Bookkeeping Informed Search AI Class 5 (Ch. 3.5-3.7) • Next lecture: • Python for AI • Eight decades of AI • (okay, 4) Based on slides by Dr. Marie desJardin. Some material also adapted from slides by Dr. Matuszek @ Villanova University, which are based on Hwee Tou Ng at Berkeley, which Dr. Cynthia Matuszek – CMSC 671 are based on Russell at Berkeley. Some diagrams are based on AIMA. Today’s Class Weak vs. Strong Methods • Heuristic search • Weak methods : “An informed search strategy—one that uses • Extremely general , not tailored to a specific situation • Best-first search problem specific • Examples • Greedy search knowledge… can find • Means-ends analysis : the current situation and goal, then look for • Beam search solutions more efficiently ways to shrink the differences between the two • A, A* then an uninformed • Space splitting : try to list possible solutions to a problem, then try • Examples strategy.” to rule out classes of these possibilities • Subgoaling : split a large problem into several smaller ones that can • Memory-conserving – R&N pg. 92 be solved one at a time. variations of A* • Called “weak” methods because they do not take • Heuristic functions advantage of more powerful domain-specific heuristics 3 4 Heuristic Heuristic Search Free On-line Dictionary of Computing* • Uninformed search is generic 1. A rule of thumb, simplification, or educated guess • Node selection depends only on shape of tree and node 2. Reduces, limits, or guides search in particular domains expansion trategy. 3. Does not guarantee feasible solutions; often used with no • Sometimes domain knowledge à Better decision theoretical guarantee • Knowledge about the specific problem WordNet (r) 1.6* 1. Commonsense rule (or set of rules) intended to increase the probability of solving some problem 5 6 *Heavily edited for clarity 1
Heuristic Search Heuristic Search • Romania: Arad à Bucharest (for example) • Romania: • Eyeballing it à certain cities first • They “look closer” to where we are going • Can domain knowledge be captured in a heuristic? 7 8 Heuristics Examples Heuristic Function • 8-puzzle: • All domain-specific knowledge is encoded in • # of tiles in wrong place heuristic function h • 8-puzzle (better): • h is some estimate of how desirable a move is • Sum of distances from goal • How “close” (we think) it gets us to our goal • Captures distance and number of nodes • Usually: • h ( n ) ≥ 0: for all nodes n • Romania: • h ( n ) = 0: n is a goal node • Straight-line distance from S • h ( n ) = ∞ : n is a dead end (no goal can be reached from n ) start node to Bucharest G • Captures “closer to Bucharest” 9 10 Informed Methods Add Example Search Space Revisited Domain-Specific Information start state • Goal: select the best path to continue searching 8 S • Define h ( n ) to estimates the “goodness” of node n arc cost 8 • h ( n ) = estimated cost (or distance) of minimal cost path 1 5 from n to a goal state A B 4 C 3 • Heuristic function is: 8 3 9 h value • An estimate of how close we are to a goal 7 4 5 • Based on domain-specific information D E ∞ G ∞ • Computable from the current state description 0 goal state 11 12 2
Straight Lines to Budapest (km) Admissible Heuristics h SLD (n) • Admissible heuristics never overestimate cost • They are optimistic – think goal is closer than it is • h ( n ) ≤ h * ( n ) • where h * ( n ) is true cost to reach goal from n • h LSD (Lugoj) = 244 • Can there be a shorter path? • Using admissible heuristics guarantees that the first solution found will be optimal 13 R&N pg. 68, 93 14 Best-First Search Best-First Search (more) • A generic way of referring to informed methods • Order nodes on the list by • Increasing value of f ( n ) • Use an evaluation function f ( n ) for each node • Expand most desirable unexpanded node à estimate of “desirability” • Implementation: • f ( n ) incorporates domain-specific information • Order nodes in frontier in decreasing order of desirability • Different f ( n ) à Different searches • Special cases: • Greedy best-first search • A* search 15 16 Greedy Best-First Search Greedy Best-First Search • Idea: always choose “closest node” to goal • Admissible? a a • Most likely to lead to a solution quickly • Why not? • So, evaluate nodes based only b g b g h=2 h=2 h=4 h=4 on heuristic function • Example: • f ( n ) = h ( n ) c • Greedy search will find: c h h h=1 h=1 h=1 h=1 • Sort nodes by increasing a à b à c à d à e à g ; cost = 5 values of f d d h=1 h=0 • Optimal solution: h=1 h=0 i i • Select node believed to be closest a à g à h à i ; cost = 3 e e to a goal node (hence “greedy”) h=1 h=1 • That is, select node with smallest f value • Not complete (why?) g g h=0 h=0 17 18 3
