🤖 AI Summary
使用Lean和切换引理形式化Hastad的PARITY下界，证明了对于固定深度d的电路计算PARITY需要指数级大小，并构造了一个多项式大小、对数深度的有界扇入公式家族。
📝 Abstract
We formalize Hastad's PARITY lower bound in Lean using the switching lemma. For every fixed d >= 2, formulas and DAG circuits of computation depth at most d computing PARITY on n inputs require size exp(Omega_d(n^(1/(d-1)))) for all sufficiently large n. This matches the classical upper bound up to constants in the exponent and implies that PARITY is not in nonuniform AC0. We also construct a polynomial-size, logarithmic-depth bounded-fan-in formula family for PARITY, providing a witness to NC1 is not a subset of AC0 for the formalized models. The Lean source code is available at https://github.com/formalcs/circuit-complexity and is checked with Lean 4.33.1 and mathlib 4.33.1.