Solving problems by searching Chapter 3 Some slide credits to Hwee - PowerPoint PPT Presentation

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