1. State Space Search I Chapter 3
The Basics

2. State Space Problem Space is a Graph Nodes: problem states
Arcs: steps in a solution process
One node corresponds to an initial state
One node corresponds to a goal state
State Space

3. State Space Solution Path
An ordered sequence of nodes from the initial state to the goal state
State Space

4. State Space Search Algorithm
Finds a solution path through a state space
State Space

5. The Water Jug Problem Die Hard: With a Vengeance (1995)
(Bruce Willis, Samuel L. Jackson, Jeremy Irons)
The Water Jug Problem

6. A Slight Variation Suppose we have An empty 4 gallon jug
A source of water
A task: put 2 gallons of water in the 4 gallon jug
A Slight Variation

7. Representation State Space Node on the graph is an ordered pair (x,y)
X is the contents of the 4 gallon jug
Y is the contents of the 3 gallon jug
Intitial State: (0,0)
Goal State: (2,N) N ε {0, 1, 2, 3}
Representation

8. Rules if x < 4, fill x : (x,y)  (4,y)
if y < 3, fill y : (x,y)  (x,3)
if x > 0, empty x : (x,y)  (0,y)
if y > 0, empty y : (x,y)  (x,0)
if (x+y) >= 4 and y > 0 : (x,y)  (4, y – (4 – x))
fill the 4 gallon jug from the 3 gallon jug
(see next slide)
if (x+y) >= 3 and x > 0 : (x,y)  (x –(3 – y), 3))
Fill the 3 gallon jug from the 4 gallon jug
if (x+y) <= 4 and y > 0 : (x,y)  (x+y), 0)
Pour the 3 gallon jug into the 4 gallon jug
if (x+y) <= 3 and x > 0 : (x,y)  (0, x + y)
pour the 4 gallon jug into the 3 gallon jug
Rules

9. 5 & 6 Redux if (x+y) >= 4 and y > 0 : (x,y)  (4, y – (4 – x))
fill the 4 gallon jug from the 3 gallon jug
if (x+y) >= 3 and x > 0 : (x,y)  (x –(3 – y), 3))
Fill the 3 gallon jug from the 4 gallon jug
4-X
3-Y
If x is the amount in the 4 gallon, 4-X is the amount necessary to fill it. This amount has to be subtracted from the 3 gallon jug (where the water came from).
5 & 6 Redux

10. if (x+y) <= 4 and y > 0 if (x+y) <= 3 and x > 0
Pour the 3 gallon jug into the 4 gallon jug: (x,y)  (x+y), 0)
if (x+y) <= 3 and x > 0
pour the 4 gallon jug into the 3 gallon jug: (x,y)  (0, x + y)

11. Is there a solution path?
Initial State: (0,0) Goal State: (2,N)
Is there a solution path?

12. Breadth First Search (0,0) 1 2 (0,3) (4,0) (0,3) 6 7 2 (3,0) 6 (4,3)
(1,3)
Breadth First Search
etc

13. Depth First (0,0) 1 (4,0) 2 3 (4,3) 7 (0,3) (3,0) 2 (3,3)
Etc. and without visiting already visited states

14. Backward/Forward Chaining
Search can proceed
From data to goal
From goal to data
Either could result in a successful search path, but one or the other might require examining more nodes depending on the circumstances
Backward/Forward Chaining

15. Data to goal is called forward chaining for data driven search Goal to data is called backward chaining or goal driven search

16. Examples Water jug was data driven Grandfather problem was goal driven
To make water jug goal driven:
Begin at (2,y)
Determine how many rules could produce this goal
Follow these rules backwards to the start state
Examples

17. Reduce the size of the search space
Object

18. if
Goal is clearly stated
Many rules match the given facts
For example: the number of rules that conlude a given theorem is much smaller than the number that may be applied to the entire axiom set
Use Goal Driven

19. If
Most data is given at the outset
Only a few ways to use the facts
Difficult to form a goal (i.e., hypothesis)
For example: DENDRAL, an expert system that finds molecular structure of organic compounds based on spectrographic data. There are lots of final possibilities, but only a few ways to use the initial data
Said another way: initial data constrains search
Use Data Driven