P-Selectivity, Immunity and the Power of One Bit

1. P-Selectivity, Immunity and the Power of One Bit Lane Hemaspaandra University of Rochester Leen Torenvliet ILLC

2. Hard to Decide • Will it beat the desert?

3. Easy to Choose • But if you had to choose

4. Hard to Decide • Terrorist?

5. Easy to Choose

6. 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 That’s P-Selective

7. Two Types of Sets Standard Left Cut Gappy Left Cut

8. The Ultimate Reference

9. 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

10. 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

11. 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]

12. 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.

13. 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.

14. Non-Uniform Measures • Advice: • Ais in C/g(n)if there exists a function f : N * such that |f(n)|=g(n) and xinAiff (x,f(|x|) inBfor someBinC. Most notably: Polynomial Size Circuits. (What's in P/poly?)

15. PSEL and Tournaments • A Pselective set can be considered as a Tournament.

16. The Lion King • Landau 1953: Every tournament has a king. That is an element k such that every other element xis beaten by kdirectly, or there is a y such that y beats x, and k beats y. • Proof, by induction.

17. k n Proof • Base Case • Induction k k n or or n

18. Deciding xis 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 2p predicate. So if A is nonempty at nthen “accept x iff x is the king of length nis a 2p algorithm that accepts only strings in A.

19. Psel is not immune • Deciding A is empty or not costs 1 bit of advice. • Conclusion: No P-selective set is 2p /1- immune.

20. 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.