Introduction to Intelligent Systems

Transcription

1 Problem Solving by Search Objectives Well-Defined Problems Tree search Uninformed search: BFS, DFS, DLS, and IDS Heuristic search: GBFS and A* Reference Russell & Norvig: Chapter 3 Y. Xiang, CIS 3700, Introduction to Intelligent Systems 1 Goal and Search A. Task of an agent can often be specified as a goal to be satisfied. B. A goal is a set of desirable env states (goal states). Example goals C. To complete the task, find a sequence of actions that brings env from initial state to a goal state. D. Search or problem-solving is the process of looking for a sequence of actions that satisfies a goal. E. Focus on env: Fully observable, deterministic, static, sequential, discrete, known, and single agent Y. Xiang, CIS 3700, Introduction to Intelligent Systems 2 Well-Defined Problems A. Problem definition 1) Initial state State space: the set of states reachable from initial state 2) Successor function The function maps each state to a set of (action, state ) pairs. 3) Goal test 4) Action (step) cost B. A solution is a path from the initial state to a goal state. What info is associated with a solution? C. An optimal solution has the lowest path cost. Search Tree A. Search tree 1) Root: Initial state 2) Expand: Apply Successor() to a node to generate its child nodes. 3) Fringe: The set of all leaf nodes B. Relation to state space a) A search tree is superimposed over state space. b) A state may reappear in a search tree. C. Search strategy: Given the current fringe, which leaf node should be expanded next? Y. Xiang, CIS 3700, Introduction to Intelligent Systems 3 4 1

2 Idea of Tree Search treesearch(problem, strategy) { 1 start tree T with root as initial state in problem; 2 loop 3 find a leaf x to visit according to strategy; 4 if no unvisited leaf, return failure; 5 if x is a goal state, 6 return corresponding solution; 7 else 8 expand x; 9 add successors of x as children of x in T; } Y. Xiang, CIS 3700, Introduction to Intelligent Systems 5 River-Crossing Problem A farmer, with a wolf, a sheep, and a cabbage, needs to cross a river using a boat. The boat can carry farmer plus only one other thing. Wolf and sheep cannot be left in the same bank unattended, nor can sheep and cabbage. How can they cross river without casualty? 1. Identify state variables and actions. 2. Identify initial state and goal. 3. Find action sequence by treesearch(). Y. Xiang, CIS 3700, Introduction to Intelligent Systems 6 Data Structure for Tree Node Components in node 1. Corresponding env state 2. Parent node on tree 3. Action that generated this node from its parent 4. Depth: path length from root 5. Path cost from root Optional data structure for fringe a) Set b) Queue Y. Xiang, CIS 3700, Introduction to Intelligent Systems 7 Algorithm for Tree Search treesearch(problem, strategy) { // fringe as queue 1 tree.makeroot(problem.initialstate); 2 fringe.insert(problem.initialstate); 3 loop 4 if ( fringe.isempty() ) return failure; 5 node = fringe.removefirst(); 6 if ( problem.isgoal(node.getstate()) ) 7 return node.getsolution(); 8 successors = node.expand(problem); 9 tree.addleaves(successors); A fringe.insert(successors, strategy); } Y. Xiang, CIS 3700, Introduction to Intelligent Systems 8 2

3 expand(problem) { // method in Node class 1 successors = ; 2 actionstatepairs = problem.succ(getstate()); 3 for each p in actionstatepairs, do 4 n = new Node(); 5 n.state = p.state; n.action = p.action; 6 n.parent = this; n.depth = this.depth + 1; 7 n.pathcost = this.pathcost + problem.getstepcost(n.action); 8 successors.add(n); 9 return successors; } 9 Measure Problem Solving Performance Before presenting strategies, we need to know how to compare them. 1. Completeness: A search algorithm is complete if it guarantees to find a solution when one exists. 2. Optimality: A search algorithm is optimal if it guarantees to find the optimal solution. 3. Time complexity: Time taken to find a solution Measured by the number of nodes generated 4. Space complexity: Memory needed during search Measured by the number of nodes stored Efficiency of solution vs efficiency of search Y. Xiang, CIS 3700, Introduction to Intelligent Systems 10 Measure Size of Problem A. To analyze complexity of a search algorithm, a measure of the size of a search problem is needed. B. Commonly measured by parameters of the fully expanded search tree 1. Branching factor (max # of successors per node): b 2. Depth of the shallowest goal node: d 3. Max path length: m Classes of Search Strategies A. The strategy adopted by a search algorithm determines the problem solving performance. B. Classes of search strategies 1. Uninformed (blind): Agent has no additional info beyond problem definition. 2. Informed (heuristic): Agent has knowledge whether one non-goal state is more promising than another. Y. Xiang, CIS 3700, Introduction to Intelligent Systems 11 Y. Xiang, CIS 3700, Introduction to Intelligent Systems 12 3

4 Breadth-First Search (BFS) A. BFS strategy Each node at a given depth is expanded, before any node at the next level is expanded. B. How to implement in TreeSearch()? FIFO queue for fringe C. Behavior of fringe.insert(successors, BFS) Convention: Insert left child first. Properties of BFS 1. Is BFS complete? 2. Is BFS optimal? 3. What is its time complexity? Theorem For integers b 2 and d 0, the following holds. 1+b+b 2 + +b d < b d What is the space complexity? Summary Y. Xiang, CIS 3700, Introduction to Intelligent Systems 13 Y. Xiang, CIS 3700, Introduction to Intelligent Systems 14 Depth-First Search (DFS) A. DFS strategy Visit the deepest node in the fringe first. B. Convention for tie breaking C. How to implement in TreeSearch()? LIFO queue (stake) for fringe 1. Is DFS complete? 2. Is DFS optimal? 3. What is the time complexity? Space Complexity of DFS A. A node can be removed from memory as soon as all descendants have been visited. B. DFS needs to store only a) a single path from root to a leaf, and b) unvisited siblings for each node on the path. C. Space complexity How many nodes must be stored? D. Summary Y. Xiang, CIS 3700, Introduction to Intelligent Systems 15 Y. Xiang, CIS 3700, Introduction to Intelligent Systems 16 4

