Lower Bounds Against Sparse Symmetric Functions of ACC Circuits: Expanding the Reach of #SAT Algorithms

Abstract

Abstract We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in

Quasi

-

NP

=

NTIME

[

n

( log n )

O ( 1 )

]

$\mathsf {Quasi}\text {-}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$ and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes

C

$\mathcal { C}$ , by showing that

C

$\mathcal { C}$ admits non-trivial satisfiability and/or # SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial # SAT algorithm for a circuit class

C

${\mathcal C}$ . Say that a symmetric Boolean function f(x1,…,xn) is sparse if it outputs 1 on O(1) values of

∑

i

x

i

${\sum }_{i} x_{i}$ . We show that for every sparse f, and for all “typical”

C

$\mathcal { C}$ , faster # SAT algorithms for

C

$\mathcal { C}$ circuits imply lower bounds against the circuit class

f ∘ C

$f \circ \mathcal { C}$ , which may be stronger than

C

$\mathcal { C}$ itself. In particular:

# SAT algorithms for nk-size

C

$\mathcal { C}$ -circuits running in 2n/nk time (for all k) imply NEXP does not have

( f ∘ C )

$(f \circ \mathcal { C})$ -circuits of polynomial size.

# SAT algorithms for

2

n

ε

$2^{n^{{\varepsilon }}}$ -size

C

$\mathcal { C}$ -circuits running in

2

n −

n

ε

$2^{n-n^{{\varepsilon }}}$ time (for some ε > 0) imply Quasi-NP does not have

( f ∘ C )

$(f \circ \mathcal { C})$ -circuits of polynomial size.

Applying # SAT algorithms from the literature, one immediate corollary of our results is that Quasi-NP does not have EMAJ ∘ ACC0 ∘ THR circuits of polynomial size, where EMAJ is the “exact majority” function, improving previous lower bounds against ACC0 [Williams JACM’14] and ACC0 ∘THR [Williams STOC’14], [Murray-Williams STOC’18]. This is the first nontrivial lower bound against such a circuit class.