This document discusses modeling and abstraction as it relates to solving the classic "water jug problem". It begins by explaining the problem and then goes through different levels of abstraction for modeling water, from a molecular level to simply representing water amounts with numbers. The key levels of abstraction for solving the water jug problem represent the jugs as placeholders for water amounts. Simple rules are defined for operations like filling and pouring between the jugs. A graph search approach uses these rules and representations to systematically find a solution by applying rules to states.

1. Abstraction
From Reality to a model
To answer questions

2. Classic AI problem:
Water Jug
Two Jugs: 2 liters and 5 liters
You can empty the water from a completely
You can pour the contents of
one jar into the other
Goal: To have a jug with exactly 1 liter of water
You can fill them up completely from
a water source

3. Water Jug Abstraction
Determine what is essential for model to solve problem
Simplify model of reality
Many levels of Abstraction

4. Modeling Water in a Jug
Levels of abstraction: Water as molecule
Absorption Spectra of water
Why microwave oven works
(IR spectra)
Why water is blue’
(visible absorbtion spectra)
QM Electronic Structure
Numerical models
But not the right level of abstraction for our problem

5. Modeling Water in a Jug
Levels of abstraction: Water as a fluid
Hydrogen bonding:
Why water is a fluid
Computational
Fluid Dynamics
flow in a pipe
Surface Tension
But still not the right level of abstraction for our problem

6. Modeling Water in a Jug
Measuring milliliters
Measuring cups
Cubic meters
Water in a lake
Weight in grams
We are closer….
We are measuring water
Do we care about units?
Do we care about the container?

7. Modeling Water in a Jug
Do we care about units?
Do we care about the container?
2 congli and 5 congli
2 uma and 5 uma
2 gallons and 5 gallons
2 ml and 5 ml
2 liters and 5 liters
2 cm3 and 5 cm3
2 sextarius and 5 sextarius

8. Modeling Water in a Jug
Do we care about units?
Do we care about the container?
NO!
2 5
Just two arithmetic numbers
Almost any type: Reals, integers, decimals, …..
We just need to add and subtract them

9. Levels of Abstraction
Modeling of water
Water can be a molecule Microscopic view of water
Water can be an electronic structure Electronic view of water
Water can be a liquid Macroscopic view of water
Water can be measured Using properties of water

10. Why Levels of Abstraction?
There is no reason why we cannot use a
Computational fluid dynamic model of water flowing
between the jugs
To solve our problem
Why not?
We are lazy
Water Jug Problem
Solution

11. Why Levels of Abstraction?
More official answer
We have only so much computing power
That is why there is no model for everything
Why is water blue
Why does a microwave work
How to solve the water jugs problem
Answers to all
my questions
The
Oracle
The modeler has to make choices of what to model

12. Two Jugs
with water
Don’t care what they are made of…..
Don’t care how big they are
…. Except that one holds 2 of something
and the other holds 5 of something
Placeholders for how much water in each jug
(Jug 1 amount, jug 2 amount)
This is (vector) representation of the
state of the system
of two jugs of water

13. Jugs
with water
(Jug 1 amount, jug 2 amount)
Both jugs are empty (0,0)
Both jugs are full (2,5)
Both jugs are partially full (1,4)
Reached goal of our problem (0,3)
Impossible situation (4,7)
Different states of the system

14. Jugs
with water
(Jug 1 amount, jug 2 amount)
Fill up jug one (X,Y) (2,Y)
Fill up jug two (X,Y) (X,5)
Empty jug one (X,Y) (0,Y)
Fill up jug two (X,Y) (X,0)
Simple Operations
Before oepration
Transformations
from one state of the system
to
another state to another state of the system After operation

15. Jugs
with water
(Jug 1 amount, jug 2 amount)
Other Operations
Pour jug one into jug two
If all of water in jug one fits in jug two
(X,Y) (0,X+Y)
With condition: X+Y <= 5
Pour jug two into jug two
(X,Y) (X+Y,0)
With condition: X+Y <= 2
If all of water in jug two fits in jug one

16. Jugs
with water
(Jug 1 amount, jug 2 amount)
Other Operations
Pour jug one into jug two
If all of water in jug one does not fit in jug two
(X,Y) (X1,5)
Jug 2 is full with 5, but how much is in jug one?
X+Y > 5
X1 = X+Y - 5
Jug one has what is left over

17. Jugs
with water
(Jug 1 amount, jug 2 amount)
Other Operations
Pour jug two into jug one
If all of water in jug two does not fit in jug one
(X,Y) (2,X1)
Jug one is full with 2, but how much is in jug two?
X+Y > 2
X1 = X+Y - 2
Jug two has what is left over

18. Abstraction
Representation of the jugs
(Jug 1 amount, jug 2 amount)
Fill up jug one (X,Y) (2,Y)
Fill up jug two (X,Y) (X,5)
Empty jug one (X,Y) (0,Y)
Fill up jug two (X,Y) (X,0)
Pour one into two (X,Y) (0,X+Y) X+Y<=5
(X,Y) (X+Y-5,5) X+Y>5
Pour two into one (X,Y) (X+Y,0) X+Y<=2
(X,Y) (2,X+Y-2) X+Y>2
Operations/Rules working on this representation
R1
R5
R6
R7
R8
R2
R3
R4

19. Use Structures
Use the data structures and rules to search for solution
A solution
Empty Jugs (0,0)
Fill up jug two (X,Y) (X,5)
(0,5)
R2
Pour two into one (X,Y) (2,X+Y-2) X+Y>2
(2,3)
R8
Empty jug one (X,Y) (0,Y)
(0,3)
R3
Pour two into one (X,Y) (2,X+Y-2) X+Y>2
(2,1)
R8
Empty jug one (X,Y) (0,Y)
(0,1) Done
R3

20. Use Structures
(0,0)
(2,3)
(0,5)
(2,1)
(0,3)
(0,1)
R2
R3
R8
R3
R8
Consecutive
Use
Of
Rules

21. Graph Search
(0,0)
(2,3)
(0,5)
(2,1)
(0,3)
(0,1)
R2
R3
R8
R3
R8
(0,2)
(0,4)
(2,2)
(2,4)
R1
R5
R1
R5
(2,0)
R1
(1,5)
R6
Every Branch
is an
Alternative
combination

22. Graph Search
Each node is the state of the system
Each connection is a rule that operates on that state
Types of graph search
Decides the order of which node to use a rule on
And
The order of which rules to apply