Encoding FIX in object calculi

We show that the FIX type theory introduced by Crole and Pitts 3] can be encoded in variants of Abadi and Cardelli's object calculi. More precisely, we show that the FIX type theory presented with judgements of both equality and operational reduction can be translated into object calculi, and the translation proved sound. In the case of operational reductions, we introduce a variant of Abadi and Cardelli's system Ob. The payoo is a programme in which the general procedure for encoding type theories in object calculi can be seen clearly|we encapsulate this as a principle; and more particularly, we show that computation types and the xpoint type can be encoded. This is a direct contribution to the theory of objects, and moreover these translations will be useful to those involved with the automation of operational semantics. The preliminary abstract for this paper appears in the symposium on Mathe-1 Motivation The results presented here are founded on the FIX type theory introduced by Crole and Pitts in 3]; and the object type theories (calculi) introduced by Abadi and Cardelli in 1]. We shall assume that readers are familiar with these theories, and more generally with type theories possessing judgements of both equality, and operational reduction. A good general reference for (dependent) type theory is the book 10]. For operational reduction, see for example 2, 4, 6, 8, 11, 12, 13]. We shall give brief summaries of the rules which generate judgements of the FIX type theory when we feel it is especially useful to do so; otherwise, to prevent undue repetition, we ask that readers consult the original references. Abadi and Cardelli have shown that object calculi are highly expressive, leading to the principle that \everything is an object". Various people (see for example 5]) have provided 1