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Additive systems for $\mathbb{Z}$ are undecidable

Andrei Zabolotskii

Abstract

What are the collections of sets ${A}_i\subset\mathbb{Z}$ such that any $n\in\mathbb{Z}$ has exactly one representation as $n=a_0+a_1+\dotsb$ with $a_i\in{A}_i$? The answer for $\mathbb{N}_0$ instead of $\mathbb{Z}$ is given by a theorem of de Bruijn. We describe a family of natural candidate collections for $\mathbb{Z}$, which we call canonical collections. Translating the problem into the language of dynamical systems, we show that the question of whether the sumset of a canonical collection covers the entire $\mathbb{Z}$ is difficult: specifically, there is a collection for which this question is equivalent to the Collatz conjecture, and there is a well-behaved family of collections for which this question is equivalent to the universal halting problem for Fractran and is therefore undecidable.

Additive systems for $\mathbb{Z}$ are undecidable

Abstract

What are the collections of sets such that any has exactly one representation as with ? The answer for instead of is given by a theorem of de Bruijn. We describe a family of natural candidate collections for , which we call canonical collections. Translating the problem into the language of dynamical systems, we show that the question of whether the sumset of a canonical collection covers the entire is difficult: specifically, there is a collection for which this question is equivalent to the Collatz conjecture, and there is a well-behaved family of collections for which this question is equivalent to the universal halting problem for Fractran and is therefore undecidable.

Paper Structure

This paper contains 4 sections, 6 theorems, 10 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Additive systems for $\mathbb{N}_0$ and de Bruijn's theorem
  3. Additive systems for $\mathbb{Z}$ and canonical collections
  4. Additive systems for $\mathbb{Z}$ are undecidable

Key Result

Theorem 1

Any additive system for $\mathbb{N}_0$ is a contraction of a British number system.

Figures (7)

  • Figure 1: Mrs. Beeton's Book of Household Management (1907), p. 126. $(b_0,b_1,b_2,b_3,b_4,\dotsc) = (16, 16, 28, 4, 20,\dotsc)$
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 2 more figures

Theorems & Definitions (25)

  • Definition 1
  • Example 1: decimal numeral system
  • Definition 2
  • Definition 3
  • Example 2
  • Example 3
  • Theorem : de Bruijn deBruijn
  • Example 4: negabinary numeral system
  • Example 5: balanced ternary numeral system
  • Definition 4
  • ...and 15 more