By V.E. Wolfengagen

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In particular, an idea to construe the systems of combinators for special classes of computations stimulates the deep and vital studies. The papers (G. -L. Curien, and M. -L. Curien, [76], [77]) contain the constructing of a special kind of combinatory logic called categorical combinatory logic. It gave 34 C HAPTER 1: P RELIMINARIES a foundation for constructing the abstract machine, which is both an enhanced notion of computation and self standing programming system, aimed to pursue the research in an area of programming.

Xn ]a, via K, S, I, where [x1 , . . , xn ] for n > 0 has a role of abstraction operator by the variables x1 , . . , xm . The principle of combinatory completeness: U b1 . . bn ≡ ([x1 , . . , xn ]a)b1 . . bn = [b1 , . . , bn /x1 , . . , xn ]a, can be proved for abstraction operator, where the expression ‘[b1 , . . , bn /x1 , . . , xn ]a’ denotes a result of simultaneous substituting the objects b1 , . . , bn in the object a instead of the corresponding occurrences of graphically distinct variables x1 , .

Deriving of basic properties of the Karoubi’s Shell A special category, called the Karoubi’s shell, makes it possible to laconically express a whole body of knowledge, related to operators, in terms of combinatory logic. In this case, the types are also encoded by objects. As a result, we implement the immersion or embedding of the typed application into the type free software environment. • Cartesian product and projections: embedding into ACS The final completion of the started process of embedding is achieved by the introduction of ordered populations of objects into consideration.

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