Compaction of Church Numerals for Higher-Order Compression

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 $\lambda$-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 $O(({\mathrm{s}\mathrm{l}\mathrm{o}\mathrm{g}}_{2}n{)}^{\log n\mathrm{/}\log \log n})$. Moreover, we quantitatively confirmed experimentally that the proposed method outperforms a binary expression of Church numerals when $n$ is less than approximately 10000.