Outline Uninformed Search Problem-solving by searching Uninformed search techniques Russell & Norvig, chapter 3 ECE457 Applied Artificial Intelligence Fall 2007 Lecture #2 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 2 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 3 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 4 1
Assumptions Goal-based agent Environment Fully observable Deterministic Sequential Static Discrete Single agent 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 5 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 6 Well-Defined Problem Example Initial state Action Move blank, right, up or, 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 7 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 (2007) Page 8 2
Example: 8-puzzle Search Tree Parent Root Child Node (state) Branching factor (b) right up Expanding a node Edge (action) Maximum depth Fringe (m) Leaf ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 9 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? 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 11 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 12 3
Breath-First Search 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 (2007) Page 13 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 14 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 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 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 16 4
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 ) 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 C*/є Number of generated nodes: O(b C*/є ) Space & time complexity: all generated nodes є є є є є є є є є є є є є є ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 17 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 18 Depth-First Search 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 (2007) Page 19 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 20 5
Depth-First Search Upper-bound case for space: goal is last node of first After that, we start deleting nodes Number of generated nodes: b nodes at each of m levels Space complexity: all generated nodes = O(bm) Depth-First Search Upper-bound case for time: goal is last node of last 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 (2007) Page 21 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 22 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 = 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 23 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 24 6
Depth-Limited Search Upper-bound case for space: goal is last node of first After that, we start deleting nodes Number of generated nodes: b nodes at each of l levels Space complexity: all generated nodes = O(bl) Depth-Limited Search Upper-bound case for time: goal is last node of last 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 (2007) Page 25 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 26 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 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 27 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 28 7
Iterative Deepening Search Upper-bound case for space: goal is last node of first After that, we start deleting nodes Number of generated nodes: b nodes at each of d levels Space complexity: all generated nodes = O(bd) Iterative Deepening Search Upper-bound case for time: goal is last node of last 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 (2007) Page 29 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 30 Depth Searches Summary of Searches Depth limit Time complexity Space complexity Depth-first search m O(b m ) O(bm) Depth-limited search l O(b l ) O(bl) Iterative deepening search d O(b d ) O(bd) 1: Assuming b finite (common in trees) 2: Assuming equal action costs 3: Assuming all costs є Breathfirst Uniform Cost Depth -first Complete Yes 1 Yes 1 No 4 Optimal Yes 2 Yes 3 No Time O(b d+1 ) O(b C*/є ) O(b m ) Space O(b d+1 ) O(b C*/є ) O(bm) Depthlimited No 5 No O(b l ) O(bl) Iterative deepening Yes 1 Yes 2 O(b d ) O(bd) 4: Unless m finite (uncommon in trees) 5: Unless l precisely selected ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 31 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 32 8
Summary / Example Going from Arad to Bucharest ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 33 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 34 Summary / Example Breath-First Search Summary / Example Uniform-Cost Search ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 35 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 36 9
Summary / Example Depth-First Search Summary / Example Depth-Limited Search, l = 4 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 37 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 38 Summary / Example Repeated Example: 8-puzzle States Iterative Deepening Search right up ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 39 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 40 10
Repeated States 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 (2007) Page 41 A B C D E B A B C C C C D D D D D D D D EEEEEEEE EEEEEEEE Each state generates a single child twice 26 different states 2 25 leaves (i.e. state Z) Over 67M nodes in the tree ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 42 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 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 43 11