Average case lower bounds for threshold circuits Ruiwen Chen, Rahul Santhanam and Srikanth Srinivasan University of Oxford and Department of Mathematics, IIT Bombay CCC 2016 Average case bounds for TC 0 Chen, Santhanam, S. May 2016 1 / 21
Boolean Circuits Circuit computing function f : { 0 , 1 } n → { 0 , 1 } . g 1 Computation proceeds through “simple” g 3 operations. g 2 x 1 x 2 x 3 x 4 Average case bounds for TC 0 Chen, Santhanam, S. May 2016 2 / 21
Boolean Circuits Circuit computing function f : { 0 , 1 } n → { 0 , 1 } . g 1 Computation proceeds through “simple” g 3 operations. g 2 g i ∈ “basic” operations. x 1 x 2 x 3 x 4 Average case bounds for TC 0 Chen, Santhanam, S. May 2016 2 / 21
Boolean Circuits Circuit computing function f : { 0 , 1 } n → { 0 , 1 } . g 1 Computation proceeds through “simple” g 3 operations. g 2 g i ∈ “basic” operations. Designated output gate x 1 x 2 x 3 x 4 computes function f . Average case bounds for TC 0 Chen, Santhanam, S. May 2016 2 / 21
Boolean Circuits Size s of the circuit: time g 1 taken by algorithm. g 3 g 2 depth = 3 x 1 x 2 x 3 x 4 # wires = 8 , # gates = 3 Average case bounds for TC 0 Chen, Santhanam, S. May 2016 3 / 21
Boolean Circuits Size s of the circuit: time g 1 taken by algorithm. Could be # edges/wires or # gates. g 3 g 2 depth = 3 x 1 x 2 x 3 x 4 # wires = 8 , # gates = 3 Average case bounds for TC 0 Chen, Santhanam, S. May 2016 3 / 21
Boolean Circuits Size s of the circuit: time g 1 taken by algorithm. Could be # edges/wires or # gates. g 3 # wires ≤ ( n + # gates) · g 2 depth = 3 # gates. x 1 x 2 x 3 x 4 # wires = 8 , # gates = 3 Average case bounds for TC 0 Chen, Santhanam, S. May 2016 3 / 21
Boolean Circuits Size s of the circuit: time g 1 taken by algorithm. Could be # edges/wires or # gates. g 3 # wires ≤ ( n + # gates) · g 2 depth = 3 # gates. Depth d of the circuit: parallelism of the x 1 x 2 x 3 x 4 algorithm. # wires = 8 , # gates = 3 Average case bounds for TC 0 Chen, Santhanam, S. May 2016 3 / 21
Boolean Circuits Size s of the circuit: time g 1 taken by algorithm. Could be # edges/wires or # gates. g 3 # wires ≤ ( n + # gates) · g 2 depth = 3 # gates. Depth d of the circuit: parallelism of the x 1 x 2 x 3 x 4 algorithm. # wires = 8 , # gates = 3 s = s ( n ) , d = O (1) . Average case bounds for TC 0 Chen, Santhanam, S. May 2016 3 / 21
Threshold circuits A threshold operation: g ( x ) = 1 iff � i w i x i ≥ θ for w i , θ ∈ R . Average case bounds for TC 0 Chen, Santhanam, S. May 2016 4 / 21
Threshold circuits A threshold operation: g ( x ) = 1 iff � i w i x i ≥ θ for w i , θ ∈ R . Some examples: OR function: OR( x ) = 1 iff � i x i ≥ 1 . Average case bounds for TC 0 Chen, Santhanam, S. May 2016 4 / 21
Threshold circuits A threshold operation: g ( x ) = 1 iff � i w i x i ≥ θ for w i , θ ∈ R . Some examples: OR function: OR( x ) = 1 iff � i x i ≥ 1 . AND function: AND( x ) = � � i x i ≥ n � . Average case bounds for TC 0 Chen, Santhanam, S. May 2016 4 / 21
