We will construct from every partial combinatory algebra (PCA, for short) A a PCA a-lim(A) s.t. (1) every representable numeric function ψ(n) of a-lim(A) is exactly of the form lim_tξ (t, n) with ξ(t, n) being a representable numeric function of A, and (2) A can be embedded into a-lim(A) which has a synchronous application operator. Here, a-lim(A.) is A equipped with a limit structure in the sense that each element of a-lim(A) is the limit of a countable sequence of A-elements. We will discuss limit structures for A in terms of Barendregt's range property. Moreover, we will repeat the construction lim(―) transfinite times to interpret infinitary λ-calculi. Finally, we will interpret affine type-free λμ-calculus by introducing another partial applicative structure which has an asynchronous application operator and allows a parallel limit operation, keywords: partial combinatory algebra, limiting recursive functions, realizability interpretation, discontinuity, infinitary lambda-calculi, λμ-calculus. In the interpretation,μ-variables(=continuations) are interpreted as streams of λ-terms.
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