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

Nikhil Vyas, Ryan Williams|arXiv (Cornell University)|Jan 1, 2020

Complexity and Algorithms in Graphs5 citations

TL;DR

This paper establishes a novel connection between efficient #SAT algorithms for ACC circuits and strong circuit lower bounds, showing that non-trivial #SAT algorithms for ACC circuits imply that Quasi-NP does not have polynomial-size circuits of the form f ◦ ACC₀, where f is a sparse symmetric function. The key contribution is a general framework that amplifies the power of #SAT algorithms to yield lower bounds against more expressive circuit classes than the original class being analyzed.

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)}] and NEXP do not have small circuits (in the worst case and/or on average) from various circuit classes C, by showing that 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 non-trivial #SAT algorithm for a circuit class {C}. Say a symmetric Boolean function f(x₁,…,x_n) is sparse if it outputs 1 on O(1) values of ∑_i x_i. We show that for every sparse f, and for all "typical" C, faster #SAT algorithms for C circuits actually imply lower bounds against the circuit class f ∘ C, which may be stronger than C itself. In particular: - #SAT algorithms for n^k-size C-circuits running in 2ⁿ/n^k time (for all k) imply NEXP does not have f ∘ C-circuits of polynomial size. - #SAT algorithms for 2^{n^ε}-size C-circuits running in 2^{n-n^ε} time (for some ε > 0) imply Quasi-NP does not have f ∘ 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.

Motivation & Objective

• To extend the algorithmic approach to circuit lower bounds beyond standard circuit classes.
• To investigate whether #SAT algorithms for a class C can imply lower bounds against a strictly more powerful class f ◦ C.
• To establish a general framework for lifting #SAT algorithmic progress into stronger circuit lower bounds.
• To prove the first nontrivial circuit lower bound against the class EMAJ ◦ ACC₀ ◦ THR.

Proposed method

• Leverages #SAT algorithms for ACC circuits to compute uniform symmetric functions of C-circuits.
• Reduces the problem of verifying large independent sets in a graph to evaluating sums over C-circuits.
• Uses a non-deterministic algorithm to simulate a symmetric function f of C-circuits via a decomposition into t sub-circuits.
• Employs a #SAT algorithm to compute the total number of satisfying assignments across all inputs.
• Applies a counting argument to distinguish between YES and NO cases based on the size of the independent set.
• Relies on the assumption that #SAT for poly-size C-circuits can be solved in time 2ⁿ / b(n) with b(n) = nω(1), leading to a contradiction if the lower bound fails.

Experimental results

Research questions

• RQ1 Can #SAT algorithms for a circuit class C imply lower bounds against a strictly more powerful class f ◦ C?
• RQ2 What is the minimal algorithmic progress required to yield nontrivial lower bounds against f ◦ C for sparse symmetric f?
• RQ3 Can the algorithmic-to-lower-bound paradigm be extended to classes beyond ACC₀?
• RQ4 Is it possible to prove the first nontrivial lower bound against EMAJ ◦ ACC₀ ◦ THR circuits?
• RQ5 How does the structure of sparse symmetric functions interact with circuit complexity and #SAT algorithms?

Key findings

• A #SAT algorithm for nk-size ACC circuits running in 2ⁿ / nk time for all k implies NEXP ∉ f ◦ ACC₀ for any sparse symmetric f.
• A #SAT algorithm for 2ⁿε-size ACC circuits running in 2ⁿ⁻ⁿε time for some ε > 0 implies Quasi-NP ∉ f ◦ ACC₀ for any sparse symmetric f.
• Applying known #SAT algorithms yields the first nontrivial lower bound: Quasi-NP does not have polynomial-size EMAJ ◦ ACC₀ ◦ THR circuits.
• The framework shows that #SAT algorithms can imply lower bounds against circuit classes strictly more powerful than the class being analyzed.
• The proof relies on a counting argument over independent sets in a graph derived from the circuit structure, verifiable via #SAT queries.
• The running time of the overall algorithm is 2ⁿ / b(n) with b(n) = nω(1), leading to a contradiction under the assumption that the lower bound fails.