Artificial Intelligence
Lecture 10
Outline
• Problem Solving using Search
• Search consists of
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state space
operators
start state
goal states
• A Search Tree is an efficient way to represent a search
• There are a variety of specific search techniques, including
– Depth-First Search
– Breadth-First Search
– Others which use heuristic knowledge (in future lectures)
What do these problems have
in common?
• Find the layout of chips on a circuit board which minimize
the total length of interconnecting wires
• Write a program which can play chess against a human
• Build a system which can find human faces in an arbitrary
digital image
• Decrypt data which has been encrypted but you do not
have the key
• Answer
– they can all be formulated as search problems
•
Setting Up a State Space Model
State-space Model is a Model for The Search Problem
– usually a set of discrete states X
• e.g., in driving, the states in the model could be towns/cities
• Start State - a state from X where the search starts
• Goal State(s)
– a goal is defined as a target state
– For now: all goal states have utility 1, and all non-goals have utility 0
– there may be many states which satisfy the goal
• e.g., drive to a town with an airport
– or just one state which satisfies the goal
• e.g., drive to Las Vagas
• Operators
– operators are mappings from X to X
• e.g. moves from one city to another that are connected with a road (legal)
Summary: Defining Search
Problems
• A statement of a Search problem has 4 components
1. A set of states
2. A set of “operators” which allow one to get from one state to another
3. A start state S
4. A set of possible goal states G, or ways to test for goal states
• Search solution consists of
– a sequence of operators which transform S into a a unique goal
state G (this is the sequence of actions the we would take to
maximize the success function)
• Representing real problems in a search framework
– may be many ways to represent states and operators
– key idea: represent only the relevant aspects of the problem
•
Example 1: Formulation of Map
Problem
Set of States
– individual cities
– e.g., San Francisco, Seattle, Las Vegas, Phoenix, Chicago
• Operators
– freeway routes from one city to another
– e.g. San Francisco to Seattle, etc
• Start State
– current city where we are, San Francisco
• Goal States
– set of cities we would like to be in
– e.g., cities which are cooler than Seattle
• Solution
– a sequence of operators which get us a specific goal city,
• e.g., Irvine to SF via 5, SF to Reno via 80, etc
State Space Graph: Map Navigation
not to be confused with Search Tree!!
A
B
C
G
S
D
E
F
S = start, G = goal, other nodes = intermediate states, links = legal transitions
Abstraction
• Definition of Abstraction:
Process of removing irrelevant detail to create an abstract
representation: ``high-level”, ignores irrelevant details
• Navigation Example: how do we define states and operators?
– First step is to abstract “the big picture”
• i.e., solve a map problem
• nodes = cities, links = freeways/roads (a high-level description)
• this description is an abstraction of the real problem
– Can later worry about details like freeway onramps, refueling, etc
• Abstraction is critical for automated problem solving
– must create an approximate, simplified, model of the world for the
computer to deal with: real-world is too detailed to model exactly
– good abstractions retain all important details
Qualities of a Good Representation
• Given a good representation/abstraction, solving a problem
can be easy!
• Conversely, a poor representation makes a problem harder
• Qualities which make a Representation useful:
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important objects and relations are emphasized
irrelevant details are suppressed
natural constraints are made explicit and clear
completeness
concise
efficient
Example 2: Puzzle-Solving as Search
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You have a 3-gallon and a 4-gallon
You have a faucet with an unlimited amount of water
You need to get exactly 2 gallons in 4-gallon jug
•
State representation: (x, y)
– x: Contents of four gallon
– y: Contents of three gallon
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•
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Start state: (0, 0)
Goal state(s) G = {(2, 0), (2, 1), (2, 2)}
Operators
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Fill 3-gallon (0,0)->(0,3), fill 4-gallon (0,0)->(4,0)
Fill 3-gallon from 4-gallon (4,0)->(1,3),
Fill 4-gallon from 3-gallon (0,3)->(3,0) or (1,3)->(4,0) or (2,2)->(4,0)….
Empty 3-gallon into 4-gallon, empty 4-gallon into 3-gallon
Dump 3-gallon down drain (0,3)->(0,0), dump 4-gallon down drain (4,0)->(0,0)
Example 3: The “8-Puzzle” Problem
Start State
1
2
4
6
7
5
8
1
2
3
4
5
6
7
Goal State
3
8
1
2
3
4
5
6
7
8
Search Basics
• When the answer is not straight foreword, you must search
for a solution. But because of the sheer magnitude of some
situations, it is hard to incorporate into AI.
