一句话：通过参数化 wires 细化 Milner 编码，统一刻画扩展与非扩展的函数作为进程 λ-理论。
本文研究函数作为进程表示时如何获得扩展 λ-理论，以及如何在扩展与非扩展表示间转换。Milner 对纯 λ-演算的编码最多只给出非扩展 λ-理论（β 成立而 η 不成立）。作者在 Internal π（Iπ，所有输出均受绑定的 π-演算子集）中细化该编码，引入参数化组件 wires，并证明当 wires 满足若干代数性质时编码产生 λ-理论。利用 Iπ 的对称性与对偶性，作者划分出三类 wires：两类顺序且互为对偶，另一类并行且自对偶。关键结果是，并行 wires 产生扩展 λ-理论，其等式与带无限 η 的 Böhm trees 相同；另外两类产生非扩展 λ-理论，分别对应 Lévy-Longo trees 与 Böhm trees。
Following Milner’s seminal paper, the representation of functions as processes has received considerable attention. For pure λ-calculus, the process representations yield (at best) non-extensional λ-theories (i.e., β rule holds, whereas η does not).In the paper, we study how to obtain extensional representations, and how to move between extensional and non-extensional representations. Using Internal π, Iπ (a subset of the π-calculus in which all outputs are bound), we develop a refinement of Milner’s original encoding of functions as processes that is parametric on certain abstract components called wires. These are, intuitively, processes whose task is to connect two end-point channels. We show that when a few algebraic properties of wires hold, the encoding yields a λ-theory. Exploiting the symmetries and dualities of Iπ, we isolate three main classes of wires. The first two have a sequential behaviour and are dual of each other; the third has a parallel behaviour and is the dual of itself. We show the adoption of the parallel wires yields an extensional λ-theory; in fact, it yields an equality that coincides with that of Böhm trees with infinite η. In contrast, the other two classes of wires yield non-extensional λ-theories whose equalities are those of the Lévy-Longo and Böhm trees.