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 Z Y X !!

all the look-ahead polynomials for T Ball

The 22 reductions: Needed for implementation ,0 1,1 2,1 2,0 3,0 3,1 3,0 3,1 2,0 2,1 22 is expanded into 6 additional relations.

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 la d,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 relations there exists an algebraic number t G For f ≤ t G : MAX-CSP(G,f) has polynomial solution For f ≥ t G +  : MAX-CSP(G,f) is NP-complete,   t G  critical transition point easy (fluid), Polynomial (finding an assignment) constant proofs (done statically using look-ahead polynomials) no clause learning hard (solid), NP-complete exponential, super-polynomial proofs ??? relies on clause learning 

Mathematical Critical Transition Point MAX-CSP({22},f): For f ≤ u: problem has always a solution For 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 relations there exists an algebraic number t G For f ≤ t G : MAX-CSP(G,f) has polynomial solution For f ≥ t G +  : MAX-CSP(G,f) is NP-complete,   t G  critical transition point easy (fluid) Polynomial hard (solid) NP-complete due to Lieberherr/Specker (1979, 1982) polynomial solution: Use optimally biased coin. Derandomize. P-Optimal. 