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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
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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 >0xi 0 mod q MAJ (x1, x2, …, xn ) = 0 iff i >0xi> 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 >0xi 0 mod q MAJ (x1, x2, …, xn ) = 0 iff i >0xi> 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-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
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
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.
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.
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)
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.