Final answer:
The statement that f-values in A* are non-decreasing along any path if the heuristic h(n) is admissible (never overestimates the true cost) is disproved with a counterexample showing a scenario where an admissible heuristic does not adhere to the Monotone Restriction (MR) and causes f-values to remain constant rather than increase.
Step-by-step explanation:
Disproving the Statement with a Counterexample
The statement in question suggests that if A* uses a heuristic function h(n) such that h(n) ≤ h*(n), where h*(n) is the true cost to reach the goal from node n, then the f-value (defined as f(n) = g(n) + h(n), where g(n) is the cost from the start node to n) is non-decreasing along any path. The statement implicitly assumes that when the heuristic is admissible (does not overestimate the true cost), it will also be consistent, or satisfy the monotone restriction (MR). However, it is possible to have an admissible heuristic that is not consistent.
Consider a counterexample with a graph consisting of three nodes A, B, and C, and two edges: A to B with a cost of 2, and B to C with a cost of 1. Let the heuristic values be h(A) = 3, h(B) = 1, and h(C) = 0 (the goal node). Here, the heuristic is admissible, but not consistent because h(B) > h(C) - c(B, C) violates the MR constraint. If we calculate the f-values: f(A) = g(A) + h(A) = 0 + 3 = 3, f(B) = g(B) + h(B) = 2 + 1 = 3, and f(C) = g(C) + h(C) = 3 + 0 = 3. You can see the f-value remains constant, rather than increasing, from A to B to C. This disproves the statement by providing a specific scenario where an admissible heuristic does not result in a non-decreasing sequence of f-values along a path.