LREC= 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 FPC - 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 LFP - fixed-point logic - is not definable in LREC= This shows that the logarithmic recursion mechanism is provably weaker than general least fixed points. The proof is based on a novel Spoiler-Duplicator game tailored for this logic.
翻译：LREC= 是一个对数递归操作器的第一阶逻辑的延伸。 它由 Grohe 等人引入, 并显示可以捕捉树和间距图形的复杂等级 L。 它一般没有捕捉L, 因为它包含在 FPC 中 - 固定点逻辑和计数中。 我们显示这个封隔是严格的。 我们特别显示路径系统问题, 一个在 LFP 中可定义的典型的P- 完整的问题- 固定点逻辑- 无法在 LREC= 中定义 。 这显示对数重现机制比一般最不固定的点弱。 证据基于为这个逻辑定制的新型的 Spoiler- Duplicator 游戏 。
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