Exercise 3.10 [negativegexercise]
On page nonnegativeg, we said that we would not consider problems with negative path costs. In this exercise, we explore this decision in more depth.

Suppose that actions can have arbitrarily large negative costs; explain why this possibility would force any optimal algorithm to explore the entire state space.

Does it help if we insist that step costs must be greater than or equal to some negative constant $c$? Consider both trees and graphs.

Suppose that a set of actions forms a loop in the state space such that executing the set in some order results in no net change to the state. If all of these actions have negative cost, what does this imply about the optimal behavior for an agent in such an environment?

One can easily imagine actions with high negative cost, even in domains such as route finding. For example, some stretches of road might have such beautiful scenery as to far outweigh the normal costs in terms of time and fuel. Explain, in precise terms, within the context of statespace search, why humans do not drive around scenic loops indefinitely, and explain how to define the state space and actions for route finding so that artificial agents can also avoid looping.

Can you think of a real domain in which step costs are such as to cause looping?