Exact bounds for lengths of reductions in typed λ-calculus

Abstract

We determine the exact bounds for the length of an arbitrary reduction sequence of a term in the typed λ-calculus with β-, ξ- and η-conversion. There will be two essentially different classifications, one depending on the height and the degree of the term and the other depending on the length and the degree of the term.

Bounding Skeletons, Locally Scoped Terms and Exact Bounds for Linear Head Reduction

“…There are multiple approaches to the complexity analysis of higher-order programs, but they seem to separate into two major families. On the one hand, Beckmann [2], extending earlier work by Schwichtenberg [14], gave exact bounds to the maximal length of β-reduction on simply-typed λ-terms. His analysis uses very basic information on the terms (their length, or height, and order), but gives bounds that are in general very rough.…”

“…using Lemma 5. This gives M such that: For a term M of height h and order n, Beckmann's results [2] predict that any β-reduction chain of M terminates in less than 2 Θ(h) n+1 steps. It might seem counter-intuitive that our bound (with linear head reduction) is smaller than Beckmann's (with β-reduction) since we substitute only one occurrence at a time, which is obviously longer.…”

“…We now set to prove the optimality of this upper bound by exhibiting a family of terms whose reduction length asymptotically reaches it. The family we describe is a variant of one used by Beckmann in [2], constructed by iterated exponentiation of Church numerals. We define higher types for Church integers by setting A −2 = o and A n+1 = A n → A n .…”

“…In the context of the λ-calculus, the first result that comes to mind is the work by Schwichtenberg [14], later improved by Beckmann [2], establishing upper bound to the length of β-reduction sequences for simply-typed λ-calculus. In the somewhat related line of work of implicit complexity, type systems have been developed to characterize extensionally certain classes of functions, such as polynomial [10] or elementary [8] time.…”

Extracting Herbrand Disjunctions by Functional Interpretation

“…When combined with known estimates ( [1]) on the size of nf (t) we 2 In the Statman example discussed below the original functional interpretation already creates as high types as the Shoenfield variant does. This is unavoidable here since the Statman example has the worst possible Herbrand complexity despite the fact that its form (2) is provable in intuitionistic logic.…”