2. Lecture 5-1CS250: Intro to AI/Lisp “If I Only had a Brain” Search Lecture 5-1 October 26 th, 1999 CS250

3. Lecture 5-1CS250: Intro to AI/Lisp Blind Search No information except –Initial state –Operators –Goal test If we want worst-case optimality, need exponential time

4. Lecture 5-1CS250: Intro to AI/Lisp “How long ‘til we get there?” Add a notion of progress to search –Not just the cost to date –How far we have to go

5. Lecture 5-1CS250: Intro to AI/Lisp Best-First Search Next node in General-Search –Queuing function –Replace with evaluation function Go with the most desirable path

6. Lecture 5-1CS250: Intro to AI/Lisp Heuristic Functions Estimate with a heuristic function, h(n) –Problem specific (Why?) Information about getting to the goal –Not just where we’ve been Examples –Route-finding? –8 Puzzle?

7. Lecture 5-1CS250: Intro to AI/Lisp Greedy Searching Take the path that looks the best right now –Lowest estimated cost Not optimal Not complete Complexity? Time: O(b m ) Space: O(b m )

8. Lecture 5-1CS250: Intro to AI/Lisp Best of Both Worlds? Greedy –Minimizes total estimated cost to goal, h(n) –Not optimal –Not complete Uniform cost –Minimizes cost so far, g(n) –Optimal & complete –Inefficient

9. Lecture 5-1CS250: Intro to AI/Lisp Greedy + Uniform Cost Evaluate with both criteria f(n) = g(n) + h(n) –What does this mean? Sounds good, but is it: –Complete? –Optimal?

10. Lecture 5-1CS250: Intro to AI/Lisp Admissible Heuristics Optimistic: Never overestimate the cost of reaching the goal A* Search = Best-first + Admissible h(n)

11. Lecture 5-1CS250: Intro to AI/Lisp A* Search Complete Optimal, if: –Heuristic is admissible

12. Lecture 5-1CS250: Intro to AI/Lisp Monotonicity Monotonic heuristic functions are nondecreasing –Why might this be an important feature? Non-monotonic? Use pathmax: –Given a node, n, and its child, n’ f(n’) = max(f(n), g(n’) + h(n’))

13. Lecture 5-1CS250: Intro to AI/Lisp A* in Action  Contoured state space  A* starts at initial node  Expands leaf node of lowest f(n) = g(n) + h(n)  Fans out to increasing contours

14. Lecture 5-1CS250: Intro to AI/Lisp A* in Perspective What if h(n) = 0 everywhere? –A* is uniform cost What if h(n) is an exact estimate of the remaining cost? –A* runs in linear time! Different errors lead to different performance factors A* is the best (in terms of expanded nodes) of optimal best-first searches

15. Lecture 5-1CS250: Intro to AI/Lisp A*’s Complexity Depends on the error of h(n) –Always 0: Breadth-first search –Exactly right: Time O(n) –Constant absolute error: Time O(n), but more than exactly right –Constant relative error: Time O(n k ), Space O(n k ) See Figure 4.8

16. Lecture 5-1CS250: Intro to AI/Lisp Branching Factors Where f ’ is the next smaller cost, after f

17. Lecture 5-1CS250: Intro to AI/Lisp Inventing Heuristics Dominant heuristics: Bigger is better, if you don’t overestimate How do you create heuristics? –Relaxed problem –Statistical approach Constraint satisfaction –Most-constrained variable –Most-constraining variable –Least-constraining value

18. Lecture 5-1CS250: Intro to AI/Lisp Improving on A* Best of both worlds with DFID Can we repeat with A*? –Successive iterations: Increasing search depth (as with DFID) Increasing total path cost

19. Lecture 5-1CS250: Intro to AI/Lisp Iterative Deepening A* Good stuff in A* Limited memory

20. Lecture 5-1CS250: Intro to AI/Lisp Iterative Improvements Loop through, trying to “zero in” on the solution Hill climbing –Climb higher –Problems? Solution? Add a touch of randomness

21. Lecture 5-1CS250: Intro to AI/Lisp Annealing an.neal vb [ME anelen to set on fire, fr. OE onaelan, fr. on + aelan to set on fire, burn, fr. al fire; akin to OE aeled fire, ON eldr] vt (1664) 1 a: to heat and then cool (as steel or glass) usu. for softening and making less brittle; also: to cool slowly usu. in a furnace b: to heat and then cool (nucleic acid) in order to separate strands and induce combination at lower temperature esp. with complementary strands of a different species 2: strengthen, toughen ~ vi: to be capable of combining with complementary nucleic acid by a process of heating and cooling

22. Lecture 5-1CS250: Intro to AI/Lisp Simulated Annealing (defun simulated-annealing-search (problem &optional(schedule (make-exp-schedule))) (let* ((current (create-start-node problem)) (successors (expand current problem)) (best current) next temp delta) (for time = 1 to infinity do (setf temp (funcall schedule time)) (when (or (= temp 0) (null successors)) (RETURN (values (goal-test problem best) best))) (when (< (node-h-cost current) (node-h-cost best)) (setf best current)) (setf next (random-element successors)) (setf delta (- (node-h-cost next) (node-h-cost current))) (when (or (< delta 0.0) ; < because we are minimizing (< (random 1.0) (exp (/ (- delta) temp)))) (setf current next successors (expand next problem))))))

23. Lecture 5-1CS250: Intro to AI/Lisp Let* (let* ((current (create-start-node problem)) (successors (expand current problem)) (best current) next temp delta) BODY )

24. Lecture 5-1CS250: Intro to AI/Lisp The Body (for time = 1 to infinity do (setf temp (funcall schedule time)) (when (or (= temp 0) (null successors)) (RETURN (values (goal-test problem best) best))) (when (< (node-h-cost current) (node-h-cost best)) (setf best current)) (setf next (random-element successors)) (setf delta (- (node-h-cost next) (node-h-cost current))) (when (or (< delta 0.0) ; < because we are minimizing (< (random 1.0) (exp (/ (- delta) temp)))) (setf current next successors (expand next problem))))))

25. Lecture 5-1CS250: Intro to AI/Lisp Proof of A*’s Optimality  Suppose that G is an optimal goal state with with path cost f*, and G 2 is a suboptimal goal state, where g(G 2 ) > f*.  Suppose A* selects G 2 from the queue, will A* terminate?  Consider a node n that is a leaf node on an optimal path to G  Since h is admissible, f*>=f(n), and since G 2 was chosen over n: f(n) >= f(G 2 )  Together, they imply f* >= f(G 2 )  But G 2 is a goal, so h(G 2 ) = 0, f(G 2 ) = g(G 2 )  Therefore, f* >= g(G 2 )