Solving problems by searching Chapter 3 Some slide credits to Hwee Tou Ng (Singapore)
Outline Problem-solving agents Problem types Problem formulation Example problems Basic search algorithms Heuristics 2013 CS 325 - Ch3 Search 2
Intelligent agent solves problems by? 2013 CS 325 - Ch3 Search 3
Problem-solving agents 2013 CS 325 - Ch3 Search 4
Example: Romania On holiday in Romania; currently in Arad. Flight leaves tomorrow from Bucharest Formulate goal: be in Bucharest Formulate problem: states: various cities actions: drive between cities Find solution: sequence of cities, e.g., Arad, Sibiu, Fagaras, Bucharest 2013 CS 325 - Ch3 Search 5
Example: Romania 2013 CS 325 - Ch3 Search 6
Romania: problem type? 2013 CS 325 - Ch3 Search 7
Romania: Problem type Deterministic, fully observable single-state problem Agent knows exactly which state it will be in; solution is a sequence Non-observable sensorless problem (conformant problem) Agent may have no idea where it is; solution is a sequence Nondeterministic and/or partially observable contingency problem percepts provide new information about current state often interleave} search, execution Unknown state space exploration problem 2013 CS 325 - Ch3 Search 8
Single-state problem formulation A problem is defined by four items: initial state e.g., "at Arad" 1. actions or successor function S(x) = set of action–state pairs 2. e.g., S(Arad) = { <Arad Zerind, Zerind>, … } goal test, can be 1. explicit, e.g., x = "at Bucharest" implicit, e.g., Checkmate(x) path cost (additive) 1. e.g., sum of distances, number of actions executed, etc. c(x,a,y) is the step cost, assumed to be ≥ 0 A solution is a sequence of actions leading from the initial state to a goal state 2013 CS 325 - Ch3 Search 9
Tree search algorithms Basic idea: offline, simulated exploration of state space by generating successors of already-explored states (a.k.a. expanding states) 2013 CS 325 - Ch3 Search 10
Tree search example 2013 CS 325 - Ch3 Search 11
Tree search example 2013 CS 325 - Ch3 Search 12
Tree search example 2013 CS 325 - Ch3 Search 13
Implementation: general tree search 2013 CS 325 - Ch3 Search 14
Uninformed search strategies Uninformed search strategies use only the information available in the problem definition Breadth-first search Uniform-cost search Depth-first search Depth-limited search Iterative deepening search 2013 CS 325 - Ch3 Search 15
Breadth-first search Expand shallowest unexpanded node Implementation: fringe is a FIFO queue, i.e., new successors go at end 2013 CS 325 - Ch3 Search 16
Breadth-first search Expand shallowest unexpanded node Implementation: fringe is a FIFO queue, i.e., new successors go at end 2013 CS 325 - Ch3 Search 17
Breadth-first search Expand shallowest unexpanded node Implementation: fringe is a FIFO queue, i.e., new successors go at end 2013 CS 325 - Ch3 Search 18
Breadth-first search Expand shallowest unexpanded node Implementation: fringe is a FIFO queue, i.e., new successors go at end 2013 CS 325 - Ch3 Search 19
Example: Romania (Q) 2013 CS 325 - Ch3 Search 20
Search strategies A search strategy is defined by picking the order of node expansion Strategies are evaluated along the following dimensions: completeness: does it always find a solution if one exists? time complexity: number of nodes generated space complexity: maximum number of nodes in memory optimality: does it always find a least-cost solution? Time and space complexity are measured in terms of b: maximum branching factor of the search tree d: depth of the least-cost solution m : maximum depth of the state space (may be ∞) 2013 CS 325 - Ch3 Search 21
Properties of breadth-first search Complete? Yes (if b is finite) Time? 1+b+b 2 +b 3 +…+ b d + b(b d -1 ) = O (b d+1 ) Space? O(b d+1 ) (keeps every node in memory) Optimal? Yes (if cost = 1 per step) Space is the bigger problem (more than time) 2013 CS 325 - Ch3 Search 22
Uniform-cost search Video 2013 CS 325 - Ch3 Search 23
Uniform-cost search Expand least-cost unexpanded node Implementation: fringe = queue ordered by path cost Equivalent to breadth-first if step costs all equal Complete? Yes, if step cost ≥ ε Time? # of nodes with g ≤ cost of optimal solution, O(b ceiling(C*/ ε) ) where C * is the cost of the optimal solution Space? # of nodes with g ≤ cost of optimal solution, O(b ceiling(C*/ ε) ) Optimal? Yes – nodes expanded in increasing order of g(n) 2013 CS 325 - Ch3 Search 24
Comparison of Searches So far: Breadth-first search Uniform-cost (cheapest) search New: Depth-first search Optimal? 2013 CS 325 - Ch3 Search 25
Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front 2013 CS 325 - Ch3 Search 26
Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front 2013 CS 325 - Ch3 Search 27
Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front 2013 CS 325 - Ch3 Search 28
Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front 2013 CS 325 - Ch3 Search 29
Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front 2013 CS 325 - Ch3 Search 30
Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front 2013 CS 325 - Ch3 Search 31
Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front 2013 CS 325 - Ch3 Search 32
Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front 2013 CS 325 - Ch3 Search 33
Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front 2013 CS 325 - Ch3 Search 34
Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front 2013 CS 325 - Ch3 Search 35
Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front 2013 CS 325 - Ch3 Search 36
Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front 2013 CS 325 - Ch3 Search 37
Why depth-first? 2013 CS 325 - Ch3 Search 38
Properties of depth-first search Complete? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path complete in finite spaces Time? O(b m ) : terrible if m >> d but if solutions are dense, may be much faster than breadth-first Space? O(bm), i.e., linear space! Optimal? No 2013 CS 325 - Ch3 Search 39
Summary of algorithms 2013 CS 325 - Ch3 Search 40
Limitations Video 2013 CS 325 - Ch3 Search 41
Best-first search Idea: use an evaluation function f(n) for each node estimate of "desirability" Expand most desirable unexpanded node Implementation: Order the nodes in fringe in decreasing order of desirability Special cases: greedy best-first search A * search
Romania with step costs in km
Greedy best-first search Evaluation function f(n) = h(n) (heuristic) = estimate of cost from n to goal e.g., h SLD (n) = straight-line distance from n to Bucharest Greedy best-first search expands the node that appears to be closest to goal
Greedy best-first search example
Greedy best-first search example
Greedy best-first search example
Greedy best-first search example
Properties of greedy best- first search • Complete? No – can get stuck in loops, e.g., Iasi Neamt Iasi Neamt • Time? O(b m ) , but a good heuristic can give dramatic improvement • Space? O(b m ) -- keeps all nodes in memory • Optimal? No
A * search Idea: avoid expanding paths that are already expensive Evaluation function f(n) = g(n) + h(n) g(n) = cost so far to reach n h(n) = estimated cost from n to goal f(n) = estimated total cost of path through n to goal
A * search example
A * search example
A * search example
A * search example
A * search example
A * search example
State Spaces
Example: vacuum world Single-state, start in #5. Solution? 2013 CS 325 - Ch3 Search 58
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