artificial intelligence chapter 3: problem solving and searching

8/18/2019 Artificial Intelligence Chapter 3: Problem Solving and Searching

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Artificial IntelligenceProblem Solving and SearchingDr. Zeeshan Bhatti

BSSW-PIVChapter 3

Institute of Information and Communication TechnologyUniversity of Sindh, Jamshoro By: Dr. Zeeshan Bhat ti 11BY: DR. ZEESHAN BHATTI


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Last time: Summary

•Definition of AI?•Turing Test?• Intelligent Agents:

•Anything that can be viewed as perceiving its environment throughsensors and acting upon that environment through its effectors tomaximize progress towards its goals .•PAGE (Percepts, Actions, Goals, Environment)•Described as a Perception (sequence) to Action Mapping: f : P * A •Using look-up-table, closed form, etc.

• Agent Types: Reflex, state-based, goal-based, utility-based

•Rational Action: The action that maximizes the expected value of

the performance measure given the percept sequence to date


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Outline: Problem solving and search

• Introduction to Problem Solving

•Complexity

•Uninformed search•Problem formulation•Search strategies: depth-first, breadth-first

• Informed search•Search strategies: best-first, A*•Heuristic functions


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Example: Measuring problem!

Problem: Using these three buckets,measure 7 liters of water.

3 l 5 l 9 l


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• (one possible) Solution:

a b c0 0 0

start3 0 00 0 33 0 30 0 63 0 60 3 63 3 61 5 60 5 7

goal

3 l 5 l 9 l

a b c


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a b c0 0 0

start3 0 00 0 33 0 30 0 63 0 60 3 63 3 61 5 60 5 7

goal

3 l 5 l 9 l

a b c


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a b c0 0 0

start3 0 00 0 33 0 30 0 63 0 60 3 63 3 61 5 60 5 7

goal

3 l 5 l 9 l

a b c


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a b c0 0 0

start3 0 00 0 33 0 30 0 63 0 60 3 63 3 61 5 60 5 7

goal

3 l 5 l 9 l

a b c


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a b c0 0 0

start3 0 00 0 33 0 30 0 63 0 60 3 63 3 61 5 60 5 7

goal

3 l 5 l 9 l

a b c


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a b c0 0 0

start3 0 00 0 33 0 30 0 63 0 60 3 63 3 61 5 60 5 7

goal

3 l 5 l 9 l

a b c


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a b c0 0 0

start3 0 00 0 33 0 30 0 63 0 60 3 63 3 61 5 60 5 7

goal

3 l 5 l 9 l

a b c


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a b c0 0 0

start3 0 00 0 33 0 30 0 63 0 60 3 63 3 61 5 60 5 7

goal

3 l 5 l 9 l

a b c


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a b c0 0 0

start3 0 00 0 33 0 30 0 63 0 60 3 63 3 61 5 60 5 7

goal

3 l 5 l 9 l

a b c


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a b c0 0 0

start3 0 00 0 33 0 30 0 63 0 60 3 63 3 61 5 60 5 7

goal

3 l 5 l 9 l

a b c


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• Another Solution:

a b c0 0 0

start0 5 00 0 3

3 0 30 0 63 0 60 3 63 3 61 5 60 5 7

goal

3 l 5 l 9 l

a b c


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a b c0 0 0

start0 5 03 2 00 0 33 0 30 0 63 0 60 3 63 3 61 5 60 5 7

goal

3 l 5 l 9 l

a b c


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a b c0 0 0

start0 5 03 2 03 0 23 0 30 0 63 0 60 3 63 3 61 5 60 5 7

goal

3 l 5 l 9 l

a b c


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a b c0 0 0

start0 5 03 2 03 0 23 5 20 0 63 0 60 3 63 3 61 5 60 5 7

goal

3 l 5 l 9 l

a b c


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a b c0 0 0

start0 5 03 2 03 0 23 5 23 0 7

goal3 0 60 3 63 3 61 5 60 5 7

goal

3 l 5 l 9 l

a b c


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Which solution do we prefer?

