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Qualitative domains were introduced by Girard [4] in order to give a model of system F, originally introduced by Girard [2]. The closed terms in system F will be interpreted as consistent objects in their corresponding domains. Girard added a notion of totality to the qualitative domains. A free type variable may be interpreted as any qualitative domain with any set of consistent sets representing the total objects. A type term with n free type variables will then be interpreted as an n-ary operator on the class of pairs (X, XTOT) where X is a qualitative domain and XTOT is any set of elements in X. These operators are extensions of the basic interpretation of system F using nice functorial operators on the category of qualitative domains. In a type abstraction the objects are the so called uniform objects. A uniform object will be total if its value is total for all interpretations of totality in the domain that is the argument of the operator. Thus when the free variables are abstracted away, each closed type-term is interpreted as a specific domain with a specific set of total objects. Since there is no restriction on the families that may be used as the total objects, there is no structural or conceptual analysis of the notion of totality in Girard's work. Even with a restriction that is weak enough to permit the interpretation of type-terms in system F as totality-spaces of the restricted form, the interpretation of totality will not neccessarily be the same. This is because the class of pairs (X, XTO~) is used negatively in Girard's interpretation of totality. A general line of interest to both authors has been the search for a conceptual understanding of totality. The first author has carried out a program for investigating the notion of totality for qualitative domains. In Normann [10] the notion of totality is discussed and axiomatized. The structures in [10] consist of two parts; a partial ordering where all initial segments are

Interpreting higher computations as types with totality