Abstract
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.
Highlights
The goal of this study is to obtain a compact lambda term (λ-term) that leads to the Church numeral of a given natural number
We propose a novel decomposition scheme called recursive tetrational partitioning (RTP), which enables to obtain a compact representation of λ-terms that leads to the Church numerals of the numbers
We addressed the problem of compaction of Church numerals
Summary
The goal of this study is to obtain a compact lambda term (λ-term) that leads to the Church numeral of a given natural number. Church numerals are unary representations of natural numbers on λ-terms. Let C(n) be the Church numeral for natural number n; C(n) is as follows: n }| { z. The length of the Church numeral increases linearly with n. We want a more compact representation for n in the scheme of λ-terms. We refer to this task as the compaction of Church numerals
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