Question

consider a search space, where each state can be red, green, blue, yellow, or black. multiple states may have the same color. the goal is to reach any black state. here are some rules on the successors of different states, based on their color (these successor functions are unidirectional): red states can only have green or blue children. blue states can only have red or black children. green states can only have blue or yellow children. yellow states can only have yellow or red children. black states can only have green or black children. define a maximally admissible heuristic that assigns a value to each state based only on the color of that state assume that any move from one state to another has cost 1.

Answer 1

Step-by-step explanation:

A maximally admissible heuristic is an admissible heuristic that never overestimates the cost of reaching the goal state. In this problem, the goal state is any black state.

For this particular problem, a maximally admissible heuristic could be to assign the following values to each state based on its color:

Red states: 2

Green states: 1

Blue states: 3

Yellow states: 2

Black states: 0

This heuristic assigns the largest value to blue states, which are the farthest from the goal state in terms of allowed transitions. The value assigned to red states is also 2 because it takes two moves to reach either a green or a blue state from a red state. The value assigned to yellow states is also 2 because it takes two moves to reach a red state from a yellow state. The value assigned to green states is 1 because it takes only one move to reach a blue state or a black state from a green state. The value assigned to black states is 0 because they are the goal state.

This heuristic satisfies the admissibility condition because the cost of reaching the goal state (a black state) from any state is never overestimated. For example, the cost of reaching a black state from a red state is 2 moves, which is equal to the value assigned to red states by the heuristic. Similarly, the cost of reaching a black state from a yellow state is 3 moves (yellow state to red state to green state to black state), which is greater than the value assigned to yellow states by the heuristic (2).

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