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1/4 Heuristic search Definitions Definitions (2) Heuristics (Greek heuriskein = find, discover): "the study For heuristic search to work, we must be able to rank Points of the methods and rules of discovery and invention". the children of a node. A heuristic function takes a Definitions state and returns a numeric value – a composite We use our knowledge of the problem to consider some Best-first search (not all) successors of the current state (preferably just assessment of this state. We then choose a child with Hill climbing one, as with an oracle). This means pruning the state the best score (this could be a maximum or minimum). Problems with hill climbing space, gaining speed, but perhaps missing the solution! A heuristic function can help gain or lose a lot, but An example: local and global In chess: consider one (apparently best) move, maybe a few – but finding the right function is not always easy. heuristic functions not all possible legal moves. The 8-puzzle: how many misplaced tiles? how many Simulated annealing In the travelling-salesman problem: select one nearest city, give up slots away from the correct place? and so on. The A* procedure complete search (the greedy technique). This gives us, in polynomial time, an approximate solution of the inherently exponential problem; it Water jugs: ??? Means-ends analysis can be proven that the approximation error is bounded. Chess: no simple counting of pieces is adequate. CSI 4106, Winter 2012 Heuristic search, page 1 CSI 4106, Winter 2012 Heuristic search, page 2 CSI 4106, Winter 2012 Heuristic search, page 3 Definitions (3) Best-first search Hill climbing The gain (often spectacular) gain is the The algorithm This is a greedy algorithm: go as high up as possible as reduction of the state space. For example, the select a heuristic function (e.g., distance to the goal); fast as possible, without looking around or keeping track. full tree for Tic-Tac-Toe has 9! = 362880 leaves. put the initial node(s) on the open list; The algorithm If we consider symmetries, the tree becomes six repeat select a heuristic function; times smaller, but it is still quite large. select N, the best node on the open list; let the current node C be the highest-valued initial node; With a fairly simple heuristic function we can get succeed if N is a goal node; otherwise put N on the closed list and add N's children to the open list; repeat the tree down to 40 states. (More on this when select N, the highest-valued child of C; until we discuss games.) we succeed or the open list becomes empty (we fail); until Heuristics can also help speed up exhaustive, A closed node reached on a different path is reopened. the goal has been reached or blind search, such as depth-first and breadth- no further improvement is possible; NOTE: "the best" only means "currently appearing the best"... first search. CSI 4106, Winter 2012 Heuristic search, page 4 CSI 4106, Winter 2012 Heuristic search, page 5 CSI 4106, Winter 2012 Heuristic search, page 6 1/4 Problems with hill climbing A hill climbing example A hill climbing example (2) 1. Local maximum, or the foothill problem: there is a A local heuristic function peak, but it is lower than the highest peak in the Count +1 for every block that sits on the correct whole space. thing. The goal state has the value +8. 2. The plateau problem: all local moves are equally Count -1 for every block that sits on an incorrect unpromising, and all peaks seem far away. thing. In the initial state blocks C, D, E, F, G, H 3. The ridge problem: almost every move takes us count +1 each. Blocks A, B count -1 each , for the down. total of +4. Random-restart hill climbing is a series of hill- Move 1 gives the value +6 (A is now on the correct climbing searches with a randomly selected start support). Moves 2a and 2b both give +4 (B and H node whenever the current search gets stuck. are wrongly situated). This means we have a local maximum of +6. See also simulated annealing – in a moment. CSI 4106, Winter 2012 Heuristic search, page 7 CSI 4106, Winter 2012 Heuristic search, page 8 CSI 4106, Winter 2012 Heuristic search, page 9 A hill climbing example (3) A hill climbing example (4) Simulated annealing anneal heat and then cool (as steel or glass) usually for softening and making less brittle; A global heuristic function Move 1 gives the value -21 (A is now on the also cool slowly usually in a furnace Count +N for every block that sits on a correct correct support). stack of N things. The goal state has the value The intuition for this search method, an Move 2a gives -16, because C, D, E, F, G, H +28. count -1, -2, -3, -4, -5, -1. improvement on hill-climbing, is the process Count -N for every block that sits on an Move 2b gives -15, because C, D, E, F, G of gradually cooling a liquid. A schedule of incorrect stack of N things. That is, there is a count -1, -2, -3, -4, -5. "temperatures" changes in time. Reaching large penalty for blocks tied up in a wrong structure. There is no local maximum! the "freezing point" (temperature 