Polynomial-Time Algorithms for Black-Box Distributive Expanded Groups
Mikhail Anokhin
TL;DR
The paper extends the black-box computation paradigm to distributive Ω-expanded groups and proves the existence of probabilistic polynomial-time algorithms that (i) extract a generating set for the additive group, (ii) generate the additive group of ideals, and (iii) decide membership in finitely based varieties with nilpotent additive groups, all with exponentially small error in the encoding length $n$. The core methods leverage random subsums and a closure-like τ-iteration to efficiently construct generators, enabling G-decidability results for broad classes such as rings, modules, and $R$-algebras. These results unify and generalize prior black-box algorithms for groups and rings, providing a versatile framework for computation in a wide range of algebraic structures. The work also highlights open questions about Burnside varieties and further boundaries of G-decidability within algebraic varieties. Overall, it delivers practical, theoretically grounded polynomial-time randomized procedures for foundational generation and decision problems in distributive Ω-expanded groups.
Abstract
Let $Ω$ be a finite set of finitary operation symbols. An $Ω$-expanded group is a group (written additively and called the additive group of the $Ω$-expanded group) with an $Ω$-algebra structure. We use the black-box model of computation in $Ω$-expanded groups. In this model, elements of a finite $Ω$-expanded group $H$ are represented (not necessarily uniquely) by bit strings of the same length, say, $n$. Given representations of elements of $H$, equality testing and the fundamental operations of $H$ are performed by an oracle. Assume that $H$ is distributive, i.e., all its fundamental operations associated with nonnullary operation symbols in $Ω$ are distributive over addition. Suppose $s=(s_1,\dots,s_m)$ is a generating system of $H$. In this paper, we present probabilistic polynomial-time black-box $Ω$-expanded group algorithms for the following problems: (i) given $(1^n,s)$, construct a generating system of the additive group of $H$, (ii) given $(1^n,s,(t_1,\dots,t_k))$ with $t_1,\dots,t_k\in H$, find a generating system of the additive group of the ideal in $H$ generated by $\{t_1,\dots,t_k\}$, and (iii) given $(1^n,s)$, decide whether $H\in\mathfrak V$, where $\mathfrak V$ is an arbitrary finitely based variety of distributive $Ω$-expanded groups with nilpotent additive groups. The error probability of these algorithms is exponentially small in $n$. In particular, this can be applied to groups, rings, $R$-modules, and $R$-algebras, where $R$ is a fixed finitely generated commutative associative ring with $1$. Rings and $R$-algebras may be here with or without $1$, where $1$ is considered as a nullary fundamental operation.