The local structure of the π-interpretation, =π, is the behavioural equivalence induced on λ-terms by their encoding into π-calculus. In the previous chapter we proved a characterization of =π as the operational equivalence of extended λ-calculi. We continue the study of =π in this chapter, with the purpose of understanding the meaning of function equality when functions are interpreted as processes. We will prove characterizations of =π in terms of tree structures that are an important part of the theory of the λ-calculus. This will show that the equivalence induced by the π-calculus encoding is a natural one. It will also show the utility of some of the π-calculus proof techniques of Section 2.4.3: using techniques such as ‘bisimulation up to context’ and ‘bisimulation up to expansion’, the proofs of the main theorems will be much easier than they would have been otherwise.

The main result says that =π coincides with LT equality, whereby two λ-terms are equal iff they have the same Lévy-Longo Trees. Lévy-Longo Trees are the lazy variant of Böhm Trees. We will also discuss modifications of the π-calculus interpretation so that its local structure is the analogous BT equality. We begin by recalling what LTs are. A reader familiar with them may go straight to Theorem 18.3.1.

We maintain Notation 17.1.1 throughout the chapter.

18.1 Sensible theories and lazy theories

Böhm Trees (BTs) are the best-known tree structure in the λ-calculus. BTs play a central role in the classical theory of the λ-calculus. The local structure of some of the most influential models of the λ-calculus, like Scott and Plotkin's Poj, Plotkin's T, and Plotkin and Engeler's DA, is precisely the BT equality; and the local structure of Scott's D (historically the first mathematical, i.e., non-svntactical, model of the untyped λ-calculus) is the equality of the ‘infinite η contraction’ of BTs.

BTs naturally give rise to a tree topology that has been used for the proof of some seminal results of the λ-calculus such as Berry's Sequentiality Theorem. BTs were introduced by Barendregt [Bar77], and so called after Böhm's theorem and proof about separability of λ-terms. The proof technique for this theorem, called the Böhm-out technique, roughly consists in finding a context capable of isolating a given subtree of a BT; in this way, certain λ-terms that have different BTs may be separated.