The identities of additive binary arithmetics

Operations of arbitrary arity expressible via addition modulo 2^n and bitwise addition modulo 2 admit a simple description. The identities connecting these two additions have finite basis. Moreover, the universal algebra with these two operations is rationally equivalent to a nilpotent ring and, therefore, generates a Specht variety.

💡 Research Summary

The paper investigates the algebraic structure formed by the set Z_q = {0,1,…,q‑1} where q = 2ⁿ, equipped with two elementary operations that are ubiquitous in computer hardware: ordinary addition modulo q (denoted “+”, often called ADD) and bitwise exclusive‑or modulo 2 (denoted “⊕”, often called XOR). The authors ask two fundamental questions: (1) which k‑ary functions f : Z_q^k → Z_q can be expressed using only these two operations, and (2) what identities relate the two operations.

The first main result, Theorem 1, gives a complete characterisation of the “algebraic” functions. For each bit i of the output, the i‑th bit of f(x₁,…,x_k) must be representable by a Zhegalkin (i.e. Boolean) polynomial g_i in the current bits of the arguments together with all lower‑order bits. Crucially, g_i must have no constant term and its weight – defined as the sum of 2^{‑ℓ} for each occurrence of a variable from bit i‑ℓ – must not exceed 1. This weight condition precisely captures the limited carry propagation that can occur when only ADD and XOR are available. Consequently, the set F_{k,q} of all algebraic functions coincides with the set of functions satisfying this bit‑wise polynomial condition.

Based on Theorem 1 the authors present an explicit decision algorithm. Starting from the most significant bit, one computes the Zhegalkin polynomial for that bit, checks the weight and constant‑term constraints, then substitutes the variables according to a shift (x₀→0, x₁→x₀, …) to obtain the polynomial for the next lower bit, and repeats down to the least significant bit. If every step succeeds, the function is realizable; otherwise it is not. The algorithm runs uniformly in the number of bits κ = log₂ q and works for any arity k.

Corollary 1 quantifies the size of the free algebra F_{k,q}. By counting the number of Zhegalkin polynomials of weight ≤ 1, the authors obtain |F_{k,q}| = 2^{(1/k!)(q²+1)(q²+2)…(q²+k)−1}. This formula shows that the number of algebraic functions grows doubly‑exponentially with the number of bits, reflecting the richness of the expressive power of ADD and XOR.

The second major contribution, Theorem 2, establishes that the algebra A_q = (Z_q, +, ⊕) possesses a finite basis of identities; in universal‑algebra terminology, the variety generated by A_q is finitely based and, moreover, it is a Specht variety (every subvariety is also finitely based). While finite bases are known for many classical structures (groups, rings, Lie algebras), A_q is neither a group nor a ring, so the result is non‑trivial.

To prove Theorem 2 the authors show in Theorem 3 that A_q is rationally equivalent (in the sense of Mal’cev) to a nilpotent, commutative, non‑associative ring R_q = (Z_q, ⊕, ◦). The ring addition is exactly the bitwise XOR, while the multiplication is defined by x ◦ y = 2·(x ⊙ y), where ⊙ denotes bitwise AND. Multiplication by 2 corresponds to a left shift of binary digits, i.e., multiplication by the integer 2 modulo q. This construction makes R_q nilpotent because any product involving more than log₂ q factors necessarily contains a factor 2^{log₂ q}=q, which is zero modulo q. It is a classical result that every nilpotent ring has a finite identity basis and generates a Specht variety; thus A_q inherits these properties via the rational equivalence.

The proof of Theorem 1 and the subsequent lemmas rely on an analysis of the “commutator”