Search. Intelligent agents. Problem-solving. Problem-solving agents. Road map of Romania. The components of a problem. that will take me to the goal!

Transcription

1 Search Intelligent agents Reflex agent Problem-solving agent T65 rtificial intelligence and Lisp Peter alenius epartment of omputer and Information Science Linköping University Percept If I see ction then I do: Problem Search 2, 3, 1, 4, 4, 1 sequence of actions Problem-solving agents Problem-solving agents 1. Where am I? 2. What is my goal? 3. Which actions can I take? 4. Find a sequence of actions that will take me to the goal! 5. xecute the actions! Goal and problem formulation Search xecution 1. Where am I? 2. What is my goal? I m in rad, Romania. 3. Which actions can I take? 4. Find a sequence of actions that will take me to the goal! 5. xecute the actions! I want to go go to ucharest. Go from town to town. Searching The components of a problem Road map of Romania STT SP Initial state What does the world look like at the beginning? ctions What actions are available, and what are the results of these actions? Goal test How do we know that a given state is a goal state? Path cost What is the cost of each action? Oradea Neamt Iasi Zerind rad 99 Fagaras Vaslui Rimnicu Vilcea Timisoara 97 Lugoj Pitesti Mehadia robeta raiova Urziceni Hirsova ucharest 90 forie Giurgiu 1

2 Romanian route finding SRH SP Some definitions The initial state fter expanding rad fter expanding rad rad Timisoara Zerind rad Oradea Fagaras Rimnicu Vilcea rad XPN Timisoara Zerind FRING GNRT The static part (problem definition) state a description of (some aspect of) the world configuration action changes the world from one state to another state space the set of all possible states The dynamic part (search for solution) search space the result of exploring all possible actions node a part of the search space, corresponding to a state, but contains more information General search algorithm xample: rossing the river function TR-SRH(problem, fringe) returns a solution or failure fringe INSRT(MK-NO(INITIL-STT[problem]), fringe) loop do if MPTY?(fringe) then return failure node RMOV-FIRST(fringe) if GOL-TST[PROLM](STT[node]) then return SOLUTION(node) fringe INSRT-LL(XPN(node, problem), fringe) verything must cross the river. Only one thing can fit in the boat, besides the farmer. The lamb will eat the cabbage, and the wolf will eat the lamb, if left alone. Problem formulation STRT FWL STT SP Initial state verything to the right ctions Farmer and maximum one passenger may cross Goal test verything to the left Path cost ach step costs 1 FWL FWL WL F FW L FL W F WL F WL L FW FWL FL W L FW W FL FWL FW L W FL FWL 2

3 Search strategies lasses of search strategies strategy is defined by picking the order of node expansion. Strategies are evaluated by: ompleteness: oes it always find a solution, if one exists? Time complexity: Number of nodes generated Space complexity: Maximum number of nodes in memory Optimality: oes it always find the cheapest solution? Time and space complexity are measured by: b (maximum branch factor of the search tree) d (depth of the cheapest solution) m (maximum depth of the state space, possibly infinite) Uninformed search (blind search) additional information besides what is in the problem definition an only generate successor states and compare against goal state Informed search (heuristic search) Strategies have additional information whether non-goal states are more promising than others 1. redth-first search redth-first search xpand all nodes at level n before expanding nodes at level n+1 Generated nodes are placed in a FIFO queue (first in, first out) FIFO Queue redth-first search redth-first search FIFO Queue FIFO Queue F G F G 3

4 nalysis of bredth-first search FS space and time complexity omplete? Yes, we will always find a solution, provided that the branching factor b is finite. Optimal? Yes, if the path cost is a nondecreasing function of the node depth (e.g. when path cost is uniform). Space complexity? O(b d+1 ) Time complexity? O(b d+1 ) ranching factor b Root: one node Level 1: b nodes Level 2: b 2 nodes Level n: b n nodes ssume that the goal is at level d The number of nodes processed is 1 + b + b 2 + b b d + (b d+1 -b) = O(b d+1 ) omplexity analysis xtension: Uniform-cost search ig O notation or Ordo notation is used in complexity analysis to indicate the most important factor. T(n) = O(f(n)) if T(n) < kf(n) for some k, for all n > n0 If time complexity is O(n 2 ) this means that for some k the time used will always be less than kn 2. ig O notation gives a rough answer to the question How bad can it be? lways expand the node with the lowest path cost omplete and optimal if the cost of each action is at least ε (i.e. no free actions) 2. epth-first search epth-first search lways expand the deepest node in the fringe Generated nodes are placed in a LIFO queue or stack (last in, first out) 4

