Can someone give me an example of admissible heuristic that is not consistent?

  • Admissibility

if you want your heuristics to be admissible then you should have that h(n) <=h*(n) for every node n where h* is the real cost to the goal. In your case you want:

h(A) <= 4
h(C) <= 3
h(G) <= 0
  • Consistency

If you want your heuristics to be consistent then you should have that h(G) = 0 and h(n) <= cost(n, c) + h(c) where the node c is a child of node c. So in your case

h(A) <= 1 + h(C)
h(C) <= 3 + h(G) = 3

If you want inconsistency and since h(C) <= 3 for the admissibility condition then you should have that h(A) > 1 + h(C). So any heristics that satisfies:

h(A) > 1 + h(C)
h(C) <= 3
h(G) = 0

is admissible and not consistent. You gave

h(A) = 4
h(C) = 1
h(G) = 0

which is a valid candidate.