Set - Theoretical Models of a - Calculus : Theories , Expansions , Isomorphisms *

This paper mainly deals with the models for type free h-calculus defined by Plotkin [IS], Engeler [12] and Scott [22]. (See also [9] for a similar construction.) Plotkin-Scott-Algebras (PSE-Algebras, in view of Engeler’s approach) are built up in a very natural set theoretic way and provide a generalization of early ideas in Scott [20,21]. Namely, the notion of application (interpreting formal application of h-terms) generalizes the classical Myhill-Shepherdson-Rogers definition of application in PO, introduced to define Enumeration Operators (see [19, p. 1433). Abstraction is defined accordingly. An interesting fact is that these definitions do not depend on codings of pairs and of finite sets, while the classical ones do. This doesn’t affect the Recursion Theory one should be able to work out on PSE-Algebras (cf. [6, 16 02, 21]), but does affect the model theory of A-calculus (see [3] and Section 5). Moreover, for various reasons which should become clear in the next sections, these structures are very ‘handy’: it is easy to grasp the intuition on which the definitions rely and to modify them for the purpose of the model theory of X-ca!culus we aim at. Section 1.1 introduces X-terms and CL-terms (terms of X-calculus, X/3, and of Combinatory Logic, CL) of various orders, corresponding to levels of functionality or number of X-abstractions. Section 1.2 discusses the consequences in Combinatory Algebras of an early remark of Wadsworth (and Scott) on how to interpret the ‘loss of information’ which is implicit in performing combinatory reductions, as in any effective process. Section 2 introduces PSE-Algebras and deals with the local analysis (according