A dichotomy result of Sevenster (2014) [29] completely classified the quantifier prefixes of regular Independence-Friendly (IF) logic according to the patterns of quantifier dependence they contain. On one hand, prefixes that contain “Henkin” or “signalling” patterns were shown to characterize fragments of IF logic that capture NP-complete problems; all the remaining prefixes were shown instead to be essentially first-order.

In the present paper we develop the machinery which is needed in order to extend the results of Sevenster to non-prenex, regular IF sentences. This involves shifting attention from quantifier prefixes to a (rather general) class of syntactical tree prefixes.

We partially classify the fragments of regular IF logic that are thus determined by syntactical trees; in particular, a) we identify four classes of tree prefixes that are neither signaling nor Henkin, and yet express NP-complete problems and other second-order concepts; and b) we give more general criteria for checking the first-orderness of an IF sentence.

Sevenster（2014）[29]的二分法结果完全根据规则所包含的量词相关性模式对常规独立友好（IF）逻辑的量词前缀进行了分类。一方面，显示了包含“ Henkin”或“信号”模式的前缀来表征捕获NP完全问题的IF逻辑片段。所有其余的前缀都显示为本质上是一阶的。

在本文中，我们开发了必要的机制，以将Sevenster的结果扩展到非前置的常规IF语句。这涉及将注意力从量词前缀转移到（而不是一般的）语法树前缀类。

我们将常规IF逻辑的片段部分分类，这些片段因此由句法树确定；尤其是：a）我们确定了既不是信令也不是Henkin的四类树前缀，但它们表达了NP完全问题和其他二阶概念；b）我们提供了更通用的标准来检查IF句子的一阶性。