This document discusses state-space representations for general problem solving. It provides examples of state-space models for various problems including the 8-queens puzzle, traveling salesman problem, sliding tile puzzle, cryptarithmetic, Boolean satisfiability, crossword puzzles, and finding common misspelled words in strings. The key components of a state-space model are the initial state, operators that change the state, and a goal test to determine when the problem is solved.

1. State-Space Representation
General Problem Solving via
simplification
Read Chapter 3

2. What you should know
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Create a state-space model
Estimate number of states
Identify goal or objective function
Identify operators
Next Lecture: how to search/use model

3. Everyday Problem Solving
• Route Planning
– Finding and navigating to a classroom seat
• Replanning if someone cuts in front
– Driving to school
• Constant updating due to traffic
• Putting the dishes away
– Spatial reasoning

4. Goal: Generality
• People are good at multiple tasks
• Same model of problem solving for all
problems
• Generality via abstraction and
simplification.
• Toy problems as benchmarks for methods,
not goal.
• AI criticism: generality is not free

5. State-Space Model
• Initial State
• Operators: maps a state into a next state
– alternative: successors of state
• Goal Predicate: test to see if goal achieved
• Optional:
– cost of operators
– cost of solution

6. Major Simplifications
• You know the world perfectly
– No one tells you how to represent the world
– Sensors always make mistakes
• You know what operators do
– Operators don’t always work
• You know the set of legal operators
– No one tells you the operators

7. 8-Queens Model 1
• Initial State: empty 8 by 8 board
• Operators:
– add a queen to empty square
– remove a queen
– [move a queen to new empty square]
• Goal: no queen attacks another queen
– Eight queens on board
• Good enough? Can a solution be found?

8. 8-Queens Model 2
• Initial State: empty 8 by 8 board
• Operators:
– add ith queen to some column (i = 1..8)
– Ith queen is in row i
• Goal: no queen attacks another queen
– 8 queens on board
• Good enough?

9. 8-Queens Model 3
• Initial State:
– random placement of 8 queens ( 1 per row)
• Operators:
– move a queen to new position (in same row)
• Goal: no queen attacks another queen
– 8 queens on board

10. Minton
• Million Queens problem
• Can’t be solved by complete methods
• Easy by Local Improvement –
– to be covered next week
• Same method works for many real-world
problems.

11. Traveling Salesman Problem
• Given: n cities and distances
• Initial State: fix a city
• Operators:
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add a city to current path
[move a city to new position]
[swap two cities]
[UNCROSS]
• Goal: cheapest path visiting all cities once and
returning.

12. TSP
• Clay prize: $1,000,000 if prove can be done
in polynomial time or not.
• Number of paths is N!
• Similar to many real-world problems.
• Often content with best achievable:
bounded rationality

13. Sliding Tile Puzzle
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8 by 8 or 15 by 15 board
Initial State:
Operators:
Goal:

14. Sliding Tile Puzzle
• 8 by 8 or 15 by 15 board
• Initial State: random (nearly) of number 1..7
or 1..14.
• Operators:
– slide tile to adjacent free square.
• Goal: All tiles in order.
• Note: Any complete information puzzle fits
this model.

15. Cryptarithmetic
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Ex: SEND+MORE = MONEY
Initial State:
Operators:
Goal:

16. Cryptarithmetic
• SEND+MORE = MONEY
• Initial State: no variable has a value
• Operators:
– assign a variable a digit (0..9) (no dups)
– unassign a variable
• Goal: arithmetic statement is true.
• Example of Constraint Satisfaction Problem

17. Boolean Satisfiability (3-sat)
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$1,000,000 problem
Problem example (a1 +~a4+a7)&(….)
Initial State:
Operators
Goal:

18. Boolean Satisfiability (3-sat)
• Problem example (a1 +~a4+a7)&(….)
• Initial State: no variables are assigned values
• Operators
– assign variable to true or false
– negate value of variable (t->f, f->t)
• Goal: boolean expression is satisfied.
• $1,000,000 problem
• Ratio of clauses to variables breaks problem into 3 classes:
– low ratio : easy to solve
– high ratio: easy to show unsolvable
– mid ratio: hard

19. CrossWord Solving
• Initial-State: empty board
• Operators:
– add a word that
• Matches definition
• Matches filled in letters
– Remove a word
• Goal: board filled

20. Most Common Word
(Misspelled) Finding
• Given: word length + set of strings
• Find: most common word to all strings
– Warning: word may be misspelled.
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length 5: hellohoutemary position 5
bargainsamhotseview
position 10
tomdogarmyprogramhomse position 17
answer: HOUSE

21. Misspelled Word Finding
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Let pi be position of word in string i
Initial state: pi = random position
Operators: assign pi to new position
Goal state: position yielding word with
fewest misspellings
• Problem derived from Bioinformatics
– finds regulatory elements; these determine
whether gene are made into proteins.