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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 P-
selective set. n-bits of advice is not enough to recongize P-
selective 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
or or
k
n
k
n
k
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
p
predicate. So if A is nonempty at n then “accept x iff x is the king of length n is a 2
p
algorithm that accepts only strings in A.
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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
p
/1- immune.
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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.