Specker Derivative Game Karl Lieberherr Spring 2009 6172021

Specker Derivative Game Karl Lieberherr Spring 2009 6/17/2021 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 6/17/2021 SDG 2

• derivative: (CSP predicate) 6/17/2021 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 6/17/2021 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 6/17/2021 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 6/17/2021 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 6/17/2021 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 6/17/2021 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 6/17/2021 SDG 9

Implication for testing Derivative Minimizer • The number of relations in the output of the minimizer must be <= MAX INDEP(3). 6/17/2021 SDG 10

Reliable Software Driving Artificial Worlds • Reliable software is important for our society: phones, trains, cars • Artificial worlds – model our own world and help to understand it better – help to teach and learn computer science • software development • empirical algorithmics • Artificial worlds are populated by robots that must be reliable in order to survive. Survival means – following the rules of the artificial world – implement optimal trading strategies 6/17/2021 SDG 11

• Artificial world – Definition of world: what the robots are allowed to do. • create a fair world – Laws: implied by definition 6/17/2021 SDG 12

Combinatorial Optimization Derivatives/Raw Materials/Finished Products • Combinatorial optimization problem range [0, 1] • Predicate language to define subsets • derivative d = (pred, price) • raw material r = (instance satisfying d. pred, secret finished product for r) • finished product f = (r, approximation to r) • quality of finished product q(f) in [0, 1] 6/17/2021 SDG 13

Important Rules • Alternating white-black/black-white mega moves. • Initial life energy • Life energy must stay positive • Only 6/17/2021 SDG 14

• John Pierce: • instead of having artificial benchmarks use artificial markets – robots need to have both skills • finding secrets • hiding secrets • being good at hiding secrets makes them better at finding secrets? • World(Rules, Opt) 6/17/2021 SDG 15

Mega moves in classic and secret Opt range [0, 1] independent of CSP • White-black mega move – white: offer derivatives 1<= – black: buy derivatives or reoffer – if buy derivaties 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: quality(FP). win if quality(FP) >= price. » secret SDG: quality(FP) wrt secret S (think of secret S as the maximum): win if quality(FP) >= price * quality(S). • pay for performance in raw material finishing: aggregate wins – if reoffer then reoffer all derivatives on sale at a lower price 6/17/2021 SDG 16

Mega moves in classic and secret SDG • White-black mega move – white: offer derivatives – black: buy derivatives or reoffer – if buy derivaties 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: quality(FP). win if quality(FP) >= price. » secret SDG: quality(FP) wrt secret S (think of secret S as the maximum): win if quality(FP) >= price * quality(S). • pay for performance in raw material finishing: aggregate wins – if reoffer then reoffer all derivatives on sale at a lower price 6/17/2021 SDG 17

Mega moves in classic and secret SDG • White-black mega move – white: offer derivatives – black: buy derivatives or reoffer – if buy derivaties 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 – if reoffer then reoffer all derivatives on sale at a lower price 6/17/2021 SDG 18

• SDG when CSP 6/17/2021 SDG 19

Mega moves in classic and secret SDG • White-black mega move – white: offer derivatives – black: buy derivatives or reoffer – if buy derivatives then • 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: quality(FP). win if quality(FP) >= price. » secret SDG: quality(FP) wrt secret S (think of secret S as the maximum): win if quality(FP) >= price * quality(S). • pay for performance in raw material finishing: aggregate wins – if reoffer then reoffer all derivatives on sale at a lower price 6/17/2021 SDG 20