10/11/2010

1

Solving problems by searchingSolving problems by searching

Chapter 3

14 Jan 2004 CS 3243 - Blind Search 1

OutlineProblem-solving agentsProblem formulationExample problemsBasic search algorithmsProblem types

14 Jan 2004 CS 3243 - Blind Search 2

10/11/2010

2

Problem-solving agents

14 Jan 2004 CS 3243 - Blind Search 3

Example: RomaniaOn holiday in Romania; currently in Arad.Flight leaves tomorrow from BucharestF l t lFormulate goal:

be in BucharestFormulate problem:

states: various citiesactions: drive between cities

14 Jan 2004 CS 3243 - Blind Search 4

Find solution:sequence of cities, e.g., Arad, Sibiu, Fagaras, Bucharest

10/11/2010

3

Example: Romania

14 Jan 2004 CS 3243 - Blind Search 5

Single-state problem formulationA problem is defined by four items:

1. initial state e.g., "at Arad"actions or successor function S(x) set of action state2. actions or successor function S(x) = set of action–state pairs

e.g., S(Arad) = {<Arad Zerind, Zerind>, … }3. goal test, can be

explicit, e.g., x = "at Bucharest"implicit, e.g., Checkmate(x)

4. path cost (additive)e g sum of distances number of actions executed etc

14 Jan 2004 CS 3243 - Blind Search 6

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

10/11/2010

4

Selecting a state spaceReal world is absurdly complex

state space must be abstracted for problem solving

(Abstract) state = set of real states(Abstract) action = complex combination of real actions

e.g., "Arad Zerind" represents a complex set of possible routes, detours, rest stops, etc.

For guaranteed realizability, any real state "in Arad“ must get to some real state "in Zerind"(Abstract) solution =

set of real paths that are solutions in the real world

14 Jan 2004 CS 3243 - Blind Search 7

set of real paths that are solutions in the real worldEach abstract action should be "easier" than the original problem

Vacuum world state space graph

states?actions?

14 Jan 2004 CS 3243 - Blind Search 8

actions?goal test?path cost?

10/11/2010

5

Vacuum world state space graph

states? integer dirt and robot location

14 Jan 2004 CS 3243 - Blind Search 9

states? integer dirt and robot locationactions? Left, Right, Suckgoal test? no dirt at all locationspath cost? 1 per action

Example: The 8-puzzle

states?actions?

14 Jan 2004 CS 3243 - Blind Search 10

goal test?path cost?

10/11/2010

6

Example: The 8-puzzle

states? locations of tiles actions? move blank left, right, up, down goal test? = goal state (given)

14 Jan 2004 CS 3243 - Blind Search 11

goal test? goal state (given)path cost? 1 per move

[Note: optimal solution of n-Puzzle family is NP-hard]

Example: robotic assembly

states?: real-valued coordinates of robot joint angles parts of the object to be assembled

14 Jan 2004 CS 3243 - Blind Search 12

actions?: continuous motions of robot jointsgoal test?: complete assemblypath cost?: time to execute

10/11/2010

7

Tree search algorithmsBasic idea:

offline, simulated exploration of state space by generating successors of already-explored statesgenerating successors of already explored states (a.k.a.~expanding states)

14 Jan 2004 CS 3243 - Blind Search 13

Tree search example

14 Jan 2004 CS 3243 - Blind Search 14

10/11/2010

8

Implementation: states vs. nodesA state is a (representation of) a physical configurationA node is a data structure constituting part of a search tree includes state, parent node, action, path cost g(x), depthdepth

14 Jan 2004 CS 3243 - Blind Search 15

The Expand function creates new nodes, filling in the various fields and using the SuccessorFn of the problem to create the corresponding states.

Search strategiesA search strategy is defined by picking the order of node expansionStrategies are evaluated along the following dimensions:

l t d it l fi d l ti if i t ?completeness: does it always find a solution if one exists?time complexity: number of nodes generatedspace complexity: maximum number of nodes in memoryoptimality: does it always find a least-cost solution?

