Correlation Bounds and #SAT Algorithms for Small Linear-Size Circuits

We revisit the gate elimination method, generalize it to prove correlation bounds of boolean circuits with Parity, and also derive deterministic #SAT algorithms for small linear-size circuits. In particular, we prove that, for boolean circuits of size \(3n - n^{0.51}\), the correlation with Parity is at most \(2^{-n^{\varOmega (1)}}\), and there is a #SAT algorithm running in time \(2^{n-n^{\varOmega (1)}}\); for circuit size 2.99n, the correlation with Parity is at most \(2^{-{\varOmega (n)}}\), and there is a #SAT algorithm running in time \(2^{n-{\varOmega (n)}}\). Similar correlation bounds and algorithms are also proved for circuits of size almost 2.5n over the full binary basis \(B_2\).

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