# Full abstraction and unnested recursion

## Abstract

*L*without identifiers (i.e., variables), which is, nonetheless, fairly expressive. We also assume

*L*has been provided with both an operational semantics and a denotational semantics. Furthermore, the denotational semantics is adequate and fully abstract with respect to the operational semantics. After clarifying exactly what these assumptions entail,

- 1.
we discuss what it means to extend

*L*algebraically to a language*L[X]*by the addition of identifiers, - 2.
we discuss the semantics of L[

*X*] in the context of possibly self-referential environments (i.e., systems of recursive definitions of the identifiers), and - 3.
we show that extremely mild topological assumptions about the operational model ensure that adequacy and full abstraction of the denotational semantics of L[

*X*] with respect to its operational semantics follow automatically from the corresponding results for*L.*

Essentially, we will assume that the operational model is a Hausdorff space, and the operational process of unwinding a recursive program converges to its operational meaning. Some work has been done on the issues we are confronting, but from the standpoint of fair unwindings of recursive constructs. That work indicates that full abstraction results are not generally to be expected in all situations. Our goal, however, is to establish that full abstraction results are indeed available in a very general setting.

## Keywords

Full abstraction adequacy algebraic semantics homomorphism algebraic poset## Preview

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