17 - Full Abstraction

Summary

17.1 The full-abstraction problem for the π-interpretation of call-by-name

An interpretation of the λ-calculus into π-calculus, as a translation of one language into another, can be considered a form of denotational semantics. The denotation of a λ-term is an equivalence class of processes. These equivalence classes are the quotient of the π-calculus processes with respect to barbed congruence, the behavioural equivalence of the π-calculus.

In the two previous chapters, we have seen various π-calculus interpretations of λ-calculi and have shown their soundness with respect to the axiomatic semantics of the calculi, where equivalence between λ-terms means provable equality from an appropriate set of axioms and inference rules. In this chapter and the next, we go further and compare the π-calculus semantics with the operational semantics of the λ-calculus. We study the important case of the untyped call-by-name λ-calculus (λN); the problem is harder in the call-by-value case, and is briefly discussed in Section 18.5. The encoding of λN into π-calculus will be that of Table 15.4, but without i/o types. We can assume that the encoding is into the plain polvadic π-calculus (π), without i/o types, because Lemma 15.4.18 shows that i/o types do not affect behavioural equivalence; in Lemma 15.4.18 we indicated the encoding without i/o types as N_#.

An interpretation of a calculus is said to be sound if it equates only operationally equivalent terms, complete if it equates all operationally equivalent terms, and fully abstract if it is sound and complete. We show in Section 17.3 that the π-calculus interpretation of λN is sound, but not complete.

When an interpretation of a calculus is not fully abstract, one may hope to achieve full abstraction by

• (1) enriching the calculus,

• (2) choosing a finer notion of operational equivalence for the calculus, or

• (3) cutting down the codomain of the interpretation.

In Section 17.4 and Chapter 18 we prove full-abstraction results for the π-interpretation by following (1) and (2), respectively. In the notes at the end of the Part, we will hint that the main theorems are, by and large, independent of the behavioural equivalence chosen for the π-calculus, which suggests that (3) is less interesting.