Non-Commutative Arithmetic Circuits

• Non-commutative arithmetic circuits are directed acyclic graphs where input nodes hold field constants or noncommuting variables and internal gates perform ordered sums and products.
• They have driven breakthroughs in algebraic complexity by establishing nearly linear and quadratic lower bounds, infinite width and depth hierarchies, and deterministic PIT methods.
• Recent techniques leverage automata-based matrix substitutions and error-tolerant approaches like SWAPPER to improve performance in polynomial identity testing and approximate computing.

A non-commutative arithmetic circuit is a directed acyclic graph in which input nodes are labeled by field constants or non-commuting variables, and internal gates compute either sums (“+”) or non-commutative products (“×”) where left and right inputs are distinguished and ordering impacts the output polynomial. The ring of non-commutative polynomials, $\F\langle x_1,\dots,x_n\rangle$, consists of formal linear combinations of words formed from the variables, and circuits compute polynomials as families of such words via their gate structure. Homogeneous circuits require every gate to compute a polynomial with all monomials of the same degree. The model is widely used for algebraic complexity lower bounds, polynomial identity testing (PIT), circuit width and depth hierarchies, and error-tolerant approximate computation.

1. Model Definitions and Formal Framework

Let $\F$ be a field and $x_1,\dots,x_n$ non-commuting variables. The ring $\F\langle x_1,\dots,x_n\rangle$ describes polynomials where monomials are words over $\{x_1,\dots,x_n\}$ and the order of appearance is significant. A non-commutative arithmetic circuit $\mathcal{C}$ is a DAG with input gates labeled by field constants or variables and internal gates labeled by “+” or “×”, each with a designated left and right child for multiplication to preserve non-commutativity. The output gate computes a polynomial in the ring.

For nonassociative variants, monomials are bracketed binary trees, i.e., elements of the free nonassociative, noncommutative ring $\F\{x_1,\dots,x_n\}$, where both the order and grouping of variables matters. Noncommutative monotone circuits restrict constants to nonnegative reals.

The size of a circuit is the count of internal gates; for homogeneous circuits, all gates compute homogeneous polynomials. Width and depth are defined via layered decompositions: width is the maximal number of gates per layer, depth is the number of layers. +-regular circuits are layered such that each layer contains only +-gates of equal syntactic degree, and every input-to-output path traverses each layer exactly once (Arvind et al., 2016).

2. Quantitative Lower Bounds and Hierarchy Theorems

Recent advances give strong lower bounds for the size of homogeneous non-commutative circuits:

• For an explicit homogeneous bivariate monomial $B_d(x_0,x_1)=\prod_{i=1}^d x_{\sigma_i}$, where $\sigma_i$ is derived from the first $d$ bits of a binary de Bruijn sequence, any homogeneous circuit computing $B_d$ requires non-scalar size $\Omega(d/\log d)$ (Chatterjee et al., 2023).
• For general explicit $n$-variate homogeneous polynomials $F_{n,d}$:

$\text{Any homogeneous non-commutative circuit computing $F_{n,d}$ has non-scalar size } \left\{ \begin{array}{ll} \Omega(nd) &\text{if $d\leq n$;}\ \Omega\bigl(nd\,\frac{\log n}{\log d}\bigr) &\text{if $d\geq n$.} \end{array} \right.$

• For the ordered symmetric polynomial $$OS^d_n(x_1,\dots,x_n)=\sum_{1\le i_1<\cdots<i_d\le n}x_{i_1}\cdots x_{i_d}$$, when $2\le d\leq n/2$, any homogeneous circuit computing it requires $\Omega(nd)$ non-scalar size, yielding $\Omega(n^2)$ for $d=\lfloor n/2\rfloor$.

Monotone noncommutative circuits of width $k$ computing polynomials $P_{k,\ell}$ (degree $2^k$, variables $2^k \ell$, monomials $\ell^k$) require exponential size in $\ell$ if restricted to width-$k$; width $2k$ circuits can realize $P_{k,\ell}$ in linear size (0907.3780). This establishes infinite constant-width and -depth hierarchies.