Threshold circuits A threshold operation: g ( x ) = 1 iff � i w i x i ≥ θ for w i , θ ∈ R . Some examples: OR function: OR( x ) = 1 iff � i x i ≥ 1 . AND function: AND( x ) = � � i x i ≥ n � . MAJ function: MAJ( x ) = � � i x i ≥ n/ 2 � . Average case bounds for TC 0 Chen, Santhanam, S. May 2016 4 / 21
Threshold circuits A threshold operation: g ( x ) = 1 iff � i w i x i ≥ θ for w i , θ ∈ R . Some examples: OR function: OR( x ) = 1 iff � i x i ≥ 1 . AND function: AND( x ) = � � i x i ≥ n � . MAJ function: MAJ( x ) = � � i x i ≥ n/ 2 � . i 2 i ( x i − y i ) ≥ 0 � . GEQ function: GEQ( x, y ) = � � Average case bounds for TC 0 Chen, Santhanam, S. May 2016 4 / 21
Threshold circuits A threshold operation: g ( x ) = 1 iff � i w i x i ≥ θ for w i , θ ∈ R . Some examples: OR function: OR( x ) = 1 iff � i x i ≥ 1 . AND function: AND( x ) = � � i x i ≥ n � . MAJ function: MAJ( x ) = � � i x i ≥ n/ 2 � . i 2 i ( x i − y i ) ≥ 0 � . GEQ function: GEQ( x, y ) = � � TC 0 g ( s, d ) : threshold circuits with s gates and depth d . Average case bounds for TC 0 Chen, Santhanam, S. May 2016 4 / 21
Threshold circuits A threshold operation: g ( x ) = 1 iff � i w i x i ≥ θ for w i , θ ∈ R . Some examples: OR function: OR( x ) = 1 iff � i x i ≥ 1 . AND function: AND( x ) = � � i x i ≥ n � . MAJ function: MAJ( x ) = � � i x i ≥ n/ 2 � . i 2 i ( x i − y i ) ≥ 0 � . GEQ function: GEQ( x, y ) = � � TC 0 g ( s, d ) : threshold circuits with s gates and depth d . TC 0 w ( s, d ) : threshold circuits with s wires and depth d . Average case bounds for TC 0 Chen, Santhanam, S. May 2016 4 / 21
Threshold circuits A threshold operation: g ( x ) = 1 iff � i w i x i ≥ θ for w i , θ ∈ R . Some examples: OR function: OR( x ) = 1 iff � i x i ≥ 1 . AND function: AND( x ) = � � i x i ≥ n � . MAJ function: MAJ( x ) = � � i x i ≥ n/ 2 � . i 2 i ( x i − y i ) ≥ 0 � . GEQ function: GEQ( x, y ) = � � TC 0 g ( s, d ) : threshold circuits with s gates and depth d . TC 0 w ( s, d ) : threshold circuits with s wires and depth d . Generalize AC 0 circuits made up of AND and OR gates. Average case bounds for TC 0 Chen, Santhanam, S. May 2016 4 / 21
The power of threshold circuits f = PARITY( x 1 , . . . , x n ) = x 1 ⊕ x 2 ⊕ · · · ⊕ x n . Average case bounds for TC 0 Chen, Santhanam, S. May 2016 5 / 21
The power of threshold circuits f = PARITY( x 1 , . . . , x n ) = x 1 ⊕ x 2 ⊕ · · · ⊕ x n . d ≥ 2 : f ∈ TC 0 g ( dn 1 / ( d − 1) , d ) (Siu-Roychowdhury-Kailath 1991) Average case bounds for TC 0 Chen, Santhanam, S. May 2016 5 / 21
The power of threshold circuits f = PARITY( x 1 , . . . , x n ) = x 1 ⊕ x 2 ⊕ · · · ⊕ x n . d ≥ 2 : f ∈ TC 0 g ( dn 1 / ( d − 1) , d ) (Siu-Roychowdhury-Kailath 1991) w ( n 1+ ε d , d ) (Beame-Brisson-Ladner, Paturi-Saks 1991) f ∈ TC 0 Average case bounds for TC 0 Chen, Santhanam, S. May 2016 5 / 21
The power of threshold circuits f = PARITY( x 1 , . . . , x n ) = x 1 ⊕ x 2 ⊕ · · · ⊕ x n . d ≥ 2 : f ∈ TC 0 g ( dn 1 / ( d − 1) , d ) (Siu-Roychowdhury-Kailath 1991) w ( n 1+ ε d , d ) (Beame-Brisson-Ladner, Paturi-Saks 1991) f ∈ TC 0 Compare with: PARITY does not have AC 0 circuits of subexponential size (H˚ astad 1986). Average case bounds for TC 0 Chen, Santhanam, S. May 2016 5 / 21
