Functions Definable by Numerical Set-Expressions

TL;DR

This work formalizes numerical set-expressions (O-circuits) that operate on sets of non-negative integers by lifting arithmetic to the level of sets, studying both additive ($\\bm{+}$) and arithmetic ($\\bm{+},\\bullet$) circuits. It provides a taxonomy of what sets and functions can be defined, offering constructive definability results for natural set-functions and predicates, and proving broad non-definability results via continuity arguments and domain-theoretic techniques. A core finding is that certain highly discontinuous functions, notably $\\Downarrow$, $\\operatorname{Max}$, and $\\operatorname{Card}$, are not definable by arithmetic circuits (even with predicate gates), while some growth-restricted numerical functions are definable by additive circuits but severely constrained by the type of circuit used. The paper also clarifies the relative expressive power when augmenting circuits with $\\Downarrow$, $\\operatorname{Max}$, and $\\operatorname{Card}$, showing, for example, that additive circuits treat $\\operatorname{Max}$ and $\\Downarrow$ differently in terms of expressivity, whereas arithmetic circuits often equate their power, and demonstrates partial limitations on Card-definability. Overall, the results map the expressive limitations and capabilities of numeric set-expressions, linking them to bounded arithmetic and topological considerations, with implications for definability in logic and complexity contexts.

Abstract

A "numerical set-expression" is a term specifying a cascade of arithmetic and logical operations to be performed on sets of non-negative integers. If these operations are confined to the usual Boolean operations together with the result of lifting addition to the level of sets, we speak of "additive circuits". If they are confined to the usual Boolean operations together with the result of lifting addition and multiplication to the level of sets, we speak of "arithmetic circuits". In this paper, we investigate the definability of sets and functions by means of additive and arithmetic circuits, occasionally augmented with additional operations.

Figures (1)

• Figure 1: Graphical depictions of two arithmetic circuits: (a) the circuit $\{2\} \bullet \mathbb{N}$, defining the set of even numbers; (b) the circuit $\overline{\{1\}} \cap \overline{\ \overline{\{1\}} \bullet \overline{\{1\}}\ }$, defining the set of primes.

Theorems & Definitions (58)

• theorem thmcountertheorem
• proof
• theorem thmcountertheorem
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof