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Specker Derivative Game

Specker Derivative Game. Karl Lieberherr Spring 2009. Mega moves in classic and secret SDG. White-black mega move white: offer derivatives black: buy derivatives or reoffer if bought then repeat r times for each bought derivative:

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Specker Derivative Game

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  1. Specker Derivative Game Karl Lieberherr Spring 2009 SDG

  2. Mega moves in classic and secret SDG • White-black mega move • white: offer derivatives • black: buy derivatives or reoffer • if bought then • repeat r times for each bought derivative: • white: deliver raw material with witness quality(S) of secret finished product S • black: deliver finished product FP • white: reveal secret S • black: check secret S against witness quality(S) • win • classic SDG: satisfaction ratio sr(FP) wrt all. win if sr(FP) >= price * 1. • secret SDG: satisfaction ratio sr(FP) wrt secret S (think of secret S as the maximum): win if sr(FP) >= price * quality(S). • pay for performance in raw material finishing: aggregate wins SDG

  3. derivative: (CSP predicate) SDG

  4. SDG Game Versions • T Ball (one relation) • Softball • Slow Pitch (recognizing noise) • one implication chain of any number of relations. • Fast Pitch • any number of relations • Level k Independent (k independent relations with no implication relationship). Note: Level 1 Independent = T Ball • Level k Reduced (any number of relations that can be reduced to Level k Independent.) Note: Slow Pitch is a special case of Level 1 Reduced. • Baseball • Classic and Secret • CSP • Any Combinatorial Maximization Problem T Ball and Softball are based on CSP SDG

  5. Fast Pitch Slow Pitch Level k Independent Level k Reduced T Ball SDG Game Versions Softball T Ball = Fast Pitch Level 1 Independent Slow Pitch = Special case of Fast Pitch Level 1 Reduced SDG

  6. Independent Relations Arity 2 15 level 3 7 11 14 13 level 2 level 1-even 12 6 10 All at level i are independent: 0 : 4 1 : 6 2: 4 1 2 level 0 8 4 level 1-odd 3 3 9 5 Level 1-odd and 2 are also independent: 7 SDG

  7. Independent Relations Arity 2 15 level 3 7 11 14 13 level 2 level 1-even 12 6 10 All at level i are independent: 0 : 4 1 : 6 2: 4 1 2 level 0 8 4 level 1-odd 3 3 9 5 Level 1-odd and 2 are also independent: 7 Red: independent set SDG

  8. Independent Relations Arity 2IS SEVEN THE MAXIMUM? level 3 15 7 11 14 13 level 2 level 1-even 12 6 10 All at level i are independent: 0 : 4 1 : 6 2: 4 1 2 level 0 8 4 level 1-odd 3 3 9 5 Level 1-odd and 2 are also independent: 7 Red: independent set SDG

  9. Alex Lemma • Consider the set of relations that are powers of 2. • Alex Lemma: Any set of relations that contain exactly k relations from PT is independent. • Example for arity 2: PT = {1 2 4 8} • k=1: PT = 4 independent • k=2: 3 5 9 6 10 12 = 6 independent • k=3: 7 11 13 14 = 4 independent • k=4: 15 = 1 independent SDG

  10. Implication for testingDerivative Minimizer • The number of relations in the output of the minimizer must be <= MAX INDEP(3). SDG

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