• Combination explosions cause the sheer magnitude of these
problems. Combination explosions are when the possibilities
for a solution explode (increase very rapidly)
• The math field of combinatorics study how these
combination explosions occur. The commonly accepted
theorem states that the number of ways N objects can be
combined equals N! (N factorial). Because of the enormous
number of possibilities, it is almost never a good idea to try
every single possibility
Search Basics
(Definitions)
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Node: A discrete point and possible goal
Terminal Node: A node that ends a path
Search Space: The set off all nodes
Goal: The node that is the object of the search
Heuristic: Information about the likelihood that any
specific node is a better choice to try next, rather than
another node
• Solution Path: A directed graph of the nodes visited that
lead to the solution
Search Space
• Most search spaces are in the form of a
tree. A tree looks like an upside down tree
that contains nodes. The following is an
example of a tree:
•
Evaluation of Searching
Algorithm
• Evaluating a search technique can
sometimes be very difficult. But the most
basic measuring techniques are:
– Speed in which the solution is found
– The quality of the solution
• Both criterions are basically equally
important, but in some case one might take
priority over the other.
DFS
• The Depth-First search explores each possible
path to find the path's conclusion or the goal.
• If the path does not contain the goal the
search tries another path by backtracking.
• The following image shows the path the search
takes when trying to find the goal ‘F':
DFS (Conclusion)
• As you can see, the search almost had to
check every node in order to find the
answer, but it did find the right answer.
• This search is very reliable but sometimes
can take a long time or even turn into an
exhaustive search (a search that looks a
every point until it reaches the goal).
• The Depth-First search can also be
modified to go from the right to the left
DFS (Pros and Cons)
• Advantages:
– Always finds the goal
• Disadvantages:
– Often can be slow
– Can result in an exhaustive search
– Can get stuck in long paths that don't
lead to the goal
BFS
• The Breadth-First search technique is exactly
the same as the Depth-First except that the
Breadth-First checks each node on a level until
it proceeds to the next level.
• The example below demonstrates this technique
with the goal of node ‘C':
BFS (Pros and Cons)
• Advantages:
– Always finds the goal
– Good for shallow trees
• Disadvantages:
– Often can be slow in deep trees
– Can result in an exhaustive search
Hill-Climbing search
• The Hill-Climbing search uses heuristics to find
the direction which leads towards the goal.
• The name "hill-climbing" comes from an analogy:
– A hiker is lost halfway up/down (depends on if you are an
optimist) a mountain at night. His camp is at the top of
the mountain. Even though it is dark, the hiker knows that
every step he takes up the mountain is a step towards his
goal.
• So a hill-climbing search always goes to the node
closest to the goal.
Traversal of graphs
• To traverse means to visit the vertices
(nodes) in some systematic order.
• Various traversal methods for trees are:
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preorder: visit each node before its children.
postorder: visit each node after its children.
inorder (for binary trees only): visit left
subtree, node, right subtree. A small portion of
a tree (Maybe in shortest path).
Examples of undirected
graphs
Observe that
1. Every free tree with n vertices contains exactly n - 1 edges
2. If we add any edge to a free tree, we get a cycle
Depth First and Breadth
First Search
Vertex
Adj. List
a
(b, c, d, e)
b
(a, d, e)
c
(a, f, g)
d
(a, b, e)
Vertex
Adj. List
e
(a, b, d)
f
(c, g)
g
(c, f)
Depth First Search
(DFS)
DFS of an undirected graph involves only two
types of arcs.
1. Tree arcs.
2. Back arcs
(there is no distinction between back arcs and forward arcs)
Cross arcs also don't exist because given any
two vertices, if there exists an arc between
them, then one of them will be an ancestor
and the other a descendant in the DFS.
Breadth First Search
Breadth First Search is also known as ForwardChaining
Graph search algorithms must eliminate loops from
solution path
Heuristic Search
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A heuristic is a method that
might not always find the best solution
but is guaranteed to find a good solution in reasonable time.
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could not be solved any other way.
solutions take an infinite time or very long time to compute.
The classic example of heuristic search methods is the travelling salesman
problem.
Heuristic Search methods Generate and Test Algorithm
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By sacrificing completeness it increases efficiency.
Useful in solving tough problems which
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generate a possible solution which can either be a point in the problem space or a
path from the initial state.
test to see if this possible solution is a real solution by comparing the state
reached with the set of goal states.
if it is a real solution, return. Otherwise repeat from 1.
This method is basically a depth first search as complete solutions must be
created before testing. It is often called the British Museum method as it
is like looking for an exhibit at random. A heuristic is needed to sharpen up
the search.
Questions & Answers
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