1. Problem Solving by Searching
Chapter 3
Fall 2009
Copyright, 1996 © Dale Carnegie & Associates, Inc.

2. 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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3. 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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4. 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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5. Well-defined problems and solutions
A problem is defined by four components
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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6. 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
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7. A simplified Romania map
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8. 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 Romania Map we can think of?
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9. 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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10. 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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11. Toy problem (1) Vacuum world (next slide)
States: 8 (2*2^2) or n*2^n
What is the first n about?
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?
n is the number of locations
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12. CSE 471/598 by H. Liu

13. Toy problem (2) The 8-puzzle States:O(9!) Initial state: Start state
Successor function: Left, Right, Up, Down
Goal test: Goal state
Path cost: Each step costs 1
O(9!/2)
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14. Toy problem (3) The 8-queens (Fig 3.5)
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 attacking queens
What is the difference between the two?
Generate first and test next
If not good, don’t generate for test
For 100 Qs
10^400 to
10^52
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15. Real-world problems Route finding
Touring and traveling salesperson problem
VLSI layout
Robot navigation
Assembly sequencing
Protein design
Internet searching
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16. Search Generating action sequences A state map
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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17. 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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18. Search Evaluation Evaluating search strategies (Criteria)
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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19. 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 a node has a zero-cost action leading back to the same node?
When is it identical to BFS?
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20. Search Strategies (2) Depth-first search The main concerns The cures
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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21. 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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22. 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
Human intelligence is still required, or not plug-and-play.
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23. 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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24. 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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25. 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
Discussed various search strategies
We haven’t answered the question – how far we are away from the goal? Is it possible to answer?
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