The textbook says that we would not consider problems with negative path costs. In this exercise, we explore this in more depth.

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

(b) 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.

(c) Suppose that there is a set of operators that form a loop, so that executing the set in some order results in no net change to the state. If all of these operators have negative cost, what does this imply about the optimal behavior for an agent in such an environment?

(d) One can easily imagine operators with high negative cost, even in domains such as route finding. For example, some stretches of a 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 state-space search, why humans do not drive around scenic loops indefinitely, and explain how to define the state space and operators for route finding so that artificial agents can also avoid looping.

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