1 / 14

Unveiling the Complexity of Robots in Relation to Constraint Satisfaction Problems

This document explores the intersection of robotic autonomy and constraint satisfaction problems (CSP), specifically focusing on Boolean relations within the realm of robotics. It discusses innovative approaches to meet the requirement of creating robots that can "win" and "survive" in complex environments. The paper elaborates on the significance of look-ahead polynomials and packed truth tables while analyzing the maximum satisfiability of Boolean CSPs. It underlines the theoretical framework surrounding the General Dichotomy Theorem, outlining the conditions under which specific CSP problems can be solved in polynomial time versus NP-completeness.

boone
Télécharger la présentation

Unveiling the Complexity of Robots in Relation to Constraint Satisfaction Problems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. T Ball (1 Relation) What Your Robots Do Karl Lieberherr CSU 670 Spring 2009

  2. 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.

  3. 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.

  4. 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-CSPa multi-set of constraints. Maximize satisfied fraction.

  5. Packed Truth Tables Z Y X !! 22 254 238 17

  6. all the look-ahead polynomials for T Ball

  7. The 22 reductions:Needed for implementation 1,0 2,0 22 60 240 3,0 2,1 3,1 1,1 2,0 3,0 3 15 255 3,1 2,1 22 is expanded into 6 additional relations. 0

  8. 3p(1-p)2 for MAX-CSP({22})

  9. Binomial Distribution

  10. 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.

  11. Some Theory • about this robotic world

  12. General Dichotomy Theorem(Discussion) MAX-CSP(G,f): For each finite set G of relations there exists an algebraic number tG For f≤ tG: MAX-CSP(G,f) has polynomial solution For f≥ tG+ e: MAX-CSP(G,f) is NP-complete, e>0. 1 hard (solid), NP-complete exponential, super-polynomial proofs ??? relies on clause learning tG = critical transition point easy (fluid), Polynomial (finding an assignment) constant proofs (done statically using look-ahead polynomials) no clause learning 0

  13. Mathematical Critical Transition Point MAX-CSP({22},f): For f ≤ u: problem has always a solution For f≥ u + e: problem has not always a solution, e>0. 1 not always (solid) u = critical transition point always (fluid) 0

  14. General Dichotomy Theorem MAX-CSP(G,f): For each finite set G of relations there exists an algebraic number tG For f ≤ tG: MAX-CSP(G,f) has polynomial solution For f≥ tG+ e: MAX-CSP(G,f) is NP-complete, e>0. 1 hard (solid) NP-complete polynomial solution: Use optimally biased coin. Derandomize. P-Optimal. tG = critical transition point easy (fluid) Polynomial 0 due to Lieberherr/Specker (1979, 1982)

More Related