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 of Mathematics at St. Petersburg) 3 n Lower Bound 1 / 11
Boolean Circuits x 1 x 2 x 3 1 Inputs: = x 1 ⊕ x 2 g 1 x 1 , x 2 , . . . , x n , 0 , 1 g 1 g 2 g 2 = x 2 ∧ x 3 ⊕ ∧ Gates: = g 1 ∨ g 2 g 3 binary functions g 3 g 4 ∨ ∨ g 4 = g 2 ∨ 1 Fan-out: = g 3 ≡ g 4 unbounded g 5 g 5 ≡ A. Kulikov (Steklov Institute of Mathematics at St. Petersburg) 3 n Lower Bound 2 / 11
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 2 2 n of all Boolean functions. A. Kulikov (Steklov Institute of Mathematics at St. Petersburg) 3 n Lower Bound 3 / 11
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 2 2 n 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) 2 � t . � 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) 3 n Lower Bound 3 / 11
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 2 2 n 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) 2 � t . � 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 = 2 n / (10 n ), F ( n , t ) is approximately 2 2 n / 5 , which is ≪ 2 2 n . A. Kulikov (Steklov Institute of Mathematics at St. Petersburg) 3 n Lower Bound 3 / 11
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 2 2 n 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) 2 � t . � 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 = 2 n / (10 n ), F ( n , t ) is approximately 2 2 n / 5 , which is ≪ 2 2 n . 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) 3 n Lower Bound 3 / 11
Known Lower Bounds circuit size formula size n 2 − o (1) full binary basis B 2 3 n − o ( n ) [Blum] [Nechiporuk] n 3 − o (1) basis U 2 = B 2 \ {⊕ , ≡} 5 n − o ( n ) [Iwama et al.] [Hastad] exponential monotone basis M 2 = {∨ , ∧} [Razborov; Alon, Boppana; Andreev; Karchmer, Wigderson] A. Kulikov (Steklov Institute of Mathematics at St. Petersburg) 3 n Lower Bound 4 / 11
Known Lower Bounds for Circuits over B 2 Known Lower Bounds 2 n − c [Kloss and Malyshev, 65] 2 n − c [Schnorr, 74] 2 . 5 n − o ( n ) [Paul, 77] 2 . 5 n − c [Stockmeyer, 77] 3 n − o ( n ) [Blum, 84] A. Kulikov (Steklov Institute of Mathematics at St. Petersburg) 3 n Lower Bound 5 / 11
Known Lower Bounds for Circuits over B 2 Known Lower Bounds 2 n − c [Kloss and Malyshev, 65] 2 n − c [Schnorr, 74] 2 . 5 n − o ( n ) [Paul, 77] 2 . 5 n − c [Stockmeyer, 77] 3 n − o ( n ) [Blum, 84] This Talk In this talk, we will present a new proof of a 3 n − 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) 3 n Lower Bound 5 / 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. A. Kulikov (Steklov Institute of Mathematics at St. Petersburg) 3 n Lower Bound 6 / 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) 3 n Lower Bound 6 / 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. 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) 3 n Lower Bound 6 / 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. 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) 3 n Lower Bound 6 / 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. 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) 3 n Lower Bound 6 / 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. 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) 3 n Lower Bound 6 / 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. 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) 3 n Lower Bound 6 / 11
Example x 1 x 2 x 3 x 4 ⊕ G 1 G 2 ∨ G 3 ∧ G 4 ⊕ ⊕ G 5 ⊕ G 6 A. Kulikov (Steklov Institute of Mathematics at St. Petersburg) 3 n Lower Bound 7 / 11
Example x 1 x 2 x 3 x 4 ⊕ G 1 G 2 ∨ G 3 ∧ G 4 ⊕ ⊕ G 5 ⊕ G 6 assign x 1 = 1 A. Kulikov (Steklov Institute of Mathematics at St. Petersburg) 3 n Lower Bound 7 / 11
Example x 2 x 3 x 4 1 ⊕ G 1 G 2 ∨ G 3 ∧ G 4 ⊕ ⊕ G 5 ⊕ G 6 G 5 now computes G 3 ⊕ 1 = ¬ G 3 A. Kulikov (Steklov Institute of Mathematics at St. Petersburg) 3 n Lower Bound 7 / 11
Example x 2 x 3 x 4 ⊕ G 1 G 2 ∨ G 3 ∧ G 4 ⊕ ¬ ⊕ G 6 A. Kulikov (Steklov Institute of Mathematics at St. Petersburg) 3 n Lower Bound 7 / 11
Example x 2 x 3 x 4 ⊕ G 1 G 2 ∨ G 3 ∧ G 4 ⊕ ¬ ⊕ G 6 now we can change the binary function assigned to G 6 A. Kulikov (Steklov Institute of Mathematics at St. Petersburg) 3 n Lower Bound 7 / 11
Example x 2 x 3 x 4 ⊕ G 1 G 2 ∨ G 3 ∧ G 4 ⊕ ≡ G 6 A. Kulikov (Steklov Institute of Mathematics at St. Petersburg) 3 n Lower Bound 7 / 11
Example x 2 x 3 x 4 ⊕ G 1 G 2 ∨ G 3 ∧ G 4 ⊕ ≡ G 6 now assign x 3 = 0 A. Kulikov (Steklov Institute of Mathematics at St. Petersburg) 3 n Lower Bound 7 / 11
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