Compaction of Church Numerals for Higher-Order Compression

Abstract

In this study, we address the problem of compacting Church numerals. Church numerals appear as a representation of the repetitive part of data in higher-order compression. We propose a novel decomposition scheme for a natural number using tetration, which leads to a compact representation of λ-terms equivalent to the original Church numerals. For natural number n, we prove that the size of the lambda term obtained by the proposed method is O((slog 2 n) log n/ log log n ). Moreover, we quantitatively confirmed experimentally that the proposed method outperforms a binary expression of Church numerals when n is less than approximately 10000.

Compaction of Church Numerals

“…By Lemma 4, #M ϕ * is bounded by Λ(T 2 [n]) since the function term of Λ(T 2 [n]) is smaller than Λ(T 2 [n]) itself. Then, by (14), we obtain #M ϕ * = O((slog 2 n) log n log log n ).…”

“…Conclusions are presented in Section 5. Previously, one page abstract of this work has been published in the proceedings of Data Compression Conference 2016 [14]. This paper is the full version article of [14].…”

Symmetric quadratic tetration interpolation using forward and backward operation combination

“…Moreover, it also can be called for a different name such as superexponentiation [40], hyperpower [41]. This mathematical operator was applied in the problem of compacting Church numerals [42], and our previous work [2]. The form of the traditional polynomial function was used to model the tetration polynomial function.…”