A new A new characterization of characterization of

ACCACC00 and and probabilistic CCprobabilistic CC00

Kristoffer ArnsfeltKristoffer Arnsfelt MichalMichal KouckýKoucký

HansenHansen Aarhus University Aarhus University Institute of Mathematics Institute of Mathematics

DenmarkDenmark Czech Republic Czech Republic

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Bounded depth Boolean circuitsBounded depth Boolean circuits

xx11 x x33 xx44 x x77

Constant depth, polynomial size circuitsConstant depth, polynomial size circuits

MOD-q (MOD-q (xx11, , xx22, …, , …, xxn n ) = 0 ) = 0 iff iff i i >0>0 xxii 0 mod q 0 mod q

MAJ (MAJ (xx11, , xx22, …, , …, xxn n ) = 0 ) = 0 iff iff i i >0>0 xxii > > nn/2/2

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Bounded depth Boolean circuitsBounded depth Boolean circuits

ACAC00: : unbounded fan-in AND, OR and unary NOT gates.unbounded fan-in AND, OR and unary NOT gates.

ACCACC00: : unbounded fan-in AND, OR, MOD-q and unary unbounded fan-in AND, OR, MOD-q and unary NOT.NOT.

CCCC00: : unbounded fan-in MOD-q gates.unbounded fan-in MOD-q gates.

TCTC00: : unbounded fan-in MAJ and unary NOT gates.unbounded fan-in MAJ and unary NOT gates.

NCNC11: : fan-in two AND, OR and unary NOT gates, O(log)-fan-in two AND, OR and unary NOT gates, O(log)-depth.depth.

Constant depth, polynomial size circuitsConstant depth, polynomial size circuits

MOD-q (MOD-q (xx11, , xx22, …, , …, xxn n ) = 0 ) = 0 iff iff i i >0>0 xxii 0 mod q 0 mod q

MAJ (MAJ (xx11, , xx22, …, , …, xxn n ) = 0 ) = 0 iff iff i i >0>0 xxii > > nn/2/2

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Known relationshipsKnown relationships

ACAC00 ACC ACC00 TC TC00 NC NC11

CCCC00 AC AC0 0 but but CCCC00 ACC ACC00

Open questions: Open questions: NP NP CC CC00 ? ?

CCCC00 ACC ACC0 0 ??

Conjecture (Barrington-Straubing-Conjecture (Barrington-Straubing-

ThThérienérien):):

AND AND CC CC00

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Our resultsOur results

Thm: Thm: ACCACC00 rand-CC rand-CC00..

Thm: Thm: ACCACC00 AND AND OR OR CC CC00..

Thm: Thm: ACCACC00 = rand-ACC = rand-ACC0 0 = rand-CC= rand-CC00 = rand( log = rand( log n n )-CC)-CC00..

Thm: Thm: ACCACC00 corresponds to planar bounded-with corresponds to planar bounded-with nondeterministic branching programs of nondeterministic branching programs of polynomial size.polynomial size.

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AND vs CCAND vs CC00

Fact: Fact: 1) For prime 1) For prime pp, CC, CC00[ [ p p ] cannot compute ] cannot compute AND.AND.

2) For prime power 2) For prime power qq, CC, CC00[ [ q q ] cannot ] cannot compute AND.compute AND.

Thm (BST): Thm (BST): MOD-MOD-pp MOD- MOD-qq circuits require circuits require exponential size to compute AND.exponential size to compute AND.

Thm (ThThm (Théérien): rien): CCCC00 circuits for AND require circuits for AND require ΩΩ( ( n n ) ) gates in their first layer.gates in their first layer.

p,qp,q co-prime integers co-prime integers

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AND vs CCAND vs CC00

Thm (BST): Thm (BST): CCCC00[ [ pq pq ] circuits of exponential size ] circuits of exponential size can compute any Boolean function, in particular can compute any Boolean function, in particular AND.AND.

Cor: Cor: CCCC00[ [ pq pq ] circuits of size 2] circuits of size 2nn and depth O(1/ and depth O(1/) )

can compute AND.can compute AND.

Thm(BBR): Thm(BBR): CCCC00[ [ q q ] circuits of size 2] circuits of size 2n n 1/1/rr and depth 4 and depth 4 can compute AND if can compute AND if qq has has rr distinct prime distinct prime factors.factors.

p,qp,q co-prime integers co-prime integers

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Thm:Thm: AND is computable by rand-CC AND is computable by rand-CC00[ [ pq pq ] circuits with ] circuits with error <1/error <1/n n log log nn if if pp and and qq are co-prime integers. are co-prime integers.

Pf: Pf: Razborov-Smolensky methodRazborov-Smolensky method

Fixed input Fixed input xx11, , xx22, …, , …, xxnn

Take a random set S Take a random set S {1, …, {1, …, n n }}

with probability at least 1/2 over random choice of Swith probability at least 1/2 over random choice of S

OR(OR(xx11, , xx22, …, , …, xxnn ) = MOD-q { ) = MOD-q { xxi i , , ii S } S }

take take kk=log=log22 nn independent random sets S independent random sets S11, S, S22, …, S, …, Skk

with probability at least 1/with probability at least 1/n n log log nn over random over random choices of S’schoices of S’s

OR(OR(xx11, , xx22, …, , …, xxnn ) = OR ) = ORjj MOD-q { MOD-q { xxi i , , ii S Sjj } }

Cor:Cor: ACC ACC00 rand-CC rand-CC00..

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Previous construction requires Previous construction requires nn log log22 nn random bits. random bits.

