Monday Oct 6

- planning with A*

WHITEBOARD: WRITE

1. Rubik's cube 10

2. STRIPS planning 10

3. Assignment 10

4. Deriving heuristics 10

DISCUSSION: representing Rubik's cube

Possible approach: position and orientation of each cubit.

Corner cubit: 8 positions, 3 orientations

Edge cubit: 8 positions, 2 orientations

Center cubit: 1 position - fixed

heuristic functions:

1. # cubits out of place (wrong color on at least one side) / 8

- must divide up 8 so is admissible

consistent = monotonic => h(n) <= c(n,n') + h(n')

WANT TO KNOW IF

h(n) - h(n') <= c(n,n') = 1

(change in # out of place) / 8 <= 1

CLEARLY TRUE! PROBLEM: TOO OPTIMISTIC BECAUSE OF DIVIDE BY 8

Better:

MAX( Sum ( Manhattan distance edge cubies ) / 4,

Sum ( Manhattan distance corner cubies ) / 4 )

USED BY KORF 1997 - TOGETHER WITH PRE-COMPUTING HEURISTIC

CLASS HANDS IN ASSIGNMENT

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1. LECTURE: STRIPS planning and A* heuristics

WHITEBOARD

PLANNING

[[ Many kinds of planning involved in intelligent behavior.

[[ We begin with

[[ the classic planning problem:

(States, Actions, initial state, goal(s))

Solution: (shortest) sequence of actions that transforms initial state into a goal state

^

optional

State = set of true propositions

examples: raining, Henry_teaching_473, Henry_jetlagged

proposition = Boolean (true / false) variable

[[ we often want to talk about sets of related propositions.

[[ we do this by writing a proposition as a list:

PREDICATE ARGUMENTS

| |

v v

(teaching Henry 473)

(holding block77)

[[ This is sometimes called LISP notation.

[[ Actions described by its preconditions and effects:

ACTION (fly amsterdam seattle)

precondition: (in amsterdam), (have_ticket amsterdam seattle)

makes true: (in seattle)

makes false: (in amsterdam), (have_ticket amsterdam seattle)

[[ We can represent a large set of actions by a SCHEMA:

[[ change constants to X, Y

[[ Because we represent a state by everything that is true in the

[[ state, making a proposition true just means adding to the set, and

[[ making it false means deleting it from the set.

[[ The GOAL specifies what must be true in the goal state.

[[ It may or may not be a complete state description.

[[ Why not complete?

Add effect: jetlagged

DISCUSS:

What assumptions does STRIPS make about the world?

[[ deterministic actions; completely observable state; only planner affects

world; actions are all the same cost ]]

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3. HAND OUT ASSIGNMENTS. DISCUSS!

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4. WHITEBOARD:

Deriving A* Heuristics

(1) Number of goal propositions not true

MAY NOT BE ADMISSIBLE!

(1') Number of goal propositions / (max # ``makes true'' effects)

VERY OPTIMISTIC - WEAK LOWER BOUND

(2) Length of solution to relaxed problem: no preconditions on actions

DISCUSS: WHAT ALGORITHM SOLVES? SUBSET COVER. WHY OKAY? SMALL; CAN USE APPROXIMATION

(3) Length of solution to relaxed problem: no makes-false effects

DISCUSS: WHAT ALGORITHM? A* SEARCH WITH HEURISTIC #1 OR #2

WHY IS EASIER? SUSSMAN ANOMOLY:

A

C B

B A ==> C

CANNOT ACHIEVE GOALS INDEPENDENTLY!

NEXT CLASS: WHAT MAKES PLANNING HARD!