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.