One can reduce the number of random bits to One can reduce the number of random bits to O(log O(log nn) while keeping the error below 1/) while keeping the error below 1/n n kk by by use of:use of:

1.1. Valiant-Vazirani isolation technique, andValiant-Vazirani isolation technique, and

2.2. Randomness efficient sampling using random Randomness efficient sampling using random walks on expanders.walks on expanders.

Similar to [AJMV]Similar to [AJMV]

logspace uniformitylogspace uniformity

Cor:Cor: ACC ACC00 rand(log rand(log nn)-CC)-CC00..

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DerandomizationDerandomization

Thm (Ajtai, Ben-Or):Thm (Ajtai, Ben-Or): 1) rand-AC1) rand-AC00 AC AC00..2) rand-ACC2) rand-ACC00 ACC ACC00..

Open: Open: rand-CC rand-CC00 CC CC00 ? ?

Claim: Claim: rand-CCrand-CC00 = CC = CC0 0 iff AND iff AND CC CC00..

Thm: Thm: ACCACC00 AND AND OR OR CC CC00..

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Thm: Thm: ACCACC00 AND AND OR OR CC CC00. . (non-(non-uniformly)uniformly)

Pf:Pf: Technique of Ajtai and Ben-Or Technique of Ajtai and Ben-Or

CCnn a rand-CC a rand-CC00 circuit computing f circuit computing fnn with error <1/3 with error <1/3nn..

Take OR of Take OR of nn independent copies of C independent copies of Cnn

if fif fn n ( ( x x ) = 1 then OR ) = 1 then OR C Cnn( ( xx ) = 0 with probability < ) = 0 with probability < ( 1/3( 1/3n n ))nn

if fif fn n ( ( x x ) = 0 then OR ) = 0 then OR C Cnn( ( xx ) = 1 with probability < ) = 1 with probability < 1/31/3

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CCnn a rand-CC a rand-CC00 circuit computing f circuit computing fnn with error <1/3 with error <1/3nn..

Take OR of Take OR of nn independent copies of C independent copies of Cnn

if fif fn n ( ( x x ) = 1 then OR ) = 1 then OR C Cnn( ( xx ) = 0 with probability < ) = 0 with probability < ( 1/3( 1/3n n ))nn

if fif fn n ( ( x x ) = 0 then OR ) = 0 then OR C Cnn( ( xx ) = 1 with probability < 1/3 ) = 1 with probability < 1/3

Take AND of Take AND of nn independent copies of OR independent copies of OR C Cnn

if fif fn n ( ( x x ) = 1 then AND ) = 1 then AND OR OR C Cnn( ( xx ) = 0 with p. < ) = 0 with p. < n n ( 1/3( 1/3n n ))nn

if fif fn n ( ( x x ) = 0 then AND ) = 0 then AND OR OR C Cnn( ( xx ) = 1 with p. < ( 1/3 ) = 1 with p. < ( 1/3 ))nn

In both cases the probability of error is less than 2In both cases the probability of error is less than 2n n so we so we can fix a particular random bits that will give the correct can fix a particular random bits that will give the correct answer for all answer for all xx..

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Previous construction requires to fix >Previous construction requires to fix >nn22 random random bits so it is non-uniform.bits so it is non-uniform.

One can get uniform construction using:One can get uniform construction using:

1.1. Lautemann’s technique, andLautemann’s technique, and

2.2. Randomness efficient sampling using random Randomness efficient sampling using random walks on expanders.walks on expanders.

Similar to [AH, V]Similar to [AH, V]

Thm: Thm: ACCACC00 AND AND OR OR CC CC00. . (uniformly) (uniformly)

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Geometric restrictions of circuits and Geometric restrictions of circuits and branching programsbranching programs

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Geometric restrictions of circuits and Geometric restrictions of circuits and branching programsbranching programs

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Constant width circuitsConstant width circuits

Thm (Barrington): Thm (Barrington): NC NC11 corresponds to constant corresponds to constant width circuits.width circuits.

Thm (Hansen’06): Thm (Hansen’06): ACC ACC00 corresponds to constant corresponds to constant width planar circuits.width planar circuits.

Thm (BLMS’99): Thm (BLMS’99): ACAC00 corresponds to constant width corresponds to constant width upward planar circuits.upward planar circuits.

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Constant width nondeterministic branching Constant width nondeterministic branching programsprograms

Thm (Barrington): Thm (Barrington): NC NC11 corresponds to constant corresponds to constant width nondeterministic branching programs.width nondeterministic branching programs.

Thm: Thm: ACC ACC00 corresponds to constant width planar corresponds to constant width planar nondeterministic branching programs.nondeterministic branching programs.

Thm (BLMS’98): Thm (BLMS’98): ACAC00 corresponds to constant width corresponds to constant width upward planar nondeterministic branching upward planar nondeterministic branching programs.programs.

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Constant width nondeterministic branching Constant width nondeterministic branching programsprograms

Thm (Hansen’08): Thm (Hansen’08): Quasi-polynomial size ACC Quasi-polynomial size ACC00 corresponds to quasi-polynomial size constant width corresponds to quasi-polynomial size constant width planar nondeterministic branching programs.planar nondeterministic branching programs.

Thm (HMV): Thm (HMV): Functions computable by constant width Functions computable by constant width planar nondeterministic branching programs are in planar nondeterministic branching programs are in ACCACC00..

Thm (Hansen’08): Thm (Hansen’08): Functions from AND Functions from AND OR OR CC CC00 are are computable by constant width planar computable by constant width planar nondeterministic branching programs.nondeterministic branching programs.

Thm: Thm: ACC ACC00 corresponds to constant width planar corresponds to constant width planar nondeterministic branching programs.nondeterministic branching programs.