We give a combinatorial analysis (using edge expansion) of a variant of the iterative expander construction due to Reingold, Vadhan, and Wigderson [44], and show that this analysis can be formalized in the bounded arithmetic system ${\mathsf{VNC}}^1$ (corresponding to the “${\mathsf{NC}}^1$ reasoning”). As a corollary, we prove the assumption made by Jeřábek [28] that a construction of certain bipartite expander graphs can be formalized in ${\mathsf{VNC}}^1$. This in turn implies that every proof in Gentzen's sequent calculus LK of a monotone sequent can be simulated in the monotone version of LK (MLK) with only polynomial blowup in proof size, strengthening the quasipolynomial simulation result of Atserias, Galesi, and Pudlák [9].

我们给出了由Reingold，Vadhan和Wigderson [44]提出的迭代扩展器构造变体的组合分析（使用边缘扩展），并表明该分析可以在有界算术系统中形式化 ${\mathsf{VNC}}^{\mathrm{1个}}$ （对应于“${\mathsf{数控}}^{\mathrm{1个}}$推理”）。作为推论，我们证明了Jeřábek[28]所作的假设，即可以将某些二分式展开图的形式化为${\mathsf{VNC}}^{\mathrm{1个}}$。反过来，这意味着可以在LK（MLK）的单调版本中模拟单调序列的Gentzen后续演算LK中的每个证明，而证明大小只有多项式爆炸，从而加强了Atserias，Galesi和Pudlák的准多项式模拟结果[9]。 ]。