Gabriela Ochoa goc/...9/12/2016 6 EXAMPLE: ROMANIA Google Map: Romania 11 ABSTRACTION: SELECTING A...

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ARTIFICIAL INTELLIGENCE (CSC9YE )

LECTURES 2 AND 3: PROBLEM SOLVING

BY SEARCH

Gabriela Ochoa

http://www.cs.stir.ac.uk/~goc/

OUTLINE

Problem solving by searching

Problem formulation

Example problems

Search algorithms (2 Lectures)

Uninformed search

Informed (heuristic) search

Artificial Intelligence: A Modern Approach

(Third edition) by Stuart Russell and Peter Norvig http://aima.cs.berkeley.edu/

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http://www.cs.stir.ac.uk/~goc/http://aima.cs.berkeley.edu/http://aima.cs.berkeley.edu/

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PRELIMINARIES

Definition of search Cambridge Dictionary

to look somewhere carefully in order to find something

to try to find the answer to a problem

Meanings of search in computing science (next slide)

More Robot Demos

Asimo. Developed by Honda Research and Development:

http://world.honda.com/ASIMO/technology/2011/intelligence/index

.html

https://www.youtube.com/watch?v=JlRPICfnmhw

Boston Robotics

http://www.bostondynamics.com/index.html

https://www.youtube.com/user/BostonDynamics

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SEARCH IN COMPUTING SCIENCE

1. Search for stored data

• Finding information stored in disc or

memory.

• Examples: Sequential search, Binary

search

2. Search for web documents

• Finding information on the world

wide web

• Results are presented as a list of

results

3. Search for paths or routes

• Finding a set of actions that will

bring us from an initial stat to a goal

stat

• Relevant to AI

• Examples: depth first search, breath

first search, branch and bound, A*,

Monte Carlo tree search.

4. Search for solutions

• Find a solution in a large space of

candidate solutions

• Relevant to AI, Optimisation, OR

• Examples: evolutionary algorithms,

Tabu search, simulated annealing,

ant colony optimisation, etc.

At least 4 meanings of the word search in CS

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http://world.honda.com/ASIMO/technology/2011/intelligence/index.htmlhttp://world.honda.com/ASIMO/technology/2011/intelligence/index.htmlhttp://world.honda.com/ASIMO/technology/2011/intelligence/index.htmlhttps://www.youtube.com/watch?v=JlRPICfnmhwhttps://www.youtube.com/watch?v=JlRPICfnmhwhttp://www.bostondynamics.com/index.htmlhttp://www.bostondynamics.com/index.html

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EXAMPLES OF SEARCH PROBLEMS

A robot vehicle would search for a route to a

given destination.

An automated air traffic controller would search

for a safe landing sequence for a set of incoming

planes

In games of strategy, such as chess or checkers:

search for a sequence of moves to beat your

opponent

More generally: trying to find a particular object

from a large number of such objects.

Search problems are common in AI: Planning and

Learning.

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SOLVING PROBLEMS BY SEARCHING

Problem-solving agents decide what to do by finding

sequences of actions that lead to desirable states

What is a problem and what is a solution?

Problem: a goal and a set of means to achieve it

Solution: a sequence of actions to achieve that goal

Given a precise definition of problem, it is possible to

construct a search process for finding solutions

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EXAMPLE: PAC-MAN

An arcade game developed by Namco

First released in Japan in May 1980

Google Doodle PAC-MAN

e-chalk version in Macromedia (PAC-MAN)

Artificial Intelligence Problem:

Develop an automatic player for PAC-MAN

Formulate it as a Search Problem

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SEARCH PROBLEMS A search problem consists of:

A state space

A successor function (with actions, costs)

A start state and a goal test

A solution is a sequence of actions (a plan) which transforms the start state to a goal state

“N”, 1.0

“E”, 1.0

https://www.google.co.uk/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=pacman gamehttps://www.google.co.uk/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=pacman gamehttps://www.google.co.uk/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=pacman gamehttp://www.echalk.co.uk/amusements/Games/pacman/pacman.htmlhttp://www.echalk.co.uk/amusements/Games/pacman/pacman.htmlhttp://www.echalk.co.uk/amusements/Games/pacman/pacman.html

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WHAT’S IN A STATE SPACE?

