The initial state, actions, and transition model together define the **state space** of the problem implicitly.

**State-space** of a problem is a set of all states which can be reached from the initial state followed by any sequence of actions.

The state space forms a directed map or graph where nodes are the states, links between the nodes are actions, and the path is a sequence of states connected by the sequence of actions.

Major components of state space representation:

- Initial State
- Goal State
- Operator or legal moves

Example:

**WATER JUG PROBLEM**

You are given two jugs, a 4-gallon one and a 3-gallon one. 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 2 gallons of water into the 4-gallon jug?

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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 the 3-gallon jug Y=0,1,2,3

Then,

**Initial State**: (0,0)

**Goal State**: (2,0)

**Operator / Production Rule**:

– Fill the 4 or 3-gallon jug.

– Empty the 4 or 3-gallon jug.

– Pour from either of the jugs to another.

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