Time and space complexity are measured in terms of b: maximum branching factor of the search treed: depth of the least-cost solution

14 Jan 2004 CS 3243 - Blind Search 16

pm: maximum depth of the state space (may be ∞)

10/11/2010

9

Uninformed search strategiesUninformed search strategies use only the information available in the problem definitiondefinitionBreadth-first searchUniform-cost searchDepth-first searchD th li it d h

14 Jan 2004 CS 3243 - Blind Search 17

Depth-limited searchIterative deepening search

Breadth-first searchExpand shallowest unexpanded nodeImplementation:

fringe is a FIFO queue, i.e., new successors go at end

14 Jan 2004 CS 3243 - Blind Search 18

10/11/2010

10

Breadth-first searchExpand shallowest unexpanded nodeImplementation:

fringe is a FIFO queue, i.e., new successors go at end

14 Jan 2004 CS 3243 - Blind Search 19

Breadth-first searchExpand shallowest unexpanded nodeImplementation:

fringe is a FIFO queue, i.e., new successors go at end

14 Jan 2004 CS 3243 - Blind Search 20

10/11/2010

11

Breadth-first searchExpand shallowest unexpanded nodeImplementation:

fringe is a FIFO queue, i.e., new successors go at end

14 Jan 2004 CS 3243 - Blind Search 21

Properties of breadth-first searchComplete? Yes (if b is finite)Time? 1+b+b2+b3+… +bd + b(bd-1) = O(bd+1)Space? O(bd+1) (keeps every node in memory)Space? O(b ) (keeps every node in memory)Optimal? Yes (if cost = 1 per step)

Space is the bigger problem (more than time)

14 Jan 2004 CS 3243 - Blind Search 22

10/11/2010

12

Uniform-cost searchExpand least-cost unexpanded nodeImplementation:

fringe = queue ordered by path costEquivalent to breadth-first if step costs all equalComplete? Yes, if step cost ≥ εTime? # of nodes with g ≤ cost of optimal solution, O(bceiling(C*/ ε)) where C* is the cost of the optimal solutionSpace? # of nodes with g ≤ cost of optimal solution, O(bceiling(C*/ ε))

14 Jan 2004 CS 3243 - Blind Search 23

O(bceiling(C / ε))Optimal? Yes – nodes expanded in increasing order of g(n)

Depth-first searchExpand deepest unexpanded nodeImplementation:

fringe LIFO queue i e put successors at frontfringe = LIFO queue, i.e., put successors at front

14 Jan 2004 CS 3243 - Blind Search 24

10/11/2010

13

Depth-first searchExpand deepest unexpanded nodeImplementation:

fringe LIFO queue i e put successors at frontfringe = LIFO queue, i.e., put successors at front

14 Jan 2004 CS 3243 - Blind Search 25

Depth-first searchExpand deepest unexpanded nodeImplementation:

fringe LIFO queue i e put successors at frontfringe = LIFO queue, i.e., put successors at front

14 Jan 2004 CS 3243 - Blind Search 26

10/11/2010

14

Depth-first searchExpand deepest unexpanded nodeImplementation:

fringe LIFO queue i e put successors at frontfringe = LIFO queue, i.e., put successors at front

14 Jan 2004 CS 3243 - Blind Search 27

Depth-first searchExpand deepest unexpanded nodeImplementation:

fringe LIFO queue i e put successors at frontfringe = LIFO queue, i.e., put successors at front

14 Jan 2004 CS 3243 - Blind Search 28

10/11/2010

15

Depth-first searchExpand deepest unexpanded nodeImplementation:

fringe LIFO queue i e put successors at frontfringe = LIFO queue, i.e., put successors at front

14 Jan 2004 CS 3243 - Blind Search 29

Depth-first searchExpand deepest unexpanded nodeImplementation:

fringe LIFO queue i e put successors at frontfringe = LIFO queue, i.e., put successors at front

14 Jan 2004 CS 3243 - Blind Search 30

10/11/2010

16

Depth-first searchExpand deepest unexpanded nodeImplementation:

fringe LIFO queue i e put successors at frontfringe = LIFO queue, i.e., put successors at front

14 Jan 2004 CS 3243 - Blind Search 31

Depth-first searchExpand deepest unexpanded nodeImplementation:

fringe LIFO queue i e put successors at frontfringe = LIFO queue, i.e., put successors at front

14 Jan 2004 CS 3243 - Blind Search 32

10/11/2010

17

Depth-first searchExpand deepest unexpanded nodeImplementation:

fringe LIFO queue i e put successors at frontfringe = LIFO queue, i.e., put successors at front

14 Jan 2004 CS 3243 - Blind Search 33

Depth-first searchExpand deepest unexpanded nodeImplementation:

fringe LIFO queue i e put successors at frontfringe = LIFO queue, i.e., put successors at front

14 Jan 2004 CS 3243 - Blind Search 34

10/11/2010

18

Depth-first searchExpand deepest unexpanded nodeImplementation:

fringe LIFO queue i e put successors at frontfringe = LIFO queue, i.e., put successors at front

14 Jan 2004 CS 3243 - Blind Search 35

Properties of depth-first searchComplete? No: fails in infinite-depth spaces, spaces with loops

Modify to avoid repeated states along pathModify to avoid repeated states along pathcomplete in finite spaces

