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An Elementary Proof of a 3 n o ( n ) Lower Bound on Circuit Complexity of Affine Dispersers E. Demenkov and A. Kulikov Steklov Institute of Mathematics at St. Petersburg Estonian Theory Days 08 October 2011 A. Kulikov (Steklov Institute

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SLIDE 1

An Elementary Proof of a 3n − o(n) Lower Bound on Circuit Complexity of Affine Dispersers

  • E. Demenkov and A. Kulikov

Steklov Institute of Mathematics at St. Petersburg

Estonian Theory Days 08 October 2011

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 1 / 11

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SLIDE 2

Boolean Circuits

Inputs: x1, x2, . . . , xn, 0, 1 Gates: binary functions Fan-out: unbounded g1 = x1 ⊕ x2 g2 = x2 ∧ x3 g3 = g1 ∨ g2 g4 = g2 ∨ 1 g5 = g3 ≡ g4 x1 x2 x3 1 ⊕ g1 ∧ g2 ∨ g3 ∨ g4 ≡ g5

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 2 / 11

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SLIDE 3

Random Functions are Complex

Shannon counting argument: count how many different Boolean functions in n variables can be computed by circuits with t gates and compare this number with the total number 22n of all Boolean functions.

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 3 / 11

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SLIDE 4

Random Functions are Complex

Shannon counting argument: count how many different Boolean functions in n variables can be computed by circuits with t gates and compare this number with the total number 22n of all Boolean functions. The number F(n, t) of circuits of size ≤ t with n input variables does not exceed

  • 16(t + n + 2)2t .

Each of t gates is assigned one of 16 possible binary Boolean functions that acts on two previous nodes, and each previous node can be either a previous gate (≤ t choices) or a variables or a constant (≤ n + 2 choices).

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 3 / 11

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SLIDE 5

Random Functions are Complex

Shannon counting argument: count how many different Boolean functions in n variables can be computed by circuits with t gates and compare this number with the total number 22n of all Boolean functions. The number F(n, t) of circuits of size ≤ t with n input variables does not exceed

  • 16(t + n + 2)2t .

Each of t gates is assigned one of 16 possible binary Boolean functions that acts on two previous nodes, and each previous node can be either a previous gate (≤ t choices) or a variables or a constant (≤ n + 2 choices). For t = 2n/(10n), F(n, t) is approximately 22n/5, which is ≪ 22n.

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 3 / 11

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SLIDE 6

Random Functions are Complex

Shannon counting argument: count how many different Boolean functions in n variables can be computed by circuits with t gates and compare this number with the total number 22n of all Boolean functions. The number F(n, t) of circuits of size ≤ t with n input variables does not exceed

  • 16(t + n + 2)2t .

Each of t gates is assigned one of 16 possible binary Boolean functions that acts on two previous nodes, and each previous node can be either a previous gate (≤ t choices) or a variables or a constant (≤ n + 2 choices). For t = 2n/(10n), F(n, t) is approximately 22n/5, which is ≪ 22n. Thus, the circuit complexity of almost all Boolean functions on n variables is exponential in n. Still, we do not know any explicit function with super-linear circuit complexity.

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 3 / 11

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SLIDE 7

Known Lower Bounds

circuit size formula size full binary basis B2 3n − o(n) n2−o(1) [Blum] [Nechiporuk] basis U2 = B2 \ {⊕, ≡} 5n − o(n) n3−o(1) [Iwama et al.] [Hastad] exponential monotone basis M2 = {∨, ∧} [Razborov; Alon, Boppana; Andreev; Karchmer, Wigderson]

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 4 / 11

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SLIDE 8

Known Lower Bounds for Circuits over B2

Known Lower Bounds 2n − c [Kloss and Malyshev, 65] 2n − c [Schnorr, 74] 2.5n − o(n) [Paul, 77] 2.5n − c [Stockmeyer, 77] 3n − o(n) [Blum, 84]

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 5 / 11

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SLIDE 9

Known Lower Bounds for Circuits over B2

Known Lower Bounds 2n − c [Kloss and Malyshev, 65] 2n − c [Schnorr, 74] 2.5n − o(n) [Paul, 77] 2.5n − c [Stockmeyer, 77] 3n − o(n) [Blum, 84] This Talk In this talk, we will present a new proof of a 3n − o(n) lower. The proof is much simpler than Blum’s proof, however the function used is much more complicated.

