ACC Circuits Research Articles

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

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 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 mathcal { C}, by showing that 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 {mathcal C}. Say that a symmetric Boolean function f(x1,…,xn) is sparse if it outputs 1 on O(1) values of {sum }_{i} x_{i}. We show that for every sparse f, and for all “typical” mathcal { C}, faster # SAT algorithms for mathcal { C} circuits imply lower bounds against the circuit class f circ mathcal { C}, which may be stronger than mathcal { C} itself. In particular: # SAT algorithms for nk-size mathcal { C}-circuits running in 2n/nk time (for all k) imply NEXP does not have (f circ mathcal { C})-circuits of polynomial size.# SAT algorithms for 2^{n^{{varepsilon }}}-size mathcal { C}-circuits running in 2^{n-n^{{varepsilon }}} time (for some ε > 0) imply Quasi-NP does not have (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.

Nonuniform ACC Circuit Lower Bounds

The class ACC consists of circuit families with constant depth over unbounded fan-in AND, OR, NOT, and MOD m gates, where m > 1 is an arbitrary constant. We prove the following. ---NEXP, the class of languages accepted in nondeterministic exponential time, does not have nonuniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasipolynomials and other less natural functions. ---E NP , the class of languages recognized in 2 O(n) time with an NP oracle, doesn’t have nonuniform ACC circuits of 2 n o(1) size. The lower bound gives an exponential size-depth tradeoff: for every d, m there is a δ > 0 such that E NP doesn’t have depth- d ACC circuits of size 2 n δ with MOD m gates. Previously, it was not known whether EXP NP had depth-3 polynomial-size circuits made out of only MOD 6 gates. The high-level strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms entail these lower bounds. The algorithms combine known properties of ACC with fast rectangular matrix multiplication and dynamic programming, while the second step requires a strengthening of the author’s prior work.

SIGACT news complexity theory column 71

I will discuss the recent proof that the complexity class NEXP (nondeterministic exponential time) lacks nonuniform ACC circuits of polynomial size. The proof will be described from the perspective of someone trying to discover it. 1 c ©R. R. Williams, 2011. Computer Science Department, Stanford University, Stanford, CA, USA. rrwilliams@gmail.com. At the time of writing, the author was supported by the Josef Raviv Memorial Fellowship at IBM Research – Almaden.

Guest column

I will discuss the recent proof that the complexity class NEXP (nondeterministic exponential time) lacks nonuniform ACC circuits of polynomial size. The proof will be described from the perspective of someone trying to discover it.

A Uniform Circuit Lower Bound for the Permanent

The authors show that uniform families of ACC circuits of subexponential size cannot compute the permanent function. This also implies similar lower bounds for certain sets in PP This is one of the very few examples of a lower bound in circuit complexity whose proof hinges on the uniformity condition; it is still unknown if there is any set in ${\operatorname{Ntime}}(2^{n^{O(1)} } )$ that does not have nonuniform ACC circuits.