State-Space Representation

1. State-Space Representation Read Chapter 3

2. Searches you use • MapQuest • road maps • Google • documents • CiteSeer • research documents

3. Abstract 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

4. Representation Matters

5. 8-Queens Problem.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 • all queens on board

6. 8-Queens Problem.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

7. 8-Queens Problem.3 • Initial State: • random placement of 8 queens ( 1 per row) • Operators: • move a queen to new position • Goal: no queen attacks another queen • 8 queens on board

8. Minton • Million Queens problem • Can’t be solved by complete methods • Easy by Local Improvement – • to be covered in 2nd week • Same method works for many real-world problems.

9. 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.

10. 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

11. 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.

12. 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

13. 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

14. Most Common Word 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

15. 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: words defined by pi with lowest entropy. (or some other measure) • Problem derived from Bioinformatics • finds regulatory elements; these determine whether gene are made into proteins.