Limiting partial combinatory algebras

From every partial combinatory algebra script A sign we construct another partial combinatory algebra a-lim(script A sign). In a-lim(script A sign), every representable partial numerical function φ(n) is exactly of the form lim_tζ(t,n) for some representable partial numerical function ζ(t,n) of script A sign. The partial combinatory algebra a-lim(script A sign) is a quotient of script A sign by a partial equivalence relation, and is equipped with a limit structure in the sense that each element of a-lim(script A sign) is the limit of a countable sequence of script A sign-elements. In this paper, we discuss the limit structures for script A sign in terms of Barendregt's range property (if the range of a combinator is finite, then it is a singleton). Moreover, we repeat the construction a-lim(-) transfinite times to interpret infinitary λ-calculi. Finally, we attempt to interpret type-free λμ-calculus by introducing another partial applicative structure which has an asynchronous application operator that allows a parallel limit operation.