State-Space Representation

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 dont 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
• Cant 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 SENDMORE MONEY
• Initial State
• Operators
• Goal

16
Cryptarithmetic

• SENDMORE 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 a4a7)(.)
• Initial State
• Operators
• Goal

18
Boolean Satisfiability (3-sat)

• Problem example (a1 a4a7)(.)
• Initial State no variables are assigned values
• Operators
• assign variable to true or false
• negate value of variable (t-gtf, f-gtt)
• 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.