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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

State-Space Representation

General Problem Solving via simplification

Read Chapter 3

what you should know
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
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
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
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
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
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
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
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
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
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.
slide12
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
Sliding Tile Puzzle
  • 8 by 8 or 15 by 15 board
  • Initial State:
  • Operators:
  • Goal:
sliding tile puzzle14
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
Cryptarithmetic
  • Ex: SEND+MORE = MONEY
  • Initial State:
  • Operators:
  • Goal:
cryptarithmetic16
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
Boolean Satisfiability (3-sat)
  • $1,000,000 problem
  • Problem example (a1 +~a4+a7)&(….)
  • Initial State:
  • Operators
  • Goal:
boolean satisfiability 3 sat18
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
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
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
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.