Uninformed Search ECE457 Applied Artificial Intelligence Spring 2007 Lecture #2

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 2 Outline Problem-solving by searching Uninformed search techniques Russell & Norvig, chapter 3

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 3 Problem-solving by searching An agent needs to perform actions to get from its current state to a goal. This process is called searching. Central in many AI systems Theorem proving, VLSI layout, game playing, navigation, scheduling, etc.

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 4 Requirements for searching Define the problem Represent the search space by states Define the actions the agent can perform and their cost Define a goal What is the agent searching for? Define the solution The goal itself? The path (i.e. sequence of actions) to get to the goal?

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 5 Assumptions Goal-based agent Environment Fully observable Deterministic Sequential Static Discrete Single agent

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 6 Formulating problems A well-defined problem has: An initial state A set of actions A goal test A concept of cost

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 7 Example: 8-puzzle Initial state Action Move blank left, right, up or down, provided it does not get out of the game Goal test Are the tiles in the “goal state” order? Cost Each move costs 1 Path cost is the sum of moves

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 8 Example: 8-puzzle leftdown left right down left up

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 9 Search Tree Parent Child Edge (action) Node (state) Expanding a node Root Leaf Fringe Branching factor (b) Maximum depth (m)

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 10 Properties of Search Algos. Completeness Is the algorithm guaranteed to find a goal node, if one exists? Optimality Is the algorithm guaranteed to find the best goal node, i.e. the one with the cheapest path cost? Time complexity How many nodes are generated? Space complexity What’s the maximum number of nodes stored in memory?

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 11 Types of Search Uninformed Search Only has the information provided by the problem formulation (initial state, set of actions, goal test, cost) Informed Search Has additional information that allows it to judge the promise of an action, i.e. the estimated cost from a state to a goal

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 12 Breath-First Search

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 13 Breath-First Search Complete, if b is finite Optimal, if path cost is equal to depth Guaranteed to return the shallowest goal (depth d) Number of generated nodes: 1+b+b²+b³+…+b d +(b d+1 -b) = O(b d+1 ) Time complexity = O(b d+1 ) Space complexity = O(b d+1 )

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 14 Uniform-Cost Search Expansion of Breath-First Search Explore the cheapest node first (in terms of path cost) Condition: No zero-cost or negative-cost edges. Minimum cost is є

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 15 Uniform-Cost Search Complete given a finite tree Optimal Time complexity = O(b  C*/є  ) ≥ O(b d+1 ) Space complexity = O(b  C*/є  ) ≥ O(b d+1 )

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 16 Depth-First Search

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 17 Depth-First Search Complete, if m is finite Not optimal Time complexity = O(b m ) Space complexity = bm+1 = O(bm) Can be reduced to O(m)

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 18 Depth-Limited Search Depth-First Search with depth limit l Avoids problems of Depth-First Search when trees are unbounded Depth-First Search is Depth-Limited Search with l = 

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 19 Depth-Limited Search Complete, if l > d Not optimal Time complexity = O(b l ) Space complexity = O(bl)

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 20 Iterative Deepening Search Depth-First Search with increasing depth limit l Repeat depth-limited search over and over, with l = l + 1 Avoids problems of Depth-First Search when trees are unbounded

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 21 Iterative Deepening Search Complete, if b is finite Optimal, if path cost is equal to depth Guaranteed to return the shallowest goal Time complexity = O(b d ) Space complexity = O(bd) Nodes on levels above d are generated multiple times

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 22 Repeated States leftdown left right down left up Example: 8-puzzle

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 23 Repeated States Unavoidable in problems where: Actions are reversible Multiple paths to the same state are possible Can greatly increase the number of nodes in a tree Or even make a finite tree infinite! Maintain a closed list of visited states Detect repeated states Increase space complexity

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 24 Summary / Example Going from Arad to Bucharest

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 25 Summary / Example Initial state Being in Arad Action Move to a neighbouring city, if a road exists. Goal test Are we in Bucharest? Cost Move cost = distance between cities Path cost = distance travelled since Arad

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 26 Summary / Example Breath-First Search

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 27 Summary / Example Uniform-Cost Search

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 28 Summary / Example Depth-First Search

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 29 Summary / Example Depth-Limited Search, l = 4

ECE457 Applied Artificial Intelligence R. Khoury (2007)Page 30 Summary / Example Iterative Deepening Search