Download presentation

Presentation is loading. Please wait.

Published byJoe Mance Modified over 3 years ago

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

Similar presentations

Presentation is loading. Please wait....

OK

INHERENT LIMITATIONS OF COMPUTER PROGRAMS CSci 4011.

INHERENT LIMITATIONS OF COMPUTER PROGRAMS CSci 4011.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on air powered car Ppt on conservation of forest in india Ppt on life study of mathematician salary Ppt on purchase order Ppt on nutrition and dietetics Ppt on chromosomes and genes pictures Ppt on regular expression syntax Ppt on online banking project in java Ppt on recycling of waste material in india Ppt on smoking is injurious to health