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T Ball (1 Relation) What Your Robots Do

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.

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T Ball (1 Relation) What Your Robots Do

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  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)

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