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Water Jug Problem in Artificial Intelligence.

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Problem: There are two jugs of volume A litre and B litre. Neither has any measuring mark on it.There is a pump that can be used to fill the jugs with water.How can you get exactly x litre of water into the A litre jug.Assuming that we have unlimited supply of water.

Note:Let's assume we have  A=4 litre and B= 3 litre jugs. And we want exactly 2 Litre water into jug A (i.e 4 litre jug) how we will do this.

Solution:

The state space for this problem can be described as the set of ordered pairs of integers (x,y)

Where,

x represents the quantity of  water in the 4-gallon jug  x= 0,1,2,3,4

y represents the quantity of water in 3-gallon jug y=0,1,2,3

Start State: (0,0)

Goal State: (2,0)

Generate production rules for the water jug problem

We basically perform three operations to achieve the goal.

  1. Fill water jug.
  2. Empty water jug 
  3. and Transfer water jug

Rule

State

Process

1

(X,Y | X<4)

(4,Y)

{Fill 4-gallon jug}

2

(X,Y |Y<3)

(X,3)

{Fill 3-gallon jug}

3

(X,Y |X>0)

(0,Y)

{Empty 4-gallon jug}

4

(X,Y | Y>0)

(X,0)

{Empty 3-gallon jug}

5

(X,Y | X+Y>=4 ^ Y>0)

(4,Y-(4-X))

{Pour water from 3-gallon jug into 4-gallon jug until 4-gallon jug is full}

6

(X,Y | X+Y>=3 ^X>0)

(X-(3-Y),3)

{Pour water from 4-gallon jug into 3-gallon jug until 3-gallon jug is full}

7

(X,Y | X+Y<=4 ^Y>0)

(X+Y,0)

{Pour all water from 3-gallon jug into 4-gallon jug}

8

(X,Y | X+Y <=3^ X>0)

(0,X+Y)

{Pour all water from 4-gallon jug into 3-gallon jug}

9

(0,2)

(2,0)

{Pour 2 gallon water from 3 gallon jug into 4 gallon jug}

Initialization:

Start State: (0,0)

Apply Rule 2:

Fill 3-gallon jug
Now the state is (x,3)

Iteration 1:

Current State: (x,3)

Apply Rule 7:

Pour all water from 3-gallon jug into 4-gallon jug

Now the state is (3,0)

Iteration 2:

Current State : (3,0)

Apply Rule 2:

Fill 3-gallon jug
Now the state is (3,3)

Iteration 3:

Current State:(3,3)

Apply Rule 5:

Pour water from 3-gallon jug into 4-gallon jug until 4-gallon jug is full

Now the state is (4,2)

Iteration 4:

Current State : (4,2)

Apply Rule 3:

Empty 4-gallon jug
Now state is (0,2)

Iteration 5:

Current State : (0,2)

Apply Rule 9:

Pour 2 gallon water from 3 gallon jug into 4 gallon jug


Now the state is (2,0)-- Goal Achieved.

Water Jug Solution using DFS (Depth First Search)

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