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Compaction of Church Numerals

*,† and
Graduate School/Faculty of Information Science and Technology, Hokkaido University, Kita 14 Nishi 9, Sapporo 060-0814, Japan
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Algorithms 2019, 12(8), 159;
Received: 28 June 2019 / Revised: 5 August 2019 / Accepted: 5 August 2019 / Published: 8 August 2019
(This article belongs to the Special Issue Data Compression Algorithms and their Applications)
PDF [4411 KB, uploaded 8 August 2019]


In this study, we address the problem of compaction of Church numerals. Church numerals are unary representations of natural numbers on the scheme of lambda terms. We propose a novel decomposition scheme from a given natural number into an arithmetic expression using tetration, which enables us to obtain a compact representation of lambda terms that leads to the Church numeral of the natural number. 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 experimentally confirmed that the proposed method outperforms binary representation of Church numerals on average, when n is less than approximately 10,000 .
Keywords: loss-less compression; lambda term; higher-order compression loss-less compression; lambda term; higher-order compression
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Furuya, I.; Kida, T. Compaction of Church Numerals. Algorithms 2019, 12, 159.

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