Circuit lower bounds Problem: Find explicit family of functions (say in NP) that have no TC 0 circuits of poly ( n ) size. Average case bounds for TC 0 Chen, Santhanam, S. May 2016 6 / 21
Circuit lower bounds Problem: Find explicit family of functions (say in NP) that have no TC 0 circuits of poly ( n ) size. Even open for depth 2 . Average case bounds for TC 0 Chen, Santhanam, S. May 2016 6 / 21
Work on threshold circuits Hajnal Maass Pudl´ ak Turan Szegedy 1987 (Polynomial Approximations) Paturi Saks 1991, Siu Roychowdhury Kailath 1992; Beigel 1994; Aspnes Beigel Furst Rudich 1994, Podolskii 2012 (Combinatorial restrictions) Impagliazzo Paturi Saks 1991 (Communication complexity) Goldmann Hastad Razborov 1992; Nisan 1992; Hansen Miltersen 2004; Chattopadhyay Hansen 2005; Lovett, S. 2012 (Analytic techniques) Gopalan Servedio 2010 Average case bounds for TC 0 Chen, Santhanam, S. May 2016 7 / 21
State-of-the-art lower bounds (Impagliazzo-Paturi-Saks 1991) PARITY not in TC 0 g ( n 1 / 2( d − 1) , d ) and w ( n 1+ ε d , d ) . TC 0 Average case bounds for TC 0 Chen, Santhanam, S. May 2016 8 / 21
State-of-the-art lower bounds (Impagliazzo-Paturi-Saks 1991) PARITY not in TC 0 g ( n 1 / 2( d − 1) , d ) and w ( n 1+ ε d , d ) . TC 0 (Kane-Williams 2015) Explicit functions not in TC 0 g ( n 1 . 5 − o (1) , 2) and TC 0 w ( n 2 . 5 − o (1) , 2) . Average case bounds for TC 0 Chen, Santhanam, S. May 2016 8 / 21
State-of-the-art lower bounds (Impagliazzo-Paturi-Saks 1991) PARITY not in TC 0 g ( n 1 / 2( d − 1) , d ) and w ( n 1+ ε d , d ) . TC 0 (Kane-Williams 2015) Explicit functions not in TC 0 g ( n 1 . 5 − o (1) , 2) and TC 0 w ( n 2 . 5 − o (1) , 2) . Also extends to a special case of depth- 3 . Average case bounds for TC 0 Chen, Santhanam, S. May 2016 8 / 21
Average case lower bounds Want to show a function f : { 0 , 1 } n → { 0 , 1 } hard on average . Average case bounds for TC 0 Chen, Santhanam, S. May 2016 9 / 21
Average case lower bounds Want to show a function f : { 0 , 1 } n → { 0 , 1 } hard on average . Trivial to compute f on half the inputs. Average case bounds for TC 0 Chen, Santhanam, S. May 2016 9 / 21
Average case lower bounds Want to show a function f : { 0 , 1 } n → { 0 , 1 } hard on average . Trivial to compute f on half the inputs. f has ε -correlation with ckt C if x [ C ( x ) = f ( x )] − 1 Corr( C, f ) := Pr 2 ≤ ε. Average case bounds for TC 0 Chen, Santhanam, S. May 2016 9 / 21
Average case lower bounds Want to show a function f : { 0 , 1 } n → { 0 , 1 } hard on average . Trivial to compute f on half the inputs. f has ε -correlation with ckt C if x [ C ( x ) = f ( x )] − 1 Corr( C, f ) := Pr 2 ≤ ε. Want to show that f hard on average against TC 0 ( s, d ) . Average case bounds for TC 0 Chen, Santhanam, S. May 2016 9 / 21
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