A Syntactic Characterization of the Equality in Some Models for the Lambda Calculus

نویسنده

  • MARTIN HYLAND
چکیده

An equality relation on the terms of the A-calculus is an equivalence relation closed under the (syntactical) operations of application and A-abstraction. We may distinguish between syntactic and semantic ways of introducing equality relations, /^-equality is introduced syntactically; it is the least equality relation satisfying the equations for aand ^-conversion. For a more subtle way of introducing equality relations syntactically, consider the relations =f and =h of §5 of this paper. To give a semantic characterization of an equality relation, we simply take the relation ' has the same value in £>', where D is some model for the A-calculus. Of course, no equality relation is of interest to the intended interpretation of the A-calculus, unless it extends /^-equality. An equality relation is inconsistent if and only if it sets all terms equal; otherwise it is consistent. It is maximal consistent if and only if it is consistent and has no consistent proper extensions. In this paper we consider a class of continuous lattice models for the A-calculus, and a particular model, the Graph model. The same equality is induced by all the continuous lattice models; we shall refer to them as the Scott models (see [3], where they were first constructed). For the history of the Graph model see [4]. We shall give, in this paper, syntactic characterizations of the equality induced by the Scott models, and by the Graph model; and we shall show that the equality induced by the Scott models is the unique maximal consistent equality relation, extending the relation = H, which was proved consistent in [1]. We use x, y,z, w ... for variables, and M,N, P ... for terms of the A-calculus (with subscripts as necessary). D will refer to whatever model or models are under consideration. The content of our Theorem 5.4 (a) has been discovered independently by C. P. Wadsworth.

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تاریخ انتشار 1976