Specker Derivative Game

1
Specker Derivative Game

• Karl Lieberherr
• Spring 2009

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) gt price 1.
• secret SDG satisfaction ratio sr(FP) wrt secret
  S (think of secret S as the maximum) win if
  sr(FP) gt price quality(S).
• pay for performance in raw material finishing
  aggregate wins

3

• derivative (CSP predicate)

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
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SDG Game Versions
Softball
T Ball Fast Pitch Level 1 Independent Slow
Pitch Special case of Fast Pitch Level 1 Reduced
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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
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
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
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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
• k1 PT 4 independent
• k2 3 5 9 6 10 12 6 independent
• k3 7 11 13 14 4 independent
• k4 15 1 independent

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Implication for testingDerivative Minimizer

• The number of relations in the output of the
  minimizer must be lt MAX INDEP(3).