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 5 / 11

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SLIDE 10

Gate Elimination Method

Gate Elimination All the proofs are based on the so-called gate elimination method. This is essentially the only known method for proving lower bounds on circuit complexity.

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 6 / 11

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SLIDE 11

Gate Elimination Method

Gate Elimination All the proofs are based on the so-called gate elimination method. This is essentially the only known method for proving lower bounds on circuit complexity. The main idea Take an optimal circuit for the function in question.

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 6 / 11

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SLIDE 12

Gate Elimination Method

Gate Elimination All the proofs are based on the so-called gate elimination method. This is essentially the only known method for proving lower bounds on circuit complexity. The main idea Take an optimal circuit for the function in question. Setting some variables to constants obtain a subfunction of the same type (in order to proceed by induction) and eliminate several gates.

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 6 / 11

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SLIDE 13

Gate Elimination Method

Gate Elimination All the proofs are based on the so-called gate elimination method. This is essentially the only known method for proving lower bounds on circuit complexity. The main idea Take an optimal circuit for the function in question. Setting some variables to constants obtain a subfunction of the same type (in order to proceed by induction) and eliminate several gates. A gate is eliminated if it computes a constant or a variable.

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 6 / 11

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SLIDE 14

Gate Elimination Method

Gate Elimination All the proofs are based on the so-called gate elimination method. This is essentially the only known method for proving lower bounds on circuit complexity. The main idea Take an optimal circuit for the function in question. Setting some variables to constants obtain a subfunction of the same type (in order to proceed by induction) and eliminate several gates. A gate is eliminated if it computes a constant or a variable. By repeatedly applying this process, conclude that the original circuit must have had many gates.

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 6 / 11

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SLIDE 15

Gate Elimination Method

Gate Elimination All the proofs are based on the so-called gate elimination method. This is essentially the only known method for proving lower bounds on circuit complexity. The main idea Take an optimal circuit for the function in question. Setting some variables to constants obtain a subfunction of the same type (in order to proceed by induction) and eliminate several gates. A gate is eliminated if it computes a constant or a variable. By repeatedly applying this process, conclude that the original circuit must have had many gates.

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 6 / 11

slide-16
SLIDE 16

Gate Elimination Method

Gate Elimination All the proofs are based on the so-called gate elimination method. This is essentially the only known method for proving lower bounds on circuit complexity. The main idea Take an optimal circuit for the function in question. Setting some variables to constants obtain a subfunction of the same type (in order to proceed by induction) and eliminate several gates. A gate is eliminated if it computes a constant or a variable. By repeatedly applying this process, conclude that the original circuit must have had many gates. Remark This method is very unlikely to produce non-linear lower bounds.

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 6 / 11

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SLIDE 17

Example

x1 x2 x3 x4 ⊕ G1 ∨ G2 ∧ G3 ⊕ G4 ⊕ G5 ⊕ G6

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 7 / 11

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SLIDE 18

Example

x1 x2 x3 x4 ⊕ G1 ∨ G2 ∧ G3 ⊕ G4 ⊕ G5 ⊕ G6 assign x1 = 1

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 7 / 11

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SLIDE 19

Example

x2 x3 x4 ⊕ G1 ∨ G2 ∧ G3 ⊕ G4 ⊕ G5 ⊕ G6 1 G5 now computes G3 ⊕ 1 = ¬G3

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 7 / 11

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SLIDE 20

Example

x2 x3 x4 ⊕ G1 ∨ G2 ∧ G3 ⊕ G4 ⊕ G6 ¬

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 7 / 11

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SLIDE 21

Example

x2 x3 x4 ⊕ G1 ∨ G2 ∧ G3 ⊕ G4 ⊕ G6 ¬ now we can change the binary function assigned to G6

