Problem Solving as Search - I

Transcription

1 Problem Solving as Search - I Shobhanjana Kalita Dept. of Computer Science & Engineering Tezpur University Slides prepared from Artificial Intelligence A Modern approach by Russell & Norvig

2 Problem-Solving Agents A goal based agent whose goal is to solve a particular problem Define problem Identify solution states goals Task: Find sequence of actions that will allow the agent to go from current state to goal state search Execute action sequence that the search returns as solution

3 Problem A problem can be defined formally by four components: The initial state that the agent starts in. A description of the possible actions available to the agent. Commonly done using a successor function. Given a particular state x, SUCCESSOR-FN(x) returns a set of (action, successor) ordered pairs. The initial state and successor function implicitly define the state space of the problem-the set of all states reachable from the initial state. The goal test determines whether a given state is a goal state. Explicit set of possible goal states or specified by an abstract property A path cost function that assigns a numeric cost to each path. Agent chooses a cost function that reflects its own performance measure Agent attempts to minimize the cost function A solution to a problem is a path from the initial state to a goal state. Solution quality is measured by the path cost An optimal solution has the lowest path cost among all solutions.

4 Problem Formulation Abstraction: process of removing detail from representation 8 puzzle formulation: States are a 3x3 matrix of numbers from 1-8, with one blank Actions: To move blank Left, Right, Up or Down Path Cost: Each step/move costs 1

5 Problem Formulation Path Finding Problem: Find a path from city A to city B on a given map States: List of cities that have been visited so far Initial state: A Goal state: List of cities starting with A and ending with B Successor function: Which cities can be visited next from the current city and the corresponding new state Path cost: Cost of going from one city to the next Shortest Path Finding Problem?? Robot Navigation??

6 Searching for Solutions The problem thus defined can be solved by searching the state space Search for goal states in a search tree generated from the initial state using the successor function Search strategy: Choice (!) of which action to be taken up in Search strategy: Choice (!) of which action to be taken up in order to continue search for the goal state Or, which node to expand next in the collection of nodes that have been generated (fringe) Information in search node: state; parent node; action; path-cost; depth

7 Searching for Solutions Branching factor: Maximum number of successor nodes on any node Effectiveness of search can be determined by: Search cost: Time taken to reach the goal state Total cost: Search cost + path-cost

8 Uninformed Search An uninformed search strategy is one in which no additional information about the states is available beyond the problem definition in order to choose the next node to be expanded Breadth First Search Uniform Cost Search Depth First Search Depth-limited Search Iterative Deepening Search Bidirectional Search Informed search or heuristic search strategy is one in which additional information is available to determine the expansion of which node is more likely to take us to the goal node

9 Breadth first search The list of nodes to be expanded is maintained in the form of a queue 1. Insert ROOT in queue Q 2. Remove head H from Q 3. If state(h) is a goal state then exit else expand H 4. Insert all successors of H in Q 5. Go back to 2 Drawback: BFS will always find the goal node at the shallowest depth Not necessarily optimal Optimal only if the path-cost is a non-decreasing function of the depth or if all actions have the same cost All generated nodes must remain in memory O(b d+l ) time and space requirements

10 Uniform cost search Similar to BFS but expand node with the lowest path cost The list of nodes to be expanded is maintained in the form of a priority queue 1. Insert ROOT in priority queue Q 2. Remove node with highest priority (lowest cost) N from Q 3. If state(n) is a goal state then exit else expand N 4. Insert all successors of N in Q 5. Go back to 2 Might get stuck in an infinite loop if it expands a node that has a zero-cost action leading back to the same state ensure no zero cost operation Greater time complexity because it searches small steps before exploring paths with larger and possibly more useful steps

11 Depth First Search Always expand the deepest node in the current fringe first The list of nodes to be expanded is maintained as a stack 1. Insert ROOT in stack S 2. Remove top node N from the stack S 3. If state(n) is a goal state then exit else expand N 4. Insert all successors of N in S 5. Go back to 2 Time complexity: O(b m ); Space complexity of O(bm) In DFS with backtracking only one successor is generated at a time rather than all successors each partially expanded node remembers which successor to generate next. Drawback: If wrong choice is made then possible to get stuck going down a very long (or even infinite) path when a different choice would lead to a solution near the root of the search tree.

12 Depth Limited Search Similar to DFS but with a predetermined depth limit l Solves the infinite-path problem 1. Insert ROOT in stack S 2. Remove top node N from the stack S 3. If state(n) is a goal state then exit else go to 4 4. If depth(n) < l expand N else go back to 2 5. Insert all successors of N in S 6. Go back to 2 Drawback: Problem of incompleteness if shallowest goal is at depth d<l

13 Iterative deepening DFS Performs a depth limited search by gradually increasing the depth limit until the goal at the shallowest depth is found 1. Perform Depth limited search for depth_limit = 0 2. If solution is found exit, else go to 3 3. Increase depth_limit by 1 4. Go back to 1 Generally, this is the preferred uninformed search method when there is a large search space and the depth of the solution is not known. Time complexity: O(b d ); Space complexity of O(bd) Iterative deepening analog to uniform-cost search Use increasing path-cost limits instead of increasing depth limits Iterative Lengthening Search

14 Bidirectional Search Run two simultaneous searches one forward from the initial state the other backward from the goal Stop when the two searches meet check each node before it is expanded to see if it is in the fringe of check each node before it is expanded to see if it is in the fringe of the other search tree Searching backwards is not so easy Predecessor(x) function that computes all states that have x as a successor Goal state description matches a large set of goal states (Checkmate states in Chess!)

15 Repeated States Possible wastage of time states that have already been encountered and expanded before, are expanded again 8-queens problem Efficient if formulated such that each new queen is placed in the leftmost empty column no possibility of repeated states Not efficient if formulated such that each new queen can be placed anywhere repeated states make it possible to reach solution using n! paths Reversible actions => repeated states Prune repeated states Compare the node about to be expanded to those that have been expanded already; discard one of the two paths Tradeoff between memory and time requirements

16 Partial Information If the agent does not know exactly which state it is in or the results of each possible action Sensorless problem: No sensors to detect current state Search in the space of belief states Contingency problem: If environment is partially Contingency problem: If environment is partially observable and effects of actions are not completely certain The agent must explore to discover them