Time? O(bm): terrible if m is much larger than d

but if solutions are dense, may be much faster than breadth-first

14 Jan 2004 CS 3243 - Blind Search 36

faster than breadth firstSpace? O(bm), i.e., linear space!Optimal? No

10/11/2010

19

Depth-limited search= depth-first search with depth limit l,i.e., nodes at depth l have no successors

Recursive implementation:Recursive implementation:

14 Jan 2004 CS 3243 - Blind Search 37

Iterative deepening search

14 Jan 2004 CS 3243 - Blind Search 38

10/11/2010

20

Iterative deepening search l =0

14 Jan 2004 CS 3243 - Blind Search 39

Iterative deepening search l =1

14 Jan 2004 CS 3243 - Blind Search 40

10/11/2010

21

Iterative deepening search l =2

14 Jan 2004 CS 3243 - Blind Search 41

Iterative deepening search l =3

14 Jan 2004 CS 3243 - Blind Search 42

10/11/2010

22

Iterative deepening searchNumber of nodes generated in a depth-limited search to depth d with branching factor b:

NDLS = b0 + b1 + b2 + … + bd-2 + bd-1 + bdNumber of nodes generated in an iterative deepeningNumber of nodes generated in an iterative deepening search to depth d with branching factor b:

NIDS = (d+1)b0 + d b^1 + (d-1)b^2 + … + 3bd-2 +2bd-1 + 1bd

For b = 10, d = 5,NDLS = 1 + 10 + 100 + 1,000 + 10,000 + 100,000 = 111,111NIDS = 6 + 50 + 400 + 3,000 + 20,000 + 100,000 = 123,456

14 Jan 2004 CS 3243 - Blind Search 43

Overhead = (123,456 - 111,111)/111,111 = 11%

Properties of iterative deepening search

Complete? YesTime? (d+1)b0 + d b1 + (d-1)b2 + … + bd

(bd)= O(bd)Space? O(bd)Optimal? Yes, if step cost = 1

14 Jan 2004 CS 3243 - Blind Search 44

10/11/2010

23

Summary of algorithms

14 Jan 2004 CS 3243 - Blind Search 45

Repeated statesFailure to detect repeated states can turn a linear problem into an exponential one!

14 Jan 2004 CS 3243 - Blind Search 46

10/11/2010

24

Graph search

14 Jan 2004 CS 3243 - Blind Search 47

Problem typesDeterministic, fully observable single-state problem

Agent knows exactly which state it will be in; solution is a sequenceis a sequence

Non-observable sensorless problem (conformant problem)

Agent may have no idea where it is; solution is a sequence

Nondeterministic and/or partially observablecontingency problem

14 Jan 2004 CS 3243 - Blind Search 48

contingency problempercepts provide new information about current stateoften interleave} search, execution

Unknown state space exploration problem

10/11/2010

25

Example: vacuum worldSingle-state, start in #5. Solution?

14 Jan 2004 CS 3243 - Blind Search 49

Example: vacuum worldSingle-state, start in #5. Solution? [Right, Suck]

Sensorless, start inSensorless, start in {1,2,3,4,5,6,7,8} e.g., Right goes to {2,4,6,8} Solution?

14 Jan 2004 CS 3243 - Blind Search 50

10/11/2010

26

Example: vacuum worldSensorless, start in

{1,2,3,4,5,6,7,8} e.g., Right goes to {2,4,6,8} Solution?[Right,Suck,Left,Suck]

ContingencyNondeterministic: Suck may dirty a clean carpetPartially observable: location, dirt at current location.

14 Jan 2004 CS 3243 - Blind Search 51

Percept: [L, Clean], i.e., start in #5 or #7Solution?

Example: vacuum worldSensorless, start in

{1,2,3,4,5,6,7,8} e.g., Right goes to {2,4,6,8} Solution?[ h k f k][Right,Suck,Left,Suck]

ContingencyNondeterministic: Suckmay dirty a clean carpetPartially observable: location, dirt at current

14 Jan 2004 CS 3243 - Blind Search 52

,location.Percept: [L, Clean], i.e., start in #5 or #7Solution? [Right, if dirt then Suck]

10/11/2010

27

SummaryProblem formulation usually requires abstracting away real-world details to define a state space that can feasibly be explored

Variety of uninformed search strategies

Iterative deepening search uses only linear space and not much more time than other uninformed algorithms

14 Jan 2004 CS 3243 - Blind Search 53

AssignmentsFirst group (8 to 10:30)

3-6

Second group (10:30 to 13)

3-133 63-7 ( only a and c)3-83-11 ( except d)3-173-18

3 133-7 ( only b and d)3-83-11 ( except d)3-173-18

14 Jan 2004 CS 3243 - Blind Search 54

3-19 ( only a,b and e) 3-19 ( only a,b and e)