Lec2 state space
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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.
![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?](https://image.slidesharecdn.com/lec2-statespace-140207090219-phpapp02/85/Lec2-state-space-7-320.jpg)
![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.](https://image.slidesharecdn.com/lec2-statespace-140207090219-phpapp02/85/Lec2-state-space-11-320.jpg)
Lec2 state space
- 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.