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 7 / 11

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SLIDE 22

Example

x2 x3 x4 ⊕ G1 ∨ G2 ∧ G3 ⊕ G4 ≡ G6

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 7 / 11

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SLIDE 23

Example

x2 x3 x4 ⊕ G1 ∨ G2 ∧ G3 ⊕ G4 ≡ G6 now assign x3 = 0

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 7 / 11

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SLIDE 24

Example

x2 x4 ⊕ G1 ∨ G2 ∧ G3 ⊕ G4 ≡ G6 G1 then is equal to x2

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 7 / 11

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SLIDE 25

Example

x2 x4 ∨ G2 ∧ G3 ⊕ G4 ≡ G6

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 7 / 11

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SLIDE 26

Example

x2 x4 ∨ G2 ∧ G3 ⊕ G4 ≡ G6 G2 = x4

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 7 / 11

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SLIDE 27

Example

x2 x4 ∧ G3 ⊕ G4 ≡ G6

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 7 / 11

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SLIDE 28

A Typical Bottleneck

x1 x2 x3 x4 x5 x6 ⊕ G1 ⊕ G2 ⊕ G3 ⊕ G4 ∧ G5 ⊕ G6

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 8 / 11

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SLIDE 29

A Typical Bottleneck

x1 x2 x3 x4 x5 x6 ⊕ G1 ⊕ G2 ⊕ G3 ⊕ G4 ∧ G5 ⊕ G6 this is how a typical bottleneck case looks like

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 8 / 11

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SLIDE 30

A Typical Bottleneck

x1 x2 x3 x4 x5 x6 ⊕ G1 ⊕ G2 ⊕ G3 ⊕ G4 ∧ G5 ⊕ G6 this is how a typical bottleneck case looks like by assigning a variable we cannot kill more than 2 gates

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 8 / 11

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SLIDE 31

A Typical Bottleneck

x1 x2 x3 x4 x5 x6 ⊕ G1 ⊕ G2 ⊕ G3 ⊕ G4 ∧ G5 ⊕ G6 this is how a typical bottleneck case looks like by assigning a variable we cannot kill more than 2 gates at the same time we cannot ex- clude that a top of a circuit looks like this

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 8 / 11

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SLIDE 32

A Typical Bottleneck

x1 x2 x3 x4 x5 x6 ⊕ G1 ⊕ G2 ⊕ G3 ⊕ G4 ∧ G5 ⊕ G6 this is how a typical bottleneck case looks like by assigning a variable we cannot kill more than 2 gates at the same time we cannot ex- clude that a top of a circuit looks like this consider a substitution x1 ⊕ x2 ⊕ x3 = 0: under it G5 trivializes

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 8 / 11

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SLIDE 33

Affine Dispersers

OK, linear substitutions do help in gate elimination, but where is a function that survives under such substitutions?

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 9 / 11

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SLIDE 34

Affine Dispersers

OK, linear substitutions do help in gate elimination, but where is a function that survives under such substitutions? Constructing a function that does not become a constant after any n − o(n) linear substitutions is non-trivial. E.g., any symmetric function may be turned into a constant after n/2 linear substitutions: x1 ⊕ x2 = 1, x3 ⊕ x4 = 1, . . . .

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 9 / 11

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SLIDE 35

Affine Dispersers

OK, linear substitutions do help in gate elimination, but where is a function that survives under such substitutions? Constructing a function that does not become a constant after any n − o(n) linear substitutions is non-trivial. E.g., any symmetric function may be turned into a constant after n/2 linear substitutions: x1 ⊕ x2 = 1, x3 ⊕ x4 = 1, . . . . An object that we are looking for is called an affine disperser.

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 9 / 11

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SLIDE 36

Affine Dispersers

OK, linear substitutions do help in gate elimination, but where is a function that survives under such substitutions? Constructing a function that does not become a constant after any n − o(n) linear substitutions is non-trivial. E.g., any symmetric function may be turned into a constant after n/2 linear substitutions: x1 ⊕ x2 = 1, x3 ⊕ x4 = 1, . . . . An object that we are looking for is called an affine disperser. Formally, an affine disperser for dimension d is a function f : {0, 1}n → {0, 1} that is not constant on any affine subspace of {0, 1}n of dimension at least d.

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 9 / 11

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SLIDE 37

Affine Dispersers

OK, linear substitutions do help in gate elimination, but where is a function that survives under such substitutions? Constructing a function that does not become a constant after any n − o(n) linear substitutions is non-trivial. E.g., any symmetric function may be turned into a constant after n/2 linear substitutions: x1 ⊕ x2 = 1, x3 ⊕ x4 = 1, . . . . An object that we are looking for is called an affine disperser. Formally, an affine disperser for dimension d is a function f : {0, 1}n → {0, 1} that is not constant on any affine subspace of {0, 1}n of dimension at least d. Only recently, an explicit affine disperser for d = o(n) was constructed [Ben-Sasson and Kopparty, 09].

