We investigate the use of the dot notation in the context of abstract types. The dot notation -- that is, a.f referring to the operation f provided by the abstraction a -- is used by programming languages such as Modula-2 and CLU. We compare this notation with the Mitchell-Plotkin approach, which draws a parallel between type abstraction and (weak) existential quantification in constructive logic. The basic operations on existentials coming from logic give new insights about the meaning of type abstraction, but differ completely from the more familiar dot notation. In this paper, we formalize simple calculi equipped with the dot notation, and relate them to a more classical calculus a la Mitchell and Plotkin. This work provides some theoretical foundations for the dot notation, and suggests some useful extensions.
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