Specker Derivative Game Karl Lieberherr Spring 2009 10202021

Specker Derivative Game Karl Lieberherr Spring 2009 10/20/2021 SDG 1

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 10/20/2021 SDG 2

• derivative: (CSP predicate) 10/20/2021 SDG 3

SDG Game Versions • T Ball (one relation) • Softball – Slow Pitch (recognizing noise) T Ball and Softball are based on CSP • 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 10/20/2021 SDG 4

SDG Game Versions Softball Fast Pitch Slow Pitch Level k Independent T Ball Level k Reduced T Ball = Fast Pitch Level 1 Independent Slow Pitch = Special case of Fast Pitch Level 1 Reduced 10/20/2021 SDG 5

Independent Relations Arity 2 15 level 3 7 level 2 11 level 1 -even level 0 6 1 level 1 -odd 10 2 3 3 8 4 5 14 13 12 All at level i are independent: 0: 4 1: 6 2: 4 9 Level 1 -odd and 2 are also independent: 7 10/20/2021 SDG 6

Independent Relations Arity 2 15 level 3 7 level 2 11 level 1 -even level 0 6 1 level 1 -odd 3 3 Red: independent set 10/20/2021 10 2 8 4 5 14 13 12 All at level i are independent: 0: 4 1: 6 2: 4 9 Level 1 -odd and 2 are also independent: 7 SDG 7

level 3 Independent Relations Arity 2 IS SEVEN THE MAXIMUM? 15 7 level 2 11 level 1 -even level 0 6 1 level 1 -odd 3 3 Red: independent set 10/20/2021 10 2 8 4 5 14 13 12 All at level i are independent: 0: 4 1: 6 2: 4 9 Level 1 -odd and 2 are also independent: 7 SDG 8

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 10/20/2021 SDG 9

Implication for testing Derivative Minimizer • The number of relations in the output of the minimizer must be <= MAX INDEP(3). 10/20/2021 SDG 10