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.