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
• 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• 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
• 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
• 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• 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
• 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
• 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
• 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
• 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.
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
• 8 by 8 or 15 by 15 board
• Initial State:
• Operators:
• Goal:
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
• Ex: SEND+MORE = MONEY
• Initial State:
• Operators:
• Goal:
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)
• $1,000,000 problem
• Problem example (a1 +~a4+a7)&(….)
• Initial State:
• Operators
• Goal:
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
• 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
• 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
• 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.