An Investigation of Compact and Efficient Number Representations in the Pure Lambda Calculus

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 .

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“…Instead of Church numerals, we can represent natural numbers on λ-terms based on binary representation as follows [15].…”

Section: Binary Expression Of Natural Numbers On λ-Termsmentioning

confidence: 99%

Compaction of Church Numerals

Furuya

Kida

2019

Algorithms

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“…For C(n), the size of the expression is Θ(log n). Earlier, Mogensen [4] proposed a binary expression of Church numerals and generalized to higher number-bases. The method introduced by Kobayashi et al is essentially the same as Mogensen's method.…”

Section: Related Workmentioning

confidence: 99%

Compaction of Church Numerals for Higher-Order Compression

Furuya

Kida

2018

2018 Data Compression Conference

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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.

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“…From a very different perspective, and a focus on compact representation, there are binary representations [7] that give logarithmic size numbers. However, as we show later, these are not linearly definable.…”

Section: Related Workmentioning

confidence: 99%

“…There are other ways to represent numbers as λ-terms, for example binary representations (see for example [7]). However, the results of this paper show that these representations are not possible in the linear λ-calculus, and we discuss this topic in more detail in Sect.…”

Section: Numbersmentioning

confidence: 99%

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Linear Numeral Systems

Mackie

2018

J Autom Reasoning

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We investigate numeral systems in the lambda calculus; specifically in the linear lambda calculus where terms cannot be copied or erased. Our interest is threefold: representing numbers in the linear calculus, finding constant time arithmetic operations when possible for successor, addition and predecessor, and finally, efficiently encoding subtraction-an operation that is problematic in many numeral systems. This paper defines systems that address these points, and in addition provides a characterisation of linear numeral systems.

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An Investigation of Compact and Efficient Number Representations in the Pure Lambda Calculus

Cited by 6 publications

References 4 publications

Compaction of Church Numerals

Compaction of Church Numerals

Compaction of Church Numerals for Higher-Order Compression

Linear Numeral Systems

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