We study the gap between the minimum size of a Boolean circuit (DAG) and the minimum size of a formula (tree circuit) over the And-Inverter Graph (AIG) basis {AND, NOT} with free inversions. We prove that this gap is always 0 or 1 (Unit Gap Theorem), that sharing requires opt(f) >= n essential variables (Threshold Theorem), and that no sharing is needed when opt(f) <= 3 (Tree Theorem). Gate counts in optimal circuits satisfy an exact decomposition formula with a binary sharing term. When the gap equals 1, it arises from exactly one gate with fan-out 2, employing either dual-polarity or same-polarity reuse; we prove that no other sharing structure can produce a unit gap.

翻译：我们研究了在与非逆变器图（AIG）基{AND, NOT}上，布尔电路（有向无环图）的最小规模与公式（树形电路）的最小规模之间的间隙。我们证明该间隙始终为0或1（单位间隙定理），共享需要opt(f) >= n个本原变量（阈值定理），且当opt(f) <= 3时无需共享（树形定理）。最优电路中的门计数满足一个包含二元共享项的精确分解公式。当间隙等于1时，它由恰好一个扇出为2的门产生，该门采用双极性或同极性重用；我们证明没有其他共享结构能产生单位间隙。