Straight Lines to Budapest (km) Greedy Best-First Search: Ex. 1 h SLD (n) What can we say about the search space? S 224 242 G 19 R&N pg. 68, 93 Greedy Best-First Search: Ex. 2 Greedy Best-First Search: Ex. 2 h SLD (n) 21 22 Greedy Best-First Search: Ex. 2 Greedy Best-First Search: Ex. 2 23 24 4
Beam Search Algorithm A • Use evaluation function • Use an evaluation function f ( n ) = h ( n ), but the f ( n ) = g ( n ) + h ( n ) maximum size of the nodes list is k , a fixed constant • g ( n ) = minimal-cost path from • Only keeps k best nodes as candidates for expansion, any S to state n and throws the rest away • Ranks nodes on search frontier by estimated cost of • More space-efficient than greedy search, but may solution throw away a node that is on a solution path • From start node, through given node, to goal • Not complete • Not complete if h ( n ) can = ∞ • Not admissible • Not admissible 25 26 Example Search Space Revisited Example Search Space Revisited start state start state parent pointer 0 8 0 8 S S arc cost arc cost 8 8 1 1 5 5 1 4 1 4 A B C A B C 5 8 3 5 8 3 8 8 3 3 9 9 h value h value 7 4 7 4 5 5 g value g value D E D E 4 8 ∞ 4 8 ∞ 9 G 9 G ∞ ∞ 0 0 goal state goal state 27 28 Algorithm A Algorithm A S • Use evaluation function 1. Put start node S on the nodes list, called OPEN f ( n ) = g ( n ) + h ( n ) 1 5 8 2. If OPEN is empty, exit with failure • g ( n ) = minimal-cost path from 1 3. Select node in OPEN with minimal f ( n ) and place on CLOSED any S to state n 5 A B C 8 4. If n is a goal node, collect path back to start; terminate 9 • Ranks nodes on search 3 5. Expand n , generating all its successors, and attach to them frontier by estimated cost of 5 1 pointers back to n . For each successor n' of n solution D 4 1. If n' is not already on OPEN or CLOSED • From start node, through given • G put n' on OPEN node, to goal • compute h ( n' ), g ( n' ) = g ( n ) + c( n , n' ), f ( n' ) = g ( n' ) + h ( n' ) 2. If n' is already on OPEN or CLOSED and if g ( n' ) is lower for the new 9 version of n' , then: • Not complete if h ( n ) can = ∞ • Redirect pointers backward from n' along path yielding lower g ( n' ). g(d)=4 • Put n' on OPEN. • Not admissible C is chosen next to expand h(d)=9 29 30 5
Some Observations on A Some Observations on A • Perfect heuristic: If h ( n ) = h * ( n ) for all n : • Better heuristic: We say If h 1 ( n ) < h 2 ( n ) ≤ h * ( n ) for all • Only nodes on the optimal solution path will be that A 2 * is expanded non-goal nodes, h 2 is a better better heuristic than h 1 The closer h • No extra work will be performed informed is to h * , the than A 1 * • If A 1 * uses h 1 , A 2 * uses h 2 , • Null heuristic: If h ( n ) = 0 for all n : fewer extra à every node expanded by A 2 * is • This is an admissible heuristic nodes will be also expanded by A 1 * • A* acts like Uniform-Cost Search expanded • So A 1 expands at least as many nodes as A 2 * 31 32 A * Search Quick Terminology Check • What is f ( n )? • What is h *( n )? • Avoid expanding paths that are already expensive • An evaluation function • A heuristic function that • Combines costs-so-far with expected-costs that gives… gives the… • A cost estimate of... • A* is complete iff • True cost to reach goal from n • The distance from n to G • Why don’t we just use that? • Branching factor is finite • What is h ( n )? • Every operator has a fixed positive cost • What is g ( n )? • A heuristic function • The path cost of getting from that… • A* is admissible iff S to n • Encodes domain • h ( n ) is admissible knowledge about... • describes the “spent” costs of the current search • The search space 34 A * Search A * Example 1 • Idea: Evaluate nodes by combining g(n), the cost of reaching the node, with h(n), the cost of getting from the node to the goal. 0 S cost • Evaluation function f(n) = g(n) + h(n) 8 h • g(n) = cost so far to reach n • h(n) = estimated cost from n to goal C 3 8 • f(n) = estimated total cost of path through n to goal 5 g 9 G 0 35 36 6
A * Example 1 A * Example 1 37 38 A * Example 1 A * Example 1 39 40 A * Example 1 Algorithm A* • Algorithm A with constraint that h ( n ) ≤ h * ( n ) • h *( n ) = true cost of the minimal cost path from n to a goal. • Therefore, h ( n ) is an underestimate of the distance to the goal • h () is admissible when h ( n ) ≤ h *( n ) • Guarantees optimality • A* is complete whenever the branching factor is finite, and every operator has a fixed positive cost • A* is admissible 41 42 7
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