5 Depth-Limited Search (DLS) A. Motivation: Avoid being stuck in unbounded branches as DFS does B. DLS strategy a) Depth-first search with a depth limit l b) Nodes at depth l are treated as having no successors. 1. Does DLS guarantee termination? 2. Space complexity 3. Time complexity 4. Is DLS complete? 5. Is DLS optimal? 17 Iterative Deepening Search (IDS) A. Motivation: Combine merits from BFS and DFS B. IDS strategy: Perform DLS with increasing depth limit l = 0, 1, 2,, until a goal node is found. C. Procedure for visiting node x if x is a goal node, return solution; else if x.depth < l, expand x; else remove x from memory; // x.depth = l if x is last child, rm y = x.parent from memory; if y is last child, rm z = y.parent from memory; D. An example 18 Properties of IDS 1. Is IDS complete? 2. Is IDS optimal? 3. What is the space complexity? 4. What is the time complexity? Theorem For integers b 2 and d 0, the following holds. 1 b d + 2 b d (d) b 1 + (d+1) b 0 < 2b d+1. Summary Avoiding Repeated States Causes of repeated states 1. Reversible actions 2. Multiple paths from a state to another Tradeoff in checking and avoiding repeated nodes A. Save time & space for processing repeated subtrees. B. Need to check each newly generated node against visited nodes (unique). Convention: Assume no checking for repeated nodes. Y. Xiang, CIS 3700, Introduction to Intelligent Systems 19 Y. Xiang, CIS 3700, Introduction to Intelligent Systems 20 5

6 Heuristic Search Strategies A. General approach: Best-First Search 1. Utilize problem-specific knowledge on how promising a node n leads to a goal. 2. Encode the knowledge with an evaluation function f(n). 3. Expand the leaf with the best f(.) value. 4. Convention: The lower the f(.) value, the better. B. Implementation: Maintain fringe as a queue sorted in ascending order of f(.) values. C. What properties of function f(n) are desirable? Can we always find such f(n)? Greedy Best-First Search (GBFS) Greedy Best-First Search 1. Specify a heuristic function h(n) as estimated cost of the cheapest path from node n to a goal node. 2. If n is a goal, what should the value of h(n) be? 3. Define f(n) = h(n). Ex Routing from city A to city B with straight-line distance heuristics Y. Xiang, CIS 3700, Introduction to Intelligent Systems 21 Y. Xiang, CIS 3700, Introduction to Intelligent Systems Area Map A T 71 Z 111 O 140 L 70 M 75 D 151 S R C F P A 366 B 0 C 160 D 242 E 161 F 176 G 77 H Y. Xiang, CIS 3700, Introduction to Intelligent Systems 23 G B 90 N Straight-line distance to B U I V H I 226 L 244 M 241 N 234 O 380 P 100 R 193 S 253 T 329 U 80 V 199 Z E Evaluation of GBFS 1. Is GBFS complete? Ex Routing from city I to F 2. Is GBFS optimal? 3. Time complexity GBFS behaves similarly to DFS. 4. Space complexity Ex Routing from A to G Summary Y. Xiang, CIS 3700, Introduction to Intelligent Systems 24 D E B G A F C 6

7 A* Tree Search A. Combine cost g(n) from root to node n with h(n). h(n): estimate cost of the best path from n to a goal B. Evaluation function: f(n) = g(n) + h(n) C. What is the interpretation? D. Ex Routing from city A to B E. Behavior of A* tree search depends on properties of heuristic function h(). Monotonicity of Heuristic Function A. Heuristic function h(n) is monotonic, if for each parent node p and its child c, h(c) h(p) - c(p,c), where c(p,c) is step cost from p to c. B. What is the interpretation? C. Ex Is straight-line distance heuristics monotonic? D. Ex An non-monotonic h(n) Y. Xiang, CIS 3700, Introduction to Intelligent Systems 25 Y. Xiang, CIS 3700, Introduction to Intelligent Systems 26 Completeness of A* 1. Proposition: If h() is monotonic, for every parent p and its child c, f(p) f(c) holds. 2. How are p and c related in a 2-D state space? 3. A* visits leaves along concentric circles of ascending f() values. 4. Theorem: A* tree search is complete, if a) h() is monotonic, b) each step cost is greater than some >0, and c) branching factor b is finite. Admissibility of h(n) & Optimality of A* A. Heuristic function h(n) is admissible if h(n) c(n), where c(n) is minimum path cost from n to a goal. B. What is the interpretation? C. What is the implication to evaluation function f(n)? D. Theorem: A* tree search is optimal if h(n) is admissible. E. Proof idea a) r: root; z: an optimal goal; b) x: a sub-optimal goal in fringe; c) n: a leaf on optimal solution path from r to z; d) Claim: x cannot be visited before n. Y. Xiang, CIS 3700, Introduction to Intelligent Systems 27 Y. Xiang, CIS 3700, Introduction to Intelligent Systems 28 7

8 Completeness, Optimality & Complexity A. Theorem If heuristic function h(n) is monotonic, it is admissible. B. Time complexity For most AI problems, the number of states inside a goal circle is exponential in the length of solution. C. Space complexity a) A* tree search needs to store all nodes expanded. b) Hence, space and time complexity are the same. D. Summary and comparison with blind strategies Y. Xiang, CIS 3700, Introduction to Intelligent Systems 29 8