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 9 / 11

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SLIDE 38

Proof Idea

x1 x2 x3 x4 x5 x6 ⊕ G1 ⊕ G2 ⊕ G3 ⊕ G4 ∧ G5 ⊕ G6

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 10 / 11

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SLIDE 39

Proof Idea

x1 x2 x3 x4 x5 x6 ⊕ G1 ⊕ G2 ⊕ G3 ⊕ G4 ∧ G5 ⊕ G6 take the first gate which is not a XOR of out-degree 1

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 10 / 11

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SLIDE 40

Proof Idea

x1 x2 x3 x4 x5 x6 ⊕ G1 ⊕ G2 ⊕ G3 ⊕ G4 ∧ G5 ⊕ G6 take the first gate which is not a XOR of out-degree 1

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 10 / 11

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SLIDE 41

Proof Idea

x1 x2 x3 x4 x5 x6 ⊕ G1 ⊕ G2 ⊕ G3 ⊕ G4 ∧ G5 ⊕ G6 take the first gate which is not a XOR of out-degree 1 both its inputs compute linear functions

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 10 / 11

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SLIDE 42

Proof Idea

x1 x2 x3 x4 x5 x6 ⊕ G1 ⊕ G2 ⊕ G3 ⊕ G4 ∧ G5 ⊕ G6 take the first gate which is not a XOR of out-degree 1 both its inputs compute linear functions

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 10 / 11

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SLIDE 43

Proof Idea

x1 x2 x3 x4 x5 x6 ⊕ G1 ⊕ G2 ⊕ G3 ⊕ G4 ∧ G5 ⊕ G6 take the first gate which is not a XOR of out-degree 1 both its inputs compute linear functions make a substitution x1 ⊕ x2 ⊕ x3 ⊕ x5 ⊕ x6 = 1

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 10 / 11

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SLIDE 44

Proof Idea

x1 x2 x3 x4 x5 x6 ⊕ G1 ⊕ G2 ⊕ G3 ⊕ G4 ∧ G5 ⊕ G6 take the first gate which is not a XOR of out-degree 1 both its inputs compute linear functions make a substitution x1 ⊕ x2 ⊕ x3 ⊕ x5 ⊕ x6 = 1 this kills the considered gate and all its successors

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 10 / 11

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SLIDE 45

Proof Idea

x1 x2 x3 x4 x5 x6 ⊕ G1 ⊕ G2 ⊕ G3 ⊕ G4 ∧ G5 ⊕ G6 take the first gate which is not a XOR of out-degree 1 both its inputs compute linear functions make a substitution x1 ⊕ x2 ⊕ x3 ⊕ x5 ⊕ x6 = 1 this kills the considered gate and all its successors

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 10 / 11

slide-46
SLIDE 46

Proof Idea

x1 x2 x3 x4 x5 x6 ⊕ G1 ⊕ G2 ⊕ G3 ⊕ G4 ∧ G5 ⊕ G6 take the first gate which is not a XOR of out-degree 1 both its inputs compute linear functions make a substitution x1 ⊕ x2 ⊕ x3 ⊕ x5 ⊕ x6 = 1 this kills the considered gate and all its successors moreover, all its predecessors are not needed any more

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 10 / 11

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SLIDE 47

Proof Idea

x1 x2 x3 x4 x5 x6 ⊕ G1 ⊕ G2 ⊕ G3 ⊕ G4 ∧ G5 ⊕ G6 take the first gate which is not a XOR of out-degree 1 both its inputs compute linear functions make a substitution x1 ⊕ x2 ⊕ x3 ⊕ x5 ⊕ x6 = 1 this kills the considered gate and all its successors moreover, all its predecessors are not needed any more

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 10 / 11

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SLIDE 48

Proof Idea

x1 x2 x3 x4 x5 x6 ⊕ G1 ⊕ G2 ⊕ G3 ⊕ G4 ∧ G5 ⊕ G6 by a short case analysis it is possible to show that this way one can always eliminate 3 gates; since we can make n − o(n) such substitutions a lower bound 3n − o(n) follows

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 10 / 11

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SLIDE 49

Thank you for your attention!

  • A. Kulikov

(Steklov Institute of Mathematics at St. Petersburg) 3n Lower Bound 11 / 11