Linear Numeral Systems

Abstract

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.

A type-assignment of linear erasure and duplication

“…Section 7.2 contributes to the problem of defining numeral systems in linear settings. In [14], Mackie has recently introduced linear variants of numeral systems. He shows that successor, addition, predecessor, and subtraction have representatives in the linear λ-calculus.…”

“…For example, K ′ = λxy.⟨x, y⟩ represents the classical K = λxy.x, the second component of ⟨x, y⟩ being garbage. Another approach is by Mackie, and can be called "erasure by data consumption" [14]. It involves a step-wise erasure process that proceeds by β-reduction, according to the following definition: In [15], Mackie proves that all closed linear λ-terms can be erased by means of very simple contexts.…”