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

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

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 >0 xi  0 mod q MAJ (x1, x2, …, xn ) = 0 iff  i >0 xi > n/2

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

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.

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-q circuits 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

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

Pf: Razborov-Smolensky method Fixed input x1, x2, …, xn 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=log2 n 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.

Previous construction requires n log2 n 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.

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.

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

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. 

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)

Geometric restrictions of circuits and branching programs

Geometric restrictions of circuits and branching programs

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.

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.

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.