Problem: Pathing States: (x,y) location

Actions: NSEW

Successor: update location only

Goal test: is (x,y)=END

Problem: Eat-All-Dots States: {(x,y), dot

booleans}

Actions: NSEW

Successor: update location and possibly a dot boolean

Goal test: dots all false

The world state includes every last detail of the environment

A search state keeps only the details needed for planning (abstraction)

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

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EXAMPLE: ROMANIA Google Map: Romania

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ABSTRACTION: 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.

(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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https://www.google.co.uk/maps/@46.5687604,24.8308311,7.42z

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PROBLEM FORMULATION

More formally, a problem is defined by 5 components:

1. Initial state where the agent starts: e.g., "at Arad"

2. Actions available to the agent e.g., Arad Zerind , Arad Sibiu, … etc.

3. Transition model: description of what each action does. State that results from doing action a in state s.

4. Goal test, determines whether a given state is a goal state. explicit, e.g., x = "at Bucharest“

implicit, e.g., Checkmate(x)

5. Path cost (additive) function that assigns a numeric cost to each path. Reflects agents performance measure

e.g., sum of distances, number of actions executed, etc.

c(x,a,y) is the step cost, assumed to be ≥ 0

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OTHER EXAMPLES • Toy problems

• Real-world problems

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THE VACUUM WORLD

States: determined by both the agent location and the dirt

locations. Question : How many states?

Initial states: Any state can be designated as the initial state.

Actions: each state has 3 actions: Left, Right, and Suck

Transition model: effects of the actions, transition among

states. Some actions have no effect. Example: sucking in a

clean square

Goal test: This checks whether all the squares are clean.

Path cost: Each step costs 1, so the path cost is the number of

steps in the path.

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THE VACUUM WORLD

16 Compared with the real world, this toy problem has discrete locations,

discrete dirt, reliable cleaning, and it never gets any dirtier

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EXAMPLE: THE 8-PUZZLE

• States

• Actions

• Transition model

• Goal test

• Path cost

Location of each tile and the white space

Move blank left, right, up, down

Given state and action, return resulting state

Checkers whether state matches the goal

1 per move, path cost is the number of steps

Solution: Sequence of actions (moves) from the start state to the goal state 17

SEARCHING FOR SOLUTIONS • Having formulated some problems, we now need to solve

them.

• A solution is an action sequence, so search algorithms

work by considering various possible action sequences.

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TREE SEARCH ALGORITHMS

The possible action sequences starting at the initial state form a tree with the initial state at the root;

Basic idea: Simulated exploration of state space by generating successors of already-explored states (a.k.a.~expanding states)

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Shaded nodes:

expanded nodes

Outlined nodes: generated

but not expanded

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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 the

various fields and using the SuccessorFn of the problem

to create the corresponding states.

Data structure for nodes

• n.State: state to which the node

corresponds

• n.Parent: parent node

• n.Action: action applied to the

parent to generate the node

• N.Path-Cost: cost of the path

from initial state to the node

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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 ∞) 22

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UNINFORMED SEARCH STRATEGIES

Uninformed, also called blind search strategies use

only the information available in the problem

definition

Breadth-first search

Depth-first search

Depth-limited search

Iterative deepening search

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BREADTH-FIRST SEARCH

Expand shallowest unexpanded node

Implementation:

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

end

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Implementation:

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

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Implementation:

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

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Implementation:

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

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PROPERTIES OF BREADTH-FIRST SEARCH

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Complete? 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)

Optimal? Yes (if cost = 1 per step)

