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State-Space Representation

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

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

  2. 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

  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: • 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 • 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 • 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) • $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. • length 5: hellohoutemary position 5 • bargainsamhotseview position 10 • tomdogarmyprogramhomse position 17 • answer: HOUSE

  21. 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.

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