•Solution 1:a b c

0 0 0start

3 0 00 0 33 0 30 0 63 0 60 3 63 3 61 5 60 5 7

goal

•Solution 2:

a b c

0 0 0start

0 5 03 2 03 0 23 5 23 0 7

goal


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Problem Solving Agents

• Intelligent agents are supposed to maximizetheir performance measure.

• Achieving this is sometimes simplified if theagent can adopt a goal and aim at satisfying it.

Goal formulation , based on the current situationand the agent’s performance measure, is the first

step in problem solving.

DZB1


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Slide 21

DZB1 Dr. Zeeshan Bhatti, 14/3/2016


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Example: Romania


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Example: Romania


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Problem-Solving Agent

Note: This is offline problem-solving. Online problem-solving involvesacting w/o complete knowledge of the problem and environment

action

// From LA to San Diego (given curr. state)// e.g., Gas usage

// What is the current state?

// If fails to reach goal, update

Restricted form of general agent:


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Example: Buckets

Measure 7 liters of water using a 3-liter, a 5-liter, and a 9-liter buckets.

•Formulate goal: Have 7 liters of waterin 9-liter bucket

•Formulate problem:•States: amount of water in the buckets•Operators: Fill bucket from source, empty bucket

•Find solution: sequence of operators that bring you

from current state to the goalstate


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Problem types


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Problem types

•Single-state problem: deterministic, accessible Agent knows everything about world, thus can calculate optimal action sequence to reach goal state.

•Multiple-state problem: deterministic, inaccessible Agent must reason about sequences of actions and states assumed while working towards goal state.

•Contingency problem: nondeterministic, inaccessible•Must use sensors during execution •Solution is a tree or policy •Often interleave search and execution

•Exploration problem: unknown state spaceDiscover and learn about environment while taking actions.


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Problem types

•Single-state problem: deterministic, accessible

•Agent knows everything about world (the exact state),

•Can calculate optimal action sequence to reach goal state.

•E.g., playing chess. Any action will result in an exact state


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Problem types

•Contingency problem: nondeterministic, inaccessible

•Must use sensors during execution•Solution is a tree or policy•Often interleave search and execution

•E.g., a new skater in an arena•Sliding problem.•Many skaters around


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Problem types

•Exploration problem: unknown state space

Discover and learn about environment whiletaking actions.

•E.g., Maze


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Example: Vacuum world

Simplified world: 2 locations, each may or not contain dirt,each may or not contain vacuuming agent.

Goal of agent: clean up the dirt.


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Example: vacuum world

• Single-state , start in #5.Solution?

•

By: Dr. Zeeshan Bhatti


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• Single-state , start in #5.Solution? [Right, Suck]

•

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


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

•

• Contingency• Nondeterministic: Suck may

dirty a clean carpet

• Partially observable: location, dirt at current location.• Percept: [L, Clean], i.e., start in #5 or #7

Solution?


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

•• Contingency

• Nondeterministic: Suck maydirty a clean carpet

• Partially observable: location, dirt at current location.• Percept: [L, Clean], i.e., start in #5 or #7

Solution? [Right, if dirt then Suck]


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Single-state problem formulation

A problem is defined by four items:

1. initial state e.g., "at Arad"2. actions or successor function S(x) = set of action–state pairs

• e.g., S(Arad) = {, … }

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.• 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 agoal state


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Selecting a state space• Real 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

• Each abstract action should be "easier" than the original problem


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Vacuum world state space graph

• states?• actions?

• goal test?• path cost?•


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Vacuum world state space graph

• states? integer dirt and robot location

• actions? Left , Right , Suck • goal test? no dirt at all locations• path cost? 1 per action


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Example: The 8-puzzle

• states?• actions?• goal test?• path cost?


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Example: The 8-puzzle

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

• path cost? 1 per move•

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


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Example: robotic assembly

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

• actions? : continuous motions of robot joints

• goal test? : complete assembly

• path cost? : time to execute


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Tree search algorithms

• Basic idea:• offline, simulated exploration of state space by generating

successors of already-explored states (a.k.a.~ expandingstates)

•


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Example: Romania

•In Romania, on vacation. Currently in Arad.•Flight leaves tomorrow from Bucharest.

