Problem Solving by Searching
Chapter 3
Fall 2005
Copyright, 1996 © Dale Carnegie & Associates, Inc.
Problem-Solving Agents
This is a kind of goal-based agents that
decide what to do by finding sequences
of actions that lead to desirable states.
Some problems cannot be solved by
reflex agents
Formulating problems
Example problems
Searching for solutions
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A simple problem-solving
agent
Goal formulation - limiting the objectives
(for any task, we may have more than one objective)

A goal is a set of world states in which the
goal is satisfied.
Problem formulation - deciding what
actions and states to consider
The above two are about focusing
Search - looking for the best sequence of
actions
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A simple agent (2)
Solutions - the results of search, so they
are sequences of actions that lead to
the goal
Execution - acting upon the world
Formulate -> Search -> Execute
An algorithm in Fig 3.1
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Well-defined problems and solutions
A problem is defined by four components

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

Initial state is where an agent starts
Possible actions, successor function <action,successor>
Goal test determines if a state is a goal state
Path cost function, step cost
A solution is a path from the initial state to a goal
state

Path - connecting sets of states
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Problems and Solutions
States and actions are the basic elements
of a problem
A world of states: initial state I, operator O
(or successor)
State space - the set of all states
reachable from I by any sequences of O
Example: a simplified Romania road
map (Fig 3.2)
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Formulating Problems
Choosing states and actions
The real art of problem solving is in what
goes into the description of the states and
operators and what does not
How to achieve those? Abstraction - removing
detail from a representation


A key problem solving skill
What are irrelevant details for the case of Fig 3.2
we can think of?
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Measuring problem-solving
performance
Performance measures
 Completeness – if there is a solution, it will
find it
 Optimality – the solution found is the best
 Time complexity – number of nodes
generated during search
 Space complexity – maximum number of
nodes stored in memory
Reach the goal
Search cost
Total cost = path cost + search cost
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Example Problems
Toy problems: concise and exact, used to
illustrate or exercise various problem-solving
methods - ideal cases
Real-world problems: more difficult and we
want solutions, but there might be many
different descriptions
Toy problems can also be used to test ideas
quickly: if it does not work for toy problems,
we have reasons to believe that it won’t work
for real-world problems.
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Toy problem (1)
Vacuum world (Fig 3.3)

States: 8 (2*2^2) or n*2^n
 What is the first n about?
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

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Initial states: any
Successor function: Left, Right, Suck
Goal test: Clean or not
Path cost: Each step costs 1
Comparing with the real world case,
how different the two are?
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Toy problem (2)
The 8-puzzle





States:O(9!)
Initial state: Start state (Fig 3.4)
Successor function: Left, Right, Up, Down
Goal test: Goal state (Fig 3.4)
Path cost: Each step costs 1
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Toy problem (3)
The 8-queens (Fig 3.5)


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States: 64*63*…*57
Initial states: Empty board
Successor function: Add a queen to any empty
square
Goal test: 8 queens on the board, none attacked
Path cost: N/A
A better formulation with constraint


One per column in the leftmost n column w/o
attacking queens
Add a queen to any square in the leftmost empty
column w/o attaching queens
What is the difference between the two?
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Real-world problems
Route finding
Touring and traveling salesperson problem
VLSI layout
Robot navigation
Assembly sequencing
Protein design
Internet searching
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Search
Generating action sequences


Expanding the current state by ...
Generating a new set of states
Let’s look at a situation we come to BY …
 PHX -> ASU -> BY
A state map
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What is search?
The essence of search is to consider one
choice at a time.
Search tree – generated by the initial state
and successor, defines the state space (two
examples - Figs 3.6 and 3.8)

root, nodes, fringe (leaf nodes)
A general search algorithm (Fig 3.9)



Initial state, test, expand, test, expand, …
How to expand is determined by a search strategy
How to implement the fringe - a queue
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Search Evaluation
Evaluating search strategies (Criteria)

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Completeness – guaranteed to find a solution if
there is one
Time complexity – big O
Space (memory) complexity – big O
Optimality – the solution is optimal
Branching factor (b), depth of the shallowest goal
(d), maximum length of any path (m)
Two types of search


Blind search (uninformed)
Heuristic search (informed)
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Uninformed Search Strategies
Breadth-first search (a binary tree example)
 Expanding the shallowest node



Complete? Optimal?
How to implement it
branching factor - a killing cost (Fig 3.11)
The main concerns


the memory requirements
exponential complexity
Uniform-cost search – expands the node with the
lowest path cost


Complete? Optimal? What if cost is non-decreasing
When is it identical to BFS?
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Search Strategies (2)
Depth-first search



A binary tree example (Fig 3.12)
Complete? Optimal?
How to implement it
The main concerns


Wrong branch
Deep branch
The cures

Depth-limited search (Fig 3.13)
 Determining depth limit based on the problem (how
many cities in the map of Romania?)

Iterative deepening search (Fig 3.15)
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Search Strategies (3)
Comparing ID Depth-First with Breadth-First

Which one is faster?
Bidirectional search


The two directions: Start, Goal (Fig 3.16)
What’s the saving?
The main concerns



How to search backwards
Multiple goal states
Efficient checking to avoid redundant search in the
other tree
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Search Strategies (4)
Which one to use and when

We need to know the strategies in terms of
the four criteria
 Comparing uninformed search strategies
(Fig 3.17)

We need to know the problem to be solved
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Avoiding Repeated States
It’s wasting time to expand states that
have already been encountered and
expanded before.
How to avoid


Systematic search
Remembering what have been visited
(open list vs. closed list)
 Graph-Search (Fig 3.19)
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Searching with partial
information
Sensorless problems

Reason about sets of states instead of a single
state (Fig 3.21)
Contingency problems


Environment is partially observable, actions are
uncertain
[Suck, Right, if [R,Dirty] then Suck] with position
and local dirt sensors
Exploration problems


Both the states and actions of E are unknown
Can be viewed as an extreme case of contingency
problems
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Summary
Problem solving is goal-based
A problem consists of 4 parts

initial state, operator, goal test, path cost
A single general search algorithm can
solve any problem, but …
Four criteria: completeness, optimality,
time complexity, space complexity
Various search strategies
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