0) stops Moral: sometimes another heuristic function is all the process. In the initial state C, D, E, F, G, H count -1, -2, -3, -4, -5, -6. A counts -7 , for the total of -28. we need. We replace random restart with a more systematic method. continued CSI 4106, Winter 2012 Heuristic search, page 10 CSI 4106, Winter 2012 Heuristic search, page 11 CSI 4106, Winter 2012 Heuristic search, page 12 1/4 Simulated annealing (2) The A* procedure A few other search ideas The algorithm, one of the versions Hill-climbing (and its clever versions) may miss an optimal solution. select a heuristic function f; Here is a search method which ensures optimality of the solution. The algorithm let the current node C be any initial node; Stochastic beam search: keep a list of partial paths (initially root to root, length 0); let the current time t be 0; select w nodes at random; nodes with higher repeat let schedule(x) be a table of "temperatures”; values have a higher probability of selection. if the first path P reaches the goal node, return success; loop Genetic algorithms: otherwise remove path P from the list; extend P in all possible ways, add new paths to the list; t = t + 1; generate nodes like in stochastic beam search, sort the list by the sum of two values: the real cost of P till now, but from two parents rather than from one. and an estimate of the remaining distance; if schedule(t) = 0, return C; select at random N, any child of C; (This topic is worthy of a separate lecture...) prune the list by leaving only the shortest path for each node reached so far; if f(N) > f(C), set C to N, until otherwise success or the list of paths becomes empty; set C to N with probability e(f(N)-f(C))/schedule(t); CSI 4106, Winter 2012 Heuristic search, page 13 CSI 4106, Winter 2012 Heuristic search, page 14 CSI 4106, Winter 2012 Heuristic search, page 15 The A* procedure (2) The A* procedure (3) The A* procedure (4) A simple search problem: S is the start node, G is Here is a sketchy proof of the optimality of the goal node the goal node, the real distances are shown. A* requires a lower-bound estimate of the found by A*. Assume that m is a non-optimal goal: 4 4 A B C distance of any node to the goal node. A 3 heuristic which never overestimates is also g(m) > g(nk) S 5 5 G called optimistic or admissible. Let n0, n1, ..., nk be a path leading to nk. 4 3.5 2 4 D E F We consider three functions with values ≥ 0: Since m is a goal, h(m) = 0 and thus h'(m) = 0. Every step of A* selects the next ni in such a way that A lower-bound estimate of the distance to G could • g(n) is the actual cost of reaching node n, be as follows (note that we do not need it for F): g(ni) + h'(ni) ≤ g(ni) + h(ni) = g(nk) < g(m) = g(m) + h'(m) • h(n) is the actual unknown remaining cost, A B 5.8 C In other words, g(ni) + h'(ni) < g(m) + h'(m), so that A* 10.1 3.4 S • h'(n) is the optimistic estimate of h(n). cannot have selected m: the value of g + h' is too high. 11.5 G 9.2 D 3.5 E 7.1 F CSI 4106, Winter 2012 Heuristic search, page 16 CSI 4106, Winter 2012 Heuristic search, page 17 CSI 4106, Winter 2012 Heuristic search, page 18 1/4 Hill-climbing The A* procedure (5) The A* procedure (6) The A* procedure (7) S S happens to succeed here: S 3+10.1 4+9.2 13.1 13.2 A*, step 1 A*, step 2 10.1 9.2 A D A D 3+4+5.8 3+5+9.2 A D 10.1 11.5 7.1 B D A S E 5.8 9.2 3.5 B D F 7.1 0.0 E G CSI 4106, Winter 2012 Heuristic search, page 19 CSI 4106, Winter 2012 Heuristic search, page 20 CSI 4106, Winter 2012 Heuristic search, page 21 S The A* procedure (8) S The A* procedure (9) S The A* procedure (10) 13.1 13.2 13.1 13.2 13.1 13.2 A*, step 3 A*, step 4 A*, step 5 A D A D A D 12.8 17.2 12.8 17.2 4+5+10.1 12.8 17.2 19.1 B D B D A B D A 3+4+4+3.4 3+4+5+7.1 14.4 19.1 4+2+7.1 14.4 19.1 13.1 C E C E E C E E 4+2+5+5.8 4+2+4+3.5 B F CSI 4106, Winter 2012 Heuristic search, page 22 CSI 4106, Winter 2012 Heuristic search, page 23 CSI 4106, Winter 2012 Heuristic search, page 24 1/4 S The A* procedure (11) The A* procedure (12) Means-ends analysis 13.1 13.2 A*, step 6 The idea is due to Newell and Simon (1957): reduce the A D difference between states, and so approach the goal 12.8 17.2 19.1 The 8-puzzle with state. There are procedures, indexed by differences, optimistic heuristics. which change states. B D A The General Problem Solver algorithm A, ..., N: states; 14.4 19.1 13.1 m = misplaced; let the current node C be any initial node; d = depth; C E E f(x) = m(x) + d(x). loop 16.8 13.5 succeed if the goal state has been reached, otherwise find the difference between C and the goal state; B F choose a procedure that reduces this difference, apply it to C to produce the new current state; G CSI 4106, Winter 2012 Heuristic search, page 25 CSI 4106, Winter 2012 Heuristic search, page 26 CSI 4106, Winter 2012 Heuristic search, page 27 Means-ends analysis (2) An incomplete toy example: planning a trip. P r o c e d u r e D Plane Train Car Taxi Walk miles i s > 1000 X t 100-1000 X X a n 10-100 X X c e <1 X Prerequisite At plane At train At car At taxi CSI 4106, Winter 2012 Heuristic search, page 28