Lecture 3-2 CS250: Intro to AI/Lisp

“If I Only had a Brain” Search II

Lecture 3-2

January 21st, 1999

CS250

Lecture 3-2 CS250: 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’))

Lecture 3-2 CS250: Intro to AI/Lisp

A* in Action

Contoured state spaceA* starts at initial nodeExpands leaf node of lowest f(n)Fans out to increasing contours

Lecture 3-2 CS250: Intro to AI/Lisp

A* in Perspective

• If h(n) = 0 everywhere, A* is uniform cost

• 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

Lecture 3-2 CS250: Intro to AI/Lisp

Proof of A*’s Optimality Suppose that G is an optimal goal state with with path

cost f*, and G2 is a suboptimal goal state, where g(G2) > f*.

Suppose A* selects G2 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 G2 was chosen over n: f(n) >= f(G2)

Together, they imply f* >= f(G2)

But G2 is a goal, so h(G2) = 0, f(G2) = g(G2)

Therefore, f* >= g(G2)

Lecture 3-2 CS250: 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(nk), Space

O(nk)

• See Figure 4.8

Lecture 3-2 CS250: Intro to AI/Lisp

Branching Factors

• Where f ’ is the next smaller cost, after f

)1(@#

)(@#

kDepthNodes

kDepthNodesAverageBlind

)'(@#

)(@#

fCostNodes

fCostNodesAverageHeuristic

Lecture 3-2 CS250: 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

Lecture 3-2 CS250: 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

Lecture 3-2 CS250: Intro to AI/Lisp

Iterative Deepening A*

• Good stuff in A*

• Limited memory

Lecture 3-2 CS250: 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

Lecture 3-2 CS250: 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

Lecture 3-2 CS250: 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))))))

Lecture 3-2 CS250: Intro to AI/Lisp

Let*

(let* ((current (create-start-node problem)) (successors (expand current problem)) (best current) next temp delta)

BODY)

Lecture 3-2 CS250: 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))))))