T Ball (1 Relation) What Your Robots Do

Karl Lieberherr

CSU 670 Spring 2009

Requirements Analysis

• Requirement: Robot wins, survives.

• To satisfy the requirement, we need to be inventive.

• Software developers are masters at hiding complexity from their users.– they want to turn on the robot: one button

press.

What do your robots think about?

• Solving CSP problems.

• Polynomials, called look-ahead polynomials.

• Packed truth tables.

• Reductions of relations and CSP formulae.

• Maximizing look-ahead polynomials.

• Generating random assignments.

Problem Snapshot

• Boolean CSP: constraint satisfaction problem– Each constraint uses a Boolean relation.– e.g. a Boolean relation 1in3(x y z) is satisfied

iff exactly one of its parameters is true.

• Boolean MAX-CSP  a multi-set of constraints. Maximize satisfied fraction.

Packed Truth Tables

22 254 238 17

Z Y X !!

all the look-ahead polynomials for T Ball

The 22 reductions:Needed for implementation

22 60

3

240

15 255

0

1,0

1,1

2,1

2,0

3,03,1

3,0

3,1

2,0

2,1

22 is expanded into 6 additionalrelations.

1in3

0

0.1

0.2

0.3

0.4

0.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Coin bias (Probability of setting a variable to true)

Fra

ctio

n o

f co

nst

rain

ts t

hat

ar

e g

uar

ante

ed t

o b

e sa

tisf

ied

3p(1-p)2 for MAX-CSP({22})

Binomial Distribution

Look-ahead Polynomial(Definition)

• R is a raw material for derivative d.

• N is an arbitrary assignment for R.

• The look-ahead polynomial lad,R,N(p) computes the expected fraction of satisfied constraints of R when each variable in N is flipped with probability p.

• We currently use N = all zero.

Some Theory

• about this robotic world

General Dichotomy Theorem(Discussion)

MAX-CSP(G,f): For each finite set G of relationsthere exists an algebraic number tG

For f ≤ tG: MAX-CSP(G,f) has polynomial solutionFor f ≥ tG+ : MAX-CSP(G,f) is NP-complete,

tG critical transition pointeasy (fluid), Polynomial (finding an assignment)constant proofs (done statically using look-ahead polynomials)no clause learning

hard (solid), NP-completeexponential, super-polynomial proofs ???relies on clause learning

Mathematical Critical Transition Point

MAX-CSP({22},f):

For f ≤ u: problem has always a solutionFor f ≥ u + : problem has not always a solution,

u critical transition point

always (fluid)

not always (solid)

General Dichotomy Theorem

MAX-CSP(G,f): For each finite set G of relationsthere exists an algebraic number tG

For f ≤ tG: MAX-CSP(G,f) has polynomial solutionFor f ≥ tG+ : MAX-CSP(G,f) is NP-complete,

tG critical transition pointeasy (fluid)Polynomial

hard (solid)NP-complete

due to Lieberherr/Specker (1979, 1982)

polynomial solution:Use optimally biased coin.Derandomize.P-Optimal.