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State-Space Representation

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State-Space Representation. General Problem Solving via simplification Read Chapter 3. What you should know. Create a state-space model Estimate number of states Identify goal or objective function Identify operators Next Lecture: how to search/use model. Everyday Problem Solving.

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State-Space Representation

General Problem Solving via simplification

Read Chapter 3

What you should know
• Create a state-space model
• Estimate number of states
• Identify goal or objective function
• Identify operators
• Next Lecture: how to search/use model
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
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
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
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
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-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?
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
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.
Traveling Salesman Problem
• Given: n cities and distances
• Initial State: fix a city
• Operators:
• 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.
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
Sliding Tile Puzzle
• 8 by 8 or 15 by 15 board
• Initial State:
• Operators:
• Goal:
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.
Cryptarithmetic
• Ex: SEND+MORE = MONEY
• Initial State:
• Operators:
• Goal:
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
Boolean Satisfiability (3-sat)
• \$1,000,000 problem
• Problem example (a1 +~a4+a7)&(….)
• Initial State:
• Operators
• Goal:
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
CrossWord Solving
• Initial-State: empty board
• Operators:
• add a word that
• Matches definition
• Matches filled in letters
• Remove a word
• Goal: board filled
Most Common Word (Misspelled) Finding
• Given: word length + set of strings
• Find: most common word to all strings
• Warning: word may be misspelled.
• length 5: hellohoutemary position 5
• bargainsamhotseview position 10
• tomdogarmyprogramhomse position 17
• answer: HOUSE
Misspelled Word Finding
• 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.