Formulation:
Each state= ordered pair {X,Y}
Where,
X= amount of water contained in Jar 1 (4 gallon capacity)
at any time.
Y=amount of water contained in Jar 2 (3 gallon capacity)
at any time.
Initial state: {0,0}
Final state: { P, 2}, Where P is any amount of water.
Example 3: Water Jug Problem

Production Rules:
RULE 1: (X,Y)(4,Y) [fill 4 gallon jug. Applicable if X<4]
RULE 2: (X,Y)(X,3) [fill 3 gallon jug. Applicable if Y<3]
RULE 3: (X,Y)(X-X1,Y) [pour some water out of 4 gallons jar]
RULE 4: (X,Y)(X,Y-X1) [pour some water out of 3 gallons jar]
RULE 5: (X,Y)(0,Y) [Empty 4 gallon jar]
RULE 6: (X,Y)(X,0) [Empty 3 gallon jar]
RULE 7: (X,Y)(4,Y-(4-X)) [Fill 4 gallon jar by pouring some water
from 3 gallon jar]
RULE 8: (X,Y)(X-(3-Y),3) [Fill 3 gallon jar by pouring some water
from 4 gallon jar]
RULE 9: (X,Y)(X+Y,0) [Empty 3 gallon jar by pouring all its water
into 4 gallon jar]
RULE 10: (X,Y)(0,X+Y) [Empty 4 gallon jar by pouring all its water
into 3 gallon jar]
RULE 11: (0,2)(2,0) [Pour 2 gallon from 3 gallon jar into 4 gallon]

RULE 12: (2,y)(0,y) [empty the 2 gallons in 3 gallon
jar on the ground]
One solution :
Water in 4-gallon
jar (X)
Water in 3-gallon
jar (Y)
Rule applied
0 0
0 3 Rule 2
3 0 Rule 9
3 3 Rule 2
4 2 Rule 7
0 2 Rule 5 or 12

Water in 4-gallon
jar (X)
Water in 3-gallon
jar (Y)
Rule applied
0 0
4 0 Rule 1
1 3 Rule 8
1 0 Rule 6
0 1 Rule 10
4 1 Rule 1
2 3 Rule 8
Another Solution :

{0,0}
{0,3}{4,0}
{4,3}{1,3} {3,0} {4,3}
{1,0}
{0,1}
{4,1}
{2,3}
{3,3}
{4,2}
{0,2}
{2,0}

 Missionaries and cannibels
 Travelling Salesperson
 8 Queens
 Combinatorial explosition- in travelling salesperson
problem –
number of routes α (no of cities -1)!
If it takes 1 hour of cpu time to solve for 30 cities
30 hours for 31 cities and
330 hours for 32 cities!!!!

Control strategies
 Helps us decide which rule to apply next.
 What to do when there are more than 1 matching
rules?
 Good control strategy should:
1. cause motion
2.Systematic

Control strategies are classified as:
1. Uninformed/blind search control strategy
Do not have additional info about states beyond problem def.
Total search space is looked for solution
No info is used to determine preference of one child over
other.
Example: 1. Breadth First Search(BFS), Depth First
Search(DFS), Depth Limited Search (DLS).
2. Informed/Directed Search Control Strategy
Some info about problem space(heuristic) is used to compute
preference among the children for exploration and
expansion.
Examples: 1. Best First Search, 2. Problem Decomposition,
A*, Mean end Analysis

Problems problem spaces and search

Problems problem spaces and search

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