Space is the bigger problem (more than time)

b: maximum branching factor of the search tree d: depth of the least-cost solution m: maximum depth of the state space (may be ∞)

DEPTH-FIRST SEARCH

Expand deepest unexpanded node

Implementation:

frontier = LIFO queue, i.e., put successors at front

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DEPTH-FIRST SEARCH


Implementation: frontier = LIFO queue, i.e., put successors at front

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DEPTH-FIRST SEARCH



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DEPTH-FIRST SEARCH


Implementation:

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

front

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DEPTH-FIRST SEARCH


Implementation:


front

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DEPTH-FIRST SEARCH


Implementation:


front

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DEPTH-FIRST SEARCH


Implementation:


front

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DEPTH-FIRST SEARCH


Implementation:


front

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DEPTH-FIRST SEARCH


Implementation:


front

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DEPTH-FIRST SEARCH


Implementation:


front

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DEPTH-FIRST SEARCH



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PROPERTIES OF DEPTH-FIRST SEARCH

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

Can be modified to avoid repeated states along path

Complete in finite spaces

Time? bm

Space? O(bm) (linear space!)

Optimal? No (if cost = 1 per step)


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DEPTH-LIMITED SEARCH

Avoid problems of DFS by imposing a maximum

depth of a path: I

Can be implemented with special algorithms or

with the general search with operators to keep

track of the path depth

Example Romania: l = 20 (# cities)

Complete but not optimal

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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

• Sidesteps the issue of choosing the depth limit

• Tries all possible depth limits: 0, 1, 2, etc

• Combines the benefits of DFS and BFS

• Optimal and complete like BFS

• Modest memory requirements like DFS

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ITERATIVE DEEPENING SEARCH L =0

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SUMMARY

Problem formulation requires abstracting

away real world details to define a state-space

that can be explored

There are several uninformed search

strategies

Iterative deepening search uses only linear

space, and not much more time than the other

strategies

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COMPARING SEARCH STRATEGIES

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


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INFORMED SEARCH

Uninformed search strategies can find

solutions to problems by systematically

generating new states, and testing them

against the goal

These strategies are inefficient in most cases

Informed search strategies: use problem-

specific knowledge

Knowledge is given by an evaluation function

that returns a number describing the

desirability (or lack thereof) of expanding a

nodes 51

BEST-FIRST SEARCH

Idea: use an evaluation function for each node –

estimate of “desirability”

⇒ Expand most desirable unexpanded node

Implementation:

frontier is a queue sorted in decreasing order of desirability

Special cases:

greedy search: expand node closest to the goal

A∗ search: expand node on the least-cost solution path

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ROMANIA WITH STEP COSTS IN KM

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GREEDY SEARCH EXAMPLE

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A* SEARCH: MINIMISING THE TOTAL

ESTIMATED SOLUTION COST

The most widely known form of best-first search

Node evaluation, combines:

g(n): path cost from the start node to node n

h(n): the cost to get from the node to the goal

f(n) = g(n) + h(n)

f(n) is the estimated cost of the cheapest solution

through n

It makes sense to try first the node with lowest f(n)

This strategy is more than just reasonable: provided

that the heuristic function h(n) satisfies certain

conditions, A∗ search is both complete and optimal. 55

A* SEARCH EXAMPLE

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A* SEARCH EXAMPLE

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A* SEARCH EXAMPLE

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A* SEARCH EXAMPLE

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A* SEARCH EXAMPLE

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A* SEARCH EXAMPLE

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SUMMARY: HEURISTIC SEARCH

Best-first search: GENERAL-SEARCH where

minimum-cost nodes (according to some

measure) are expanded first.

Greedy search:

Expands the node closest to the goal

More efficient (in time) that uninformed algorithms

Not optimal and not complete

A* search:

Expands the node on the least-cost solution path

Complete, optimal, and most efficient among all

optimal algorithms. State space still prohibitive 62

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