Download P-selective Sets and the Power of One Bit

P-Selectivity,
Immunity and the
Power of One Bit
Lane Hemaspaandra
University of Rochester
Leen Torenvliet
ILLC
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Hard to Decide

Will it beat the desert?
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Easy to Choose

But if you had to choose
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Hard to Decide

Terrorist?
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Easy to Choose
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That’s P-Selective

A is P-selective if there is a function f such that
for all x and y
f(x,y) = x or y
if {x,y}  A ≠ then f(x,y) is in A
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Two Types of Sets
Standard Left Cut
Gappy Left Cut
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The Ultimate Reference
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Immunity

Separation of Complexity Classes


A is in C, but not in D separates C from D .
A is in C, but no infinite part of A is in D
Separates C from D , but much stronger
This is Immunity
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Immunity Separation

Fact: For most complexity classes,
separation implies separation with immunity.


Example P≠EXP, and separates with immunity
LOGSPACE ≠ PSPACE and separate with
immunity
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P-Selective sets

P-selective sets exist of arbitrary complexity.



[Selman79] Every Tally Set P-reduces to a Pselective set.
n-bits of advice is not enough to recongize Pselective sets in any recursive time bound [HT]
P-selective sets are immune to every
subrecursive complexity class [this paper]
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How?


Use gappy left-cuts
At certain lengths set boundary b such that
all x ≤ b are in A. The b are easily computable.
f(x,y)=if |x|=|y| then the lexicographically least.
o.w. the smaller length is computable in linear time in the
larger length. So decide which is the case and then
return the most likely candidate.
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Immune Gappy Left Cuts



Take any recursive time bound
Create large enough gaps
Use a wait-and-see argument:


If one of the machines accepts, then put nothing
in at appropriate length otherwise put everything
in.
Letting new requirements in slowly guarantees
infinity.
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Non-Uniform Measures

Advice:

A is in C/g(n) if there exists a function
f : N * such that |f(n)|=g(n) and
x in A iff (x,f(|x|) in B for some B in C.
Most notably: Polynomial Size Circuits.
(What's in P/poly?)
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PSEL and Tournaments

A Pselective set can be considered as a
Tournament.
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The Lion King

Landau 1953: Every tournament has a king.
That is an element k such that every other
element x is beaten by k directly, or there is a
y such that y beats x, and k beats y.

Proof, by induction.
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Proof

Base Case

Induction
k
n
k
k
n
or
or
n
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Deciding x is King


x is king if and only if for every y either
f(x,y)=x or there is a z such that f(x,z)=z and
f(z,y)=z .
This is a 2 predicate. So if A is nonempty at
n then “accept x iff x is the king of length n is
p
a 2 algorithm that accepts only strings in A.
p
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Psel is not immune


Deciding A is empty or not costs 1 bit of
advice.
Conclusion: No P-selective set is 2 /1immune.
p
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Remaining open

Is PSEL immune to REC?


Requires different type of PSEL sets.
Is PSEL bi-immune to anything


Remarkably hard problem.
Clue: if P=PP then every PSEL set is equivalent
to a standard left cut. Standard left-cuts are
definitey not immune.
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