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

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LIPIcs.STACS.2020.59.pdf
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## Cite As

Nikhil Vyas and R. Ryan Williams. Lower Bounds Against Sparse Symmetric Functions of ACC Circuits: Expanding the Reach of #SAT Algorithms. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 59:1-59:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.STACS.2020.59

## 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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Circuit complexity
##### Keywords
• #SAT
• satisfiability
• circuit complexity
• exact majority
• ACC

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## References

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