Complex Algebras of Arithmetic
Ivo Düntsch, Ian Pratt-Hartmann
TL;DR
The paper develops an algebraic framework for arithmetic circuits by embedding them into complex algebras of natural numbers and analyzing them with Boolean algebras with operators. It characterizes the complex algebras Cm$(\mathbb{N})$ under both additive and multiplicative structures, identifies finite quotients $\mathfrak{B}_n$ that generate the corresponding varieties, and establishes decidability results at the level of these quotients while proving undecidability for the full theories. Key results include a chain of congruences in Cm$(\mathbb{N}^{+})$, an isomorphism between Cm$_0(\mathbb{N}^{+})$ and Cm$_0(\mathbb{N}^{\bullet})$, and the co-r.e.-completeness of EqSat for certain operator languages. The work clarifies the logical and algebraic landscape of complex algebras arising from arithmetic circuits, highlighting both decidability prospects and fundamental barriers due to the interpretability of computation within these algebras.
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.