5 epth-first search epth-first search H I H I epth-first search nalysis of depth-first search H I I omplete?, if the tree has infinite depth we will never get an answer. Optimal?. If the optimal goal is in the right subtree, we will find any non-optimal goals in the left part first. Space complexity? O(bm) Time complexity? O(b m ) FS time and space complexity xtension: epth-limited search How many nodes must be kept in memory? ll unexpanded nodes, b nodes at each level. In total: bm + 1 = O(bm) How many nodes must be accessed? In the worst case, all nodes in the tree. In total: b m = O(b m ) Introduce a depth limit l and treat nodes at depth l as if they have no successors. t complete or optimal, but at least we can guarantee that the algorithm does not get stuck in an infinite loop. an be useful if we know something about the problem that can help us choose l. 5

6 3. Iterative deepening search ombines the benefits of bredth-first and depth-first. oes a depth-limited search with increasing depth limit l until a goal is found. 4. idirectional search Two simultaneous searches: one from the intitial state, one from the goal state and backwards Reduced complexity How can we search backwards? Uninformed search strategies Informed search strategies omplete Optimal Time Space Uniformcost redthfirst Yes 1 Yes 3 O(b d+1 ) O(b d+1 ) epthfirst Yes 1,2 Yes O(b 1+k ) O(b 1+k ) epthlimited O(b m ) O(bm) idirectional O(b l ) O(bl) Iterative deepening Yes 1 Yes 3 O(b d ) O(bd) Yes 1,4 Yes 3,4 O(b d/2 ) O(b d/2 ) 1) if b is finite 2) if step cost ε 3) if uniform step cost m = max depth, l = depth limit 4) if both directions use bredth-first search b = branching factor, d = depth of goal Use problem-specific knowledge beyond the problem definition itself. lways select the best node, based on an evaluation function f(n). The additional information encoded in the evaluation function is called a heuristic (rule of thumb, way of making educated guesses). ifferent informed search strategies use different heuristics. 5. Greedy best-first search Greedy best-first search Tries to expand the node that we currently believe is closest to a goal. Uses the following heuristic: h(n) = estimated cost of the cheapest path from node n to a goal node Priority queue (sorted) 6

7 Greedy best-first search xample: Romanian road trip rad F 44 G 51 Priority queue F G (sorted) 253 Timisoara 329 Zerind 374 rad Oradea Fagaras Rimnicu Vilcea ucharest 0 Stages in a greedy best-first search for ucharest using the straight-line distance heuristic. nalysis of greedy best-first search omplete? Optimal?. Just like depth-first search we will try to follow a single path and backtrack when we hit a dead end. Space complexity? Time complexity? O(b m ) With a good heuristic this can be reduced, but the worst case is the same as for depth-first search. 6. * search Same basic principle as greedy bestfirst search Uses the heuristic g(n) + h(n) g(n) = the cost to reach node n h(n) = estimated cost of the cheapest path from node n to a goal node Pronounced -star xample: Romanian road trip rad 366=0+366 Proof that * is optimal Will n or G 2 be expanded? 393= rad 646= Oradea 671= Fagaras 415= Rimnicu Vilcea = = ucharest 450= Stages in an * search for ucharest. Timisoara Zerind 447= = raiova 526= ucharest 418= Pitesti 417= raiova 615= = R. Vilcea 607= f(n) = g(n)+h(n) g(g 1 ) n G 2 G 1 optimal goal suboptimal goal f(g 2 ) = g(g 2 )+h(g 2 ) = g(g 2 ) > g(g 1 ) If h(n) never overestimates the cost, then * is optimal. We call h(n) an admissible heuristic. 7