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Space Hierarchy Results for Randomized Models

Space Hierarchy Results for Randomized Models. Jeff Kinne Dieter van Melkebeek University of Wisconsin-Madison. Time Hierarchy Theorems. Does allowing more resources yield strictly more computational power?. …. n 3. n 2. n. Time Hierarchy Theorems. Deterministic Machines

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Space Hierarchy Results for Randomized Models

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

  2. Time Hierarchy Theorems Does allowing more resourcesyield strictly more computational power? … n3 n2 n STACS 2008, Feb 21-23

  3. Time Hierarchy Theorems • 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 Machineswith Advice STACS 2008, Feb 21-23

  4. Time Hierarchy Theorems:Randomized Machines • Two-sided error machines • TIME(poly)/1 ⊈ TIME(nc)/O(log n)[FS04, GST04, MP07] • One-sided error machines • TIME(poly)/1 ⊈ TIME(nc)/O(log1/c n), for all c >1 [FST05, MP07] • Zero-sided error machines • TIME(poly)/1 ⊈ TIME(nc)/O(1) [MP07] • Two-sided error machines • TIME(poly)/1 ⊈ TIME(nc)/1[FS04, GST04, MP07] STACS 2008, Feb 21-23

  5. Space Hierarchy Theorems • 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? STACS 2008, Feb 21-23

  6. Space Hierarchy Theorems – Randomized Machines • Randomized space s⊆ Deterministic space s2[S70,J81,BCP83] • Randomized space s⊈ Randomized space s, for s= (s2) • Randomized space s⊈Randomized space s,for s= (s1+), any  > 0 [KV87] • Would like space s ⊈ space sfor any s = (s) STACS 2008, Feb 21-23

  7. Our Results – Randomized Machines • 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) STACS 2008, Feb 21-23

  8. Two-sided error machines:first attempt Enumeration of all randomized machines • Diagonalization M1 M2 M3 … N x1 M1(x1) ¬ M1(x1) x2 M2(x2) ¬ M2(x2) x3 M3(x3) ¬ M3(x3) … … Pr[N(x3) = 1] = ½ Pr[M3(x3) = 1] = ½ STACS 2008, Feb 21-23

  9. Mi N Two-sided error machines:high level approach “Hard”LanguageL Advice Advice InputLength xa xa ¬ Mi(x) ¬ Mi(xa)/a n x x n+1 … … y y … N(y)=L(y) … … Recovery Procedure … 0ℓ-1y 0ℓ-1y N(0ℓ-1y)=Mi(0ℓy) 0ℓy 0ℓy N(0ℓy)=L(y) What if Pr[Mi(0ℓy) = 1] = ½ ? STACS 2008, Feb 21-23

  10. 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 STACS 2008, Feb 21-23

  11. 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 STACS 2008, Feb 21-23

  12. Recovery Procedure • Input: ‹M,x,t,j›, {P1, P2, P3, …} • 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 STACS 2008, Feb 21-23

  13. 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 STACS 2008, Feb 21-23

  14. 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 STACS 2008, Feb 21-23

  15. Other Results • AnyReasonable Semantic Model • SPACE(s)/1 ⊈ SPACE(log n)/O(1),for any s = (log n) • Ifefficient deterministic simulation exists • SPACE(s)/1 ⊈ SPACE(s)/O(1),for typical s from log(n) to polynomial,any s = (s) STACS 2008, Feb 21-23

  16. Merci Thank you STACS 2008, Feb 21-23

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