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Punctually Standard and Nonstandard Models of Natural Numbers

Nikolay Bazhenov, Ivan Georgiev, Dariusz Kalociński, Stefan Vatev, Michał Wrocławski

Abstract

Abstract models of computation often treat the successor function $S$ on $\mathbb{N}$ as a primitive operation, even though its low-level implementations correspond to non-trivial programs operating on specific numerical representations. This behaviour can be analyzed without referring to notations by replacing the standard interpretation $(\mathbb{N}, S)$ with an isomorphic copy ${\mathcal A} = (\mathbb{N}, S^{\mathcal A})$, in which $S^{\mathcal A}$ is no longer computable by a single instruction. While the class of computable functions on $\mathcal{A}$ is standard if $S^{\mathcal{A}}$ is computable, existing results indicate that this invariance fails at the level of primitive recursion. We investigate which sets of operations have the property that if they are primitive recursive on $\mathcal A$ then the class of primitive recursive functions on $\mathcal A$ remains standard. We call such sets of operations \emph{bases for punctual standardness}. We exhibit a series of non-basis results which show how the induced class of primitive recursive functions on $\mathcal A$ can deviate substantially from the standard one. In particular, we demonstrate that a wide range of natural operations, including large subclasses of primitive recursive functions studied by Skolem and Levitz, fail to form such bases. On the positive side, we exhibit natural finite bases for punctual standardness. Our results answer a question recently posed by Grabmayr and establish punctual categoricity for certain natural finitely generated structures.

Punctually Standard and Nonstandard Models of Natural Numbers

Abstract

Abstract models of computation often treat the successor function on as a primitive operation, even though its low-level implementations correspond to non-trivial programs operating on specific numerical representations. This behaviour can be analyzed without referring to notations by replacing the standard interpretation with an isomorphic copy , in which is no longer computable by a single instruction. While the class of computable functions on is standard if is computable, existing results indicate that this invariance fails at the level of primitive recursion. We investigate which sets of operations have the property that if they are primitive recursive on then the class of primitive recursive functions on remains standard. We call such sets of operations \emph{bases for punctual standardness}. We exhibit a series of non-basis results which show how the induced class of primitive recursive functions on can deviate substantially from the standard one. In particular, we demonstrate that a wide range of natural operations, including large subclasses of primitive recursive functions studied by Skolem and Levitz, fail to form such bases. On the positive side, we exhibit natural finite bases for punctual standardness. Our results answer a question recently posed by Grabmayr and establish punctual categoricity for certain natural finitely generated structures.
Paper Structure (14 sections, 41 theorems, 34 equations, 1 figure)

This paper contains 14 sections, 41 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contributions
  3. Preliminaries
  4. Examples for Punctually Nonstandard Structures
  5. Exploring Images of Functions in the Model $\mathcal{A}$
  6. Exploring Images of Relations in the Model $\mathcal{A}$
  7. Enforcing Primitive Recursive Images of Linear Functions
  8. Enforcing Primitive Recursive Addition and Multiplication in the Model
  9. Mainland--Island Technique and the Metatheorem
  10. Showcasing the Mainland--Island Technique
  11. Metatheorem
  12. Corollaries of Theorem \ref{['theo:nonstandardness-meta']}
  13. An Application to the Levitz's Class
  14. Sufficient Condition for Punctual Standardness

Key Result

Proposition 2

There is $\mathcal{A}=(\mathbb{N},S^{\mathcal{A}})$, an isomorphic copy of $\mathcal{S}=(\mathbb{N},S)$, such that $S^{\mathcal{A}}$ is primitive recursive, but $c_\mathcal{A}^{-1}$ is not.

Figures (1)

  • Figure 1: Visualization of the mainland (top chain) and three islands with corresponding marks. Arrows correspond to $S^\mathcal{B}$.

Theorems & Definitions (82)

  • Definition 1
  • Proposition 2: bazhenov_foundations_2019, Example 4.1(3), p. 86
  • Definition 3: punctual standardness
  • Definition 4: kalimullin_algebraic_2017
  • proof
  • Definition 6: basis for punctual standardness
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Corollary 9
  • ...and 72 more