•Formulate goal:be in Bucharest

•Formulate problem:states: various citiesoperators: drive between cities

•Find solution:

sequence of cities, such that total driving distance isminimized.


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Example: Traveling from Arad To Bucharest


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Tree search example


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Tree search example


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Tree search example


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Implementation: general tree search


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By: Dr. Zeeshan Bhatti 52

Implementation: states vs. nodes

• A state is a (representation of) a physical configuration• A node is a data structure constituting part of a search tree

includes state , parent node , action , path cost g(x) , depth

• The Expand function creates new nodes, filling in thevarious fields and using the SuccessorFn of the problem tocreate the corresponding states.

•


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Binary Tree Example

root

N1 N2

N3 N4 N5 N6

Number of nodes: n = 2 max depthNumber of levels (max depth) = log(n) (could be n)

Depth = 0

Depth = 1

Depth = 2


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Uninformed search strategies

• Uninformed search strategies use only theinformation available in the problem definition

•• Breadth-first search•• Uniform-cost search•• Depth-first search•• Depth-limited search•


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Breadth-first search

• Expand shallowest unexpanded node•• Implementation :

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


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

•


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Properties of breadth-first search

• Complete? Yes (if b is finite)•• Time? 1+b+b 2 +b 3+… + bd + 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)


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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) )


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Depth-first search

• Expand deepest unexpanded node•• Implementation :

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

•


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Depth-first search



•


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Depth-first search



•


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Depth-first search



•


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Depth-first search



•


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Depth-first search



•


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Depth-first search



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Depth-first search



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Depth-first search



•


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Properties of depth-first search

• Complete? No: fails in infinite-depth spaces, spaceswith loops

• Modify to avoid repeated states along path•

complete in finite spaces

• Time? O(b m ): terrible if m is much larger than d • but if solutions are dense, may be much faster than

breadth-first•

• Space? O(bm), i.e., linear space!•• O timal? No


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Depth-limited search

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

• Recursive implementation :


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Iterative deepening search


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Iterative deepening search l =0


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Iterative deepening search

• Number of nodes generated in a depth-limited search todepth d with branching factor b:N DLS = b 0 + b 1 + b 2 + … + b d-2 + b d-1 + b d

• Number of nodes generated in an iterative deepening search

to depth d with branching factor b:NIDS = (d+1)b 0 + d b^ 1 + (d-1)b^ 2 + … + 3b d-2 +2b d-1 + 1b d

• For b = 10 , d = 5 ,•

• NDLS = 1 + 10 + 100 + 1,000 + 10,000 + 100,000 = 111,111•• N IDS = 6 + 50 + 400 + 3,000 + 20,000 + 100,000 = 123,456•

d h


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deepening search

• Complete? Yes•• Time? (d+1)b 0 + d b 1 + (d-1)b 2 + … + b d = O(b d )•• Space? O(bd)•• Optimal? Yes, if step cost = 1

f l h


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Summary of algorithms

R d


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Repeated states

• Failure to detect repeated states can turn a linearproblem into an exponential one!

•

G h h


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Graph search

S


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Summary

• Problem formulation usually requires abstracting away real-world details to define a state space that can feasibly beexplored

• Variety of uninformed search strategies

• Iterative deepening search uses only linear space and notmuch more time than other uninformed algorithms


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Complexity

•Why worry about complexity of algorithms?

because a problem may be solvable in principle but maytake too long to solve in practice


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Complexity example: Traveling Salesman Problem

This is a hard problem: the only known algorithms (so far)to solve it have exponential complexity, that is, the numberof operations required to solve it grows as exp(n) for n cities.


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Why is exponential complexity “hard”?

It means that the number of operations necessary tocompute the exact solution of the problem growsexponentially with the size of the problem (here, thenumber of cities).

•exp(1) = 2.72

•exp(10) = 2.20 10 4 (daily salesman trip)•exp(100) = 2.69 10 43 (monthly salesman planning)•exp(500) = 1.40 10 217 (music band worldwide tour)•exp(250,000) = 10 108,573 (fedex, postal services)•Fastest

computer = 10 12 operations/second


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So…

In general, exponential-complexity problems cannot besolved for any but the smallest instances!

artificial intelligence chapter 3: problem solving and searching

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