Uninformed Search ECE457 Applied Artificial Intelligence Spring 2008 Lecture #2

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 2 Outline Problem-solving by searching Uninformed search techniques Russell & Norvig, chapter 3

ECE457 Applied Artificial Intelligence R. Khoury (2008)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 (2008)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 (2008)Page 5 Assumptions Goal-based agent Environment Fully observable Deterministic Sequential Static Discrete Single agent

ECE457 Applied Artificial Intelligence R. Khoury (2008)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 (2008)Page 7 Well-Defined Problem Example 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 (2008)Page 8 Well-Defined Problem Example Travelling salesman problem Find the shortest round trip to visit each city exactly once Initial state Any city Set of actions Move to an unvisited city Goal test Is the agent in the initial city after having visited every city? Concept of cost Action cost: distance between cities Path cost: total distance travelled

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 9 Example: 8-puzzle leftdown left right down left up

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 10 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 (2008)Page 11 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 (2008)Page 12 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 (2008)Page 13 Breath-First Search

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 14 Breath-First Search Complete, if b is finite Optimal, if path cost is equal to depth Guaranteed to return the shallowest goal (depth d) Time complexity = O(b d+1 ) Space complexity = O(b d+1 )

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 15 Breath-First Search Upper-bound case: goal is last node of depth d Number of generated nodes: b+b²+b³+…+b d +(b d+1 -b) = O(b d+1 ) Space & time complexity: all generated nodes

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 16 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 Breath-First Search is Uniform-Cost Search with constant-cost edges

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 17 Uniform-Cost Search Complete given a finite tree Optimal Time complexity = O(b  C*/є  +1 ) ≥ O(b d+1 ) Space complexity = O(b  C*/є  +1 ) ≥ O(b d+1 )

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 18 Uniform-Cost Search Upper-bound case: goal has path cost C*, all other actions have minimum cost of є Depth explored before taking action C*:  C*/є  Depth of fringe nodes:  C*/є  + 1 Space & time complexity: all generated nodes: O(b  C*/є  +1 ) C* є єє єє єєєє єє єєєє

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 19 Depth-First Search

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 20 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) with recursive algorithm

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 21 Depth-First Search Upper-bound case for space: goal is last node of first branch After that, we start deleting nodes Number of generated nodes: b nodes at each of m levels Space complexity: all generated nodes = O(bm)

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 22 Depth-First Search Upper-bound case for time: goal is last node of last branch Number of nodes generated: b nodes for each node of m levels (entire tree) Time complexity: all generated nodes O(b m )

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 23 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 (2008)Page 24 Depth-Limited Search Complete, if l > d Not optimal Time complexity = O(b l ) Space complexity = O(bl)

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 25 Depth-Limited Search Upper-bound case for space: goal is last node of first branch After that, we start deleting nodes Number of generated nodes: b nodes at each of l levels Space complexity: all generated nodes = O(bl)

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 26 Depth-Limited Search Upper-bound case for time: goal is last node of last branch Number of nodes generated: b nodes for each node of l levels (entire tree to depth l) Time complexity: all generated nodes O(b l )

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 27 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 Avoids problem of Depth-Limited Search when goal depth d > l

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 28 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 (2008)Page 29 Iterative Deepening Search Upper-bound case for space: goal is last node of first branch After that, we start deleting nodes Number of generated nodes: b nodes at each of d levels Space complexity: all generated nodes = O(bd)

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 30 Iterative Deepening Search Upper-bound case for time: goal is last node of last branch Number of nodes generated: b nodes for each node of d levels (entire tree to depth d) Time complexity: all generated nodes O(b d )

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 31 Depth Searches Depth-first search Depth-limited search Iterative deepening search Depth limit mld Time complexity O(b m )O(b l )O(b d ) Space complexity O(bm)O(bl)O(bd)

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 32 Summary of Searches Breath- first Uniform Cost Depth -first Depth- limited Iterative deepening CompleteYes 1 No 4 No 5 Yes 1 OptimalYes 2 Yes 3 No Yes 2 TimeO(b d+1 )O(b  C*/є  +1 )O(b m )O(b l )O(b d ) SpaceO(b d+1 )O(b  C*/є  +1 )O(bm)O(bl)O(bd) 1: Assuming b finite (common in trees) 2: Assuming equal action costs 3: Assuming all costs  є 4: Unless m finite (uncommon in trees) 5: Unless l precisely selected

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 33 Summary / Example Going from Arad to Bucharest

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 34 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 (2008)Page 35 Summary / Example Breath-First Search

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 36 Summary / Example Uniform-Cost Search Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Fagaras 239 Rimnicu 220 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Fagaras 239 Rimnicu 220 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Fagaras 239 Rimnicu 220 Craiova 366 Pietsti 317 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Fagaras 239 Rimnicu 220 Craiova 366 Pietsti 317 Mehadia 299 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Fagaras 239 Rimnicu 220 Craiova 366 Pietsti 317 Mehadia 299 Bucharest 450 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Fagaras 239 Rimnicu 220 Craiova 366 Pietsti 317 Mehadia 299 Bucharest 450 Dobreta 374 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Fagaras 239 Rimnicu 220 Craiova 366 Pietsti 317 Mehadia 299 Bucharest 418 Dobreta 374 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Fagaras 239 Rimnicu 220 Craiova 366 Pietsti 317 Mehadia 299 Bucharest 418 Dobreta 374 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Fagaras 239 Rimnicu 220 Craiova 366 Pietsti 317 Mehadia 299 Bucharest 418 Dobreta 374 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Fagaras 239 Rimnicu 220 Craiova 366 Pietsti 317 Mehadia 299 Bucharest 418 Dobreta 374

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 37 Summary / Example Depth-First Search

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 38 Summary / Example Depth-Limited Search, l = 4

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 39 Summary / Example Iterative Deepening Search

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 40 Repeated States leftdown left right down left up Example: 8-puzzle

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 41 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!

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 42 Repeated States Each state generates a single child twice 26 different states 2 25 leaves (i.e. state Z) Over 67M nodes in the tree A B C D E A BB CCCC DDDDDDDD EEEEEEEEEEEEEEEE

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 43 Repeated States Maintain a closed list of visited states Closed list (for expanded nodes) vs. open list (for fringe nodes) Detect and discard repeated states upon generation Increases space complexity