Functions Definable by Numerical Set-Expressions

TL;DR

This paper explores the definability of sets and functions of non-negative integers using additive and arithmetic circuits, which involve Boolean, addition, and multiplication operations on sets, with some extensions.

Contribution

It introduces the framework of additive and arithmetic circuits for defining sets and functions, analyzing their expressive power and potential extensions.

Findings

Additive circuits use Boolean and lifted addition operations.

Arithmetic circuits extend additive circuits with lifted multiplication.

The paper characterizes the sets and functions definable by these circuits.

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.