Some results on the full abstraction problem for restricted lambda calculi

Abstract

Issues in the mathematical semantics of two restrictions of the λ-calculus, i.e. λI-calculus and λv-calculus, are discussed. A fully abstract model for the natural evaluation of the former is defined using complete partial orders and strict Scott-continuous functions. A correct, albeit non-fully abstract, model for the SECD evaluation of the latter is denned using Girard's coherence spaces and stable functions. These results are used to illustrate the interest of the analysis of the fine structure of mathematical models of programming languages.

In conclusion we have that indeed the “coherent” model succeeds in equating observationally equivalent terms that the “continuous” model tells apart, as our original intuition suggested. And this is because the Scott-continuous functions which separate those terms are indeed parallel and hence not stable. But some-what surprisingly Berry's order introduces perverse stable functions which tell apart observationally equivalent terms which are instead equated in the continuous model. Although neither the coherent model nor the continuous one are fully abstract, nevertheless the investigation carried out was shown to be rewarding.