Limiting Partial Combinatory Algebras towards Infinitary Lambda-Calculi and Classical Logic

Abstract

We will construct from every partial combinatory algebra (pca, for short) A a pca a-lim(A) s.t. (1) every representable numeric function ϕ(n) of a-lim(A) is exactly of the form lim_t ξ(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.

The author acknowledges Susumu Hayashi, Mariko Yasugi, Stefano Berardi, and Ken-etsu Fujita. The comment by anonymous referees was useful to partly improve the presentation.