1 / 18

A new characterization of ACC 0 and probabilistic CC 0

A new characterization of ACC 0 and probabilistic CC 0. Kristoffer Arnsfelt Michal Koucký Hansen Aarhus University Institute of Mathematics Denmark Czech Republic. Bounded depth Boolean circuits     

jovita
Download Presentation

A new characterization of ACC 0 and probabilistic CC 0

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. A new characterization of ACC0 and probabilistic CC0 Kristoffer Arnsfelt Michal Koucký Hansen Aarhus University Institute of Mathematics Denmark Czech Republic

  2. Bounded depth Boolean circuits      x1 x3 x4 x7 Constant depth, polynomial size circuits MOD-q (x1, x2, …, xn ) = 0 iff  i >0xi 0 mod q MAJ (x1, x2, …, xn ) = 0 iff  i >0xi> n/2

  3. Bounded depth Boolean circuits AC0: unbounded fan-in AND, OR and unary NOT gates. ACC0: unbounded fan-in AND, OR, MOD-q and unary NOT. CC0: unbounded fan-in MOD-q gates. TC0: unbounded fan-in MAJ and unary NOT gates. NC1: fan-in two AND, OR and unary NOT gates, O(log)-depth. Constant depth, polynomial size circuits MOD-q (x1, x2, …, xn ) = 0 iff  i >0xi 0 mod q MAJ (x1, x2, …, xn ) = 0 iff  i >0xi> n/2

  4. Known relationships • AC0 ACC0 TC0 NC1 • CC0 AC0 but CC0 ACC0 • Open questions: NP  CC0 ? CC0 ACC0 ? • Conjecture (Barrington-Straubing-Thérien): AND  CC0

  5. Our results Thm: ACC0 rand-CC0. Thm: ACC0 AND  OR  CC0. Thm: ACC0= rand-ACC0 = rand-CC0= rand( log n )-CC0. Thm: ACC0 corresponds to planar bounded-with nondeterministic branching programs of polynomial size.

  6. AND vs CC0 Fact: 1) For prime p, CC0[ p ] cannot compute AND. 2) For prime power q, CC0[ q ] cannot compute AND. Thm (BST): MOD-p MOD-qcircuits require exponential size to compute AND. Thm (Thérien): CC0 circuits for AND require Ω( n ) gates in their first layer. p,q co-prime integers

  7. AND vs CC0 Thm (BST): CC0[ pq ] circuits of exponential size can compute any Boolean function, in particular AND. Cor: CC0[ pq ] circuits of size 2n and depth O(1/) can compute AND. Thm(BBR): CC0[ q ] circuits of size 2n 1/r and depth 4 can compute AND if q has r distinct prime factors. p,q co-prime integers

  8. Thm: AND is computable by rand-CC0[ pq ] circuits with error <1/n log n if p and q are co-prime integers. Pf: Razborov-Smolensky method Fixed input x1, x2, …, xn Take a random set S  {1, …, n } with probability at least 1/2 over random choice of S OR(x1, x2, …, xn ) = MOD-q { xi , i S } take k=log2n independent random sets S1, S2, …, Sk with probability at least 1/n log n over random choices of S’s OR(x1, x2, …, xn ) = ORj MOD-q { xi , i Sj }  Cor: ACC0 rand-CC0.

  9. Previous construction requires n log2n random bits. • One can reduce the number of random bits to O(log n) while keeping the error below 1/n k by use of: • Valiant-Vazirani isolation technique, and • Randomness efficient sampling using random walks on expanders. Similar to [AJMV]  logspace uniformity Cor: ACC0 rand(log n)-CC0.

  10. Derandomization Thm (Ajtai, Ben-Or): 1) rand-AC0 AC0. 2) rand-ACC0 ACC0. Open: rand-CC0 CC0 ? Claim: rand-CC0= CC0 iff AND  CC0. Thm: ACC0 AND  OR  CC0.

  11. Thm: ACC0 AND  OR  CC0. (non-uniformly) Pf: Technique of Ajtai and Ben-Or Cn a rand-CC0 circuit computing fn with error <1/3n. Take OR of n independent copies of Cn  if fn ( x ) = 1 then OR  Cn( x ) = 0 with probability < ( 1/3n )n if fn ( x ) = 0 then OR  Cn( x ) = 1 with probability < 1/3

  12. Cn a rand-CC0 circuit computing fn with error <1/3n. Take OR of n independent copies of Cn  if fn ( x ) = 1 then OR  Cn( x ) = 0 with probability < ( 1/3n )n if fn ( x ) = 0 then OR  Cn( x ) = 1 with probability < 1/3 Take AND of n independent copies of OR  Cn  if fn ( x ) = 1 then AND  OR  Cn( x ) = 0 with p. < n ( 1/3n )n if fn ( x ) = 0 then AND  OR  Cn( x ) = 1 with p. < ( 1/3)n In both cases the probability of error is less than 2n so we can fix a particular random bits that will give the correct answer for all x.

  13. Previous construction requires to fix >n2 random bits so it is non-uniform. • One can get uniform construction using: • Lautemann’s technique, and • Randomness efficient sampling using random walks on expanders. Similar to [AH, V] Thm: ACC0 AND  OR  CC0. (uniformly)

  14. Geometric restrictions of circuits and branching programs

  15. Geometric restrictions of circuits and branching programs

  16. Constant width circuits Thm (Barrington): NC1 corresponds to constant width circuits. Thm (Hansen’06): ACC0 corresponds to constant width planar circuits. Thm (BLMS’99): AC0 corresponds to constant width upward planar circuits.

  17. Constant width nondeterministic branching programs Thm (Barrington): NC1 corresponds to constant width nondeterministic branching programs. Thm: ACC0 corresponds to constant width planar nondeterministic branching programs. Thm (BLMS’98): AC0 corresponds to constant width upward planar nondeterministic branching programs.

  18. Constant width nondeterministic branching programs Thm (Hansen’08): Quasi-polynomial size ACC0 corresponds to quasi-polynomial size constant width planar nondeterministic branching programs. Thm (HMV): Functions computable by constant width planar nondeterministic branching programs are in ACC0. Thm (Hansen’08): Functions from AND  OR  CC0 are computable by constant width planar nondeterministic branching programs.  Thm: ACC0 corresponds to constant width planar nondeterministic branching programs.

More Related