Complex Algebras of Arithmetic

TL;DR

This paper explores the algebraic structure of complex algebras derived from natural numbers and examines the complexity of their associated theories, providing insights into their foundational properties.

Contribution

It investigates the algebraic properties of complex algebras of natural numbers and analyzes the complexity of their theories, offering new theoretical insights.

Findings

Complex algebras form a minimal subalgebra of the semiring of natural numbers.

Theories of these algebras exhibit varying levels of computational complexity.

Algebraic and logical properties of these structures are characterized and analyzed.

Abstract

An 'arithmetic circuit' is a labeled, acyclic directed graph specifying a sequence of arithmetic and logical operations to be performed on sets of natural numbers. Arithmetic circuits can also be viewed as the elements of the smallest subalgebra of the complex algebra of the semiring of natural numbers. In the present paper, we investigate the algebraic structure of complex algebras of natural numbers, and make some observations regarding the complexity of various theories of such algebras.