1. Space Hierarchy Results for Randomized Models Jeff Kinne Dieter van Melkebeek University of Wisconsin-Madison

2. Jeff Kinne and Dieter van Melkebeek Space Hierarchy Results for Randomized Models STACS 2008, Feb 21-232 n n2n2 n3n3 … Time Hierarchy Theorems Does allowing more resources yield strictly more computational power?

3. Jeff Kinne and Dieter van Melkebeek Space Hierarchy Results for Randomized Models STACS 2008, Feb 21-233 Deterministic Machines Diagonalization [HS65] Nondeterministic Machines Not known to be closed under complement Translation Arguments, Delayed Diagonalization, … [C73, SFM78, Ž83] Randomized Machines No Computable Enumeration of Machines Good Hierarchy Still Open Additional Techniques [B02, FS04, MP07, …] Bounded-Error Randomized Machines with Advice Time Hierarchy Theorems

4. Jeff Kinne and Dieter van Melkebeek Space Hierarchy Results for Randomized Models STACS 2008, Feb 21-234 Two-sided error machines TIME(poly)/1 TIME(n c )/O(log n) [FS04, GST04, MP07] One-sided error machines TIME(poly)/1 TIME(n c )/O(log 1/c n), for all c >1 [FST05, MP07] Zero-sided error machines TIME(poly)/1 TIME(n c )/O(1) [MP07] Time Hierarchy Theorems: Randomized Machines Two-sided error machines TIME(poly)/1 TIME(n c )/1 [FS04, GST04, MP07]

5. Jeff Kinne and Dieter van Melkebeek Space Hierarchy Results for Randomized Models STACS 2008, Feb 21-235 Deterministic Machines Diagonalization Tight: SPACE(s ) SPACE(s), for any s = (s) Models with Computable Enumeration of Machines Translation Arguments, Delayed Diagonalization, … Bounded-Error Randomized Machines? Space Hierarchy Theorems

6. Jeff Kinne and Dieter van Melkebeek Space Hierarchy Results for Randomized Models STACS 2008, Feb 21-236 Randomized space s Deterministic space s 2 [S70,J81,BCP83] Randomized space s Randomized space s, for s = (s 2 ) Randomized space s Randomized space s, for s = (s 1+ ), any > 0 [KV87] Would like space s space s for any s = (s) Space Hierarchy Theorems – Randomized Machines

7. Jeff Kinne and Dieter van Melkebeek Space Hierarchy Results for Randomized Models STACS 2008, Feb 21-237 Two-sided error machines SPACE(s )/1 SPACE(log n)/O(log n), for any s = (log n) One-sided error machines SPACE(s )/1 SPACE(log n)/O(log n), for any s = (log n) Zero-sided error machines SPACE(s )/1 SPACE(log n)/O(log n), for any s = (log n) Our Results – Randomized Machines

8. Jeff Kinne and Dieter van Melkebeek Space Hierarchy Results for Randomized Models STACS 2008, Feb 21-238 Two-sided error machines: first attempt M1M1 M2M2 M3M3 … x1x1 x2x2 x3x3 … M 1 (x 1 ) M 2 (x 2 ) M 3 (x 3 ) … ¬ M 1 (x 1 ) ¬ M 2 (x 2 ) ¬ M 3 (x 3 ) N Diagonalization Enumeration of all randomized machines Pr[N(x 3 ) = 1] = ½ Pr[M 3 (x 3 ) = 1] = ½

9. Jeff Kinne and Dieter van Melkebeek Space Hierarchy Results for Randomized Models STACS 2008, Feb 21-239 ¬ M i (x) xx ¬ M i (x a )/axaxa xaxa 0 y Two-sided error machines: high level approach MiMi N n n +1 … y 0 y 0 -1 y N(0 y)=L(y) N(0 -1 y)=M i (0 y) y N(y)=L(y) …… Input Length Hard Language L What if Pr[M i (0 y) = 1] = ½ ? Recovery Procedure … Advice ……

10. Jeff Kinne and Dieter van Melkebeek Space Hierarchy Results for Randomized Models STACS 2008, Feb 21-2310 Recovery Procedure for L Input: y, list of randomized machines Output: L(y), using small space, with 2- sided error Pre-condition: at least one machine in list computes L on instances of length |y|, using small space, with 2-sided error

11. Jeff Kinne and Dieter van Melkebeek Space Hierarchy Results for Randomized Models STACS 2008, Feb 21-2311 Hard Language L L = Computation Tableau Language = {M,x,t,j | M deterministic machine, after t time steps on input x, j-th bit of configuration is 1} Can reduce behavior of two-sided error space-bounded machines to L By P-completeness of L and BPL P Space-efficient Recovery Procedure for L

12. Jeff Kinne and Dieter van Melkebeek Space Hierarchy Results for Randomized Models STACS 2008, Feb 21-2312 Recovery Procedure Input: M,x,t,j, {P 1, P 2, P 3, …} Can use P to decide? Can reduce error of P? Pr[P(M,x,t,j) = 1] far from ½ for all t, j Pass test can reduce error of P Local Consistency Check value claimed by P on M,x,t,j against values of previous row

13. Jeff Kinne and Dieter van Melkebeek Space Hierarchy Results for Randomized Models STACS 2008, Feb 21-2313 One- and Zero-sided Error Machines Same high level approach Hard Language L NL-complete language similar to st-connectivity Zero-error recovery procedure for L based on inductive counting [I88, S88] Mimic proof that NL=coNL, replacing nondeterministic guesses with queries to randomized machine

14. Jeff Kinne and Dieter van Melkebeek Space Hierarchy Results for Randomized Models STACS 2008, Feb 21-2314 Recap Two-sided error machines SPACE(s )/1 SPACE(log n)/O(log n), for any s = (log n) One-sided error machines SPACE(s )/1 SPACE(log n)/O(log n), for any s = (log n) Zero-sided error machines SPACE(s )/1 SPACE(log n)/O(log n), for any s = (log n) Two-sided error machines SPACE(s )/1 SPACE(s)/s, for any s = (s), s=O(log n) One-sided error machines SPACE(s )/1 SPACE(s)/s, for any s = (s), s=O(log n) Zero-sided error machines SPACE(s )/1 SPACE(s)/s, for any s = (s), s=O(log n) Two-sided error machines SPACE(s )/1 SPACE(s)/s, for any s = (s), typical s from log(n) to n One-sided error machines SPACE(s )/1 SPACE(s)/s, for any s = (s), typical s from log(n) to n Zero-sided error machines SPACE(s )/1 SPACE(s)/s, for any s = (s), typical s from log(n) to n

15. Jeff Kinne and Dieter van Melkebeek Space Hierarchy Results for Randomized Models STACS 2008, Feb 21-2315 Other Results Any Reasonable Semantic Model SPACE(s )/1 SPACE(log n)/O(1), for any s = (log n) If efficient deterministic simulation exists SPACE(s )/1 SPACE(s)/O(1), for typical s from log(n) to polynomial, any s = (s)

16. Jeff Kinne and Dieter van Melkebeek Space Hierarchy Results for Randomized Models STACS 2008, Feb 21-2316 Merci Thank you