一句话：证明 LREC 严格弱于 LFP：路径系统问题在 LFP 可定义但 LREC 不可定义。
本文研究 Grohe 等人提出的对数递归逻辑 LREC 与最小不动点逻辑 LFP 的表达力关系。LREC 在树和区间图上捕获对数空间复杂度类 L，但一般情形下不捕获 L，且已知包含于带计数的（最小）不动点逻辑。作者证明这一包含关系是严格的。核心方法是利用经典 P-完全问题——路径系统问题：该问题可在 LFP 中定义，却无法在 LREC 中定义。由此得到 LREC 不可定义路径系统问题，说明对数递归机制严格弱于一般最小不动点，从而完成 LREC 与 LFP 的分离。
is an extension of first-order logic with a logarithmic recursion operator. It was introduced by Grohe et al. and shown to capture the complexity class L over trees and interval graphs. It does not capture L in general as it is contained in —fixed-point logic with counting. We show that this containment is strict. In particular, we show that the path systems problem, a classic P-complete problem which is definable in —fixed-point logic—is not definable in . This shows that the logarithmic recursion mechanism is provably weaker than general least fixed points.