I’m currently taking an AI class this semester. Admissible Heuristic Let h*(N) be the cost of the optimal path from N to a goal node The heuristic function h(N) is admissible 16 if: 0 ≤h(N) ≤h*(N) An admissible heuristic function is always optimistic ! UCS is a special case (h = 0) Graph search: A* optimal if heuristic is consistent UCS optimal (h = 0 is consistent) Consistency implies admissibility In general, most natural admissible heuristics tend to be consistent, especially if from relaxed problems I All consistent heuristics are admissible. Heuristic Accuracy • Let h 1 and h 2 be two consistent heuristics such that for all nodes N: h 1 (N) h 2 (N) • h 2 is said to be more accurate (or more informed) than h 1 h 1 (N) = number of misplaced tiles h 2 (N) = sum of distances of every tile to its goal position h 2 is more accurate than … A prime example is the difference between admissible and consistent heuristics. 3.Use a heuristic that’s not only admissible, but also consistent. A* is optimal if heuristic is admissible. Consistent (monotonic) heuristic Definition: A consistent heuristic is one for which, for every pair of nodes It has so long been thought that HS yields minimal cost solution graphs only if the heuristic satisfies the so-called ‘consistency condition’. Admissible Heuristic: A heuristic function h(n) is said to be admissible on (G,Γ) iﬀ h(n) ≤ h∗(n) for every n ∈ G Consistent Heuristic: A heuristic function h(n) is said to be consistent (or monotone) on G iﬀ for any pair of nodes, n0 and n, the triangle inequality holds: h(n0) ≤ k(n0,n)+h(n) I find the topic extremely interesting and fun to learn, but that isn’t to say that there aren’t topics that confuse me. This … For example, we know that the eucledian distance is admissible for searching the shortest path (in terms of actual distance, not path cost). For your example, there is no additional information available regarding the two heuristics. Admissible heuristics A heuristic h(n) is admissible if for every node n, h(n) ≤ h*(n), where h*(n) is the true cost to reach the goal state from n. An admissible heuristic never overestimates the cost to reach the goal, i.e., it is optimistic Example: h SLD(n) (never overestimates the actual road distance) A heuristic function h is consistent or monotone if it satisﬁes the following: h(u) ≤e(u,v)+h(v) where e(u,v) is the edge distance from u to v. In the absence of obstacles, and on terrain that has the minimum movement cost D, moving one step closer to the goal should increase g by D and decrease h by D. For the best paths, and an “admissible” heuristic, set D to the lowest cost between adjacent squares. Consistent Heuristics I Suppose two nodes u and v are connected by an edge. 2 3 Admissible Heuristics • A heuristic h(n) is admissible if for every node n, h(n) ≤ h*(n) where h*(n) is the true cost to reach the goal state from n. • An admissible heuristic never overestimates the cost to reach the goal Admissible Heuristics • Is the Straight Line Distance heuristic h SLD (1 mark for clarity of description, mentioning some of the other info here, or giving an example) (3 marks) iv.Manhattan distance, Euclidean Distance, Tiles-out-of-place are three examples. (Proof left to the reader.) It is shown here that the requirement that the heuristic be consistent can be relaxed to the one that the heuristic be merely admissible. If the heuristic is admissible and consistent A* nds a solution with the fewest number of exapansions. 1 Admissible vs Consistent Heuristics. Note also that any consistent heuristic is admissible (but not always vice-versa).