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Describe the water-jug problem. Also, give a suitable state space representation for this problem.

Consider a scenario where you have a 3-litre jug and a 5-litre jug, and you need to measure precisely 4 litres of water. Visualize the scenario by imagining the two jugs and an infinite water source to fill them. The challenge here is to determine a sequence of actions that will allow you to reach the desired measurement of 4 litres, taking into account the constraints and capacities of the jugs.

The Water Jug Problem in AI involves constraints and objectives that make it a puzzle:

  • Constraint 1: The jugs have limited capacities.

  • Constraint 2: You can only fill or pour water between the jugs or from the source.

  • Objective: The goal is to measure a specific quantity of water accurately, typically by combining and transferring water between the jugs.

In AI problem-solving, we work with a state space (all possible states) and an action space (all possible actions). In the Water Jug Problem, the state space comprises all possible configurations of water levels in the jugs. The action space includes the actions you can take, such as filling a jug, emptying a jug, or pouring water from one jug to another.

  • Initial State: The initial state is where you start. In the classic scenario, this typically means both jugs are empty.

  • Goal State: The goal state is where you want to reach, representing the desired water level, e.g., 4 liters.

  • Actions: Actions are the operations you can perform on the jugs, such as filling, emptying, or pouring water between them.

Now, let us formally state the problem.

Suppose you have two jugs: a small one with a capacity of 𝑀 litres, and a large one with a capacity of 𝑁 liters (𝑀<𝑁). You are also given a target volume of water 𝑉 liters that you need to measure out using these jugs. The objective is to determine a sequence of pour operations that will result in obtaining exactly 𝑉 liters of water in one of the jugs or both.

Here's an example of a suitable state space representation for this problem:

A suitable state representation could be a tuple (𝑥,𝑦), where 𝑥 represents the amount of water in the small jug and 𝑦 represents the amount of water in the large jug. The initial state would be (0,0) (both jugs empty), and the goal state would be any state where 𝑥=𝑉 or 𝑦=𝑉.

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