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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.