3. Polynomial Identity Testing and Factorization

PIT for non-commutative circuits is a central algorithmic problem:

• For $+-$regular circuits, deterministic polynomial-time PIT is possible in the white-box setting: basis-finding among products of linear forms via word-equivalence algorithms and linear-algebraic pruning at each sum layer (Arvind et al., 2016).
• For $\Sigma\Pi^*\Sigma$ circuits (exactly two layers of +), randomized black-box PIT suffices via matrix-substitutions: evaluating the circuit on $s\times s$ matrices yields nonzero output unless the polynomial is identically zero, regardless of exponential degree and double-exponential sparsity (Arvind et al., 2016).

Randomized black-box PIT for $+-$regular circuits of constant depth employs nondeterministic substitution automata, polynomial sparsification, commutativity conversion, and coefficient-modification techniques. For size $s$ and depth $d$, substitution of $n$ variables with $s^{O(d^2)}\times s^{O(d^2)}$ matrices allows non-zeroness detection in $s^{O(d^2)}$ time (Bharadwaj et al., 10 Nov 2024).

Nonassociative, noncommutative circuits admit deterministic PIT in polynomial time; for a circuit of size $s$, degree $d$, over $n$ variables, testing whether $f\equiv0$ requires $O((nsd)^{O(1)})$ field operations (Arvind et al., 2017). Complete deterministic factorization into irreducibles is possible in polynomial time over $\mathbb{Q}$ or finite fields.

4. Proof Techniques and Structural Measures

Dimension-measure arguments on “interval-projections” of homogeneous polynomials give upper bounds on the family of polynomials computable by non-commutative circuits: for interval size $\ell$, the span dimension is at most $(\ell-1)s$, where $s$ is the non-scalar gate count. Baur–Strassen–type results generalize: first-position partial derivatives can be computed simultaneously in essentially the same circuit size as the original polynomial (Chatterjee et al., 2023).

Automata-based matrix substitution techniques—where each variable is replaced by a carefully chosen matrix and entry extraction identifies polynomial structure—are critical in proving PIT results for high-depth $+-$regular circuits and their subcases (Bharadwaj et al., 10 Nov 2024, Arvind et al., 2016).

In nonassociative settings, each bracketed monomial’s coefficient vector is tracked through layers; unique root-split decompositions and bilinear combinations through $\times$-gates mirror associative techniques, subject to rigid bracket constraints (Arvind et al., 2017).

5. Applications in Approximate Computing and Error Reduction

Non-commutative operators arise naturally in approximate computing, where logic simplifications can destroy commutativity and induce input-order-dependent errors. SWAPPER methodology demonstrates that a single-bit controlled multiplexer immediately before a non-commutative approximate block, powered by an offline-tuned decision rule on operand bits, can dynamically select the operand order to minimize output error.

At the component level, SWAPPER achieves MAE reductions up to $50\%$ on EvoApproxLib multipliers; at the application level (AxBench), accuracy improvements exceed $90\%$ for image and numerical kernels (Traiola et al., 4 Mar 2025). SWAPPER generalizes to all non-commutative approximate blocks, with negligible hardware or software overhead.

6. Implications, Open Problems, and Theoretical Significance

The classical Baur–Strassen lower bound ($\Omega(n \log d)$ for commutative circuits) remains unbeaten in unrestricted models, but in homogeneous non-commutative settings, new constructions yield nearly linear and quadratic lower bounds for explicit polynomials (Chatterjee et al., 2023). In monotone constant-width and -depth circuits, infinite hierarchies have been established (0907.3780). PIT for general non-commutative circuits, beyond ABPs and low-depth $+-$regular circuits, is an active frontier; deterministic black-box algorithms are widely open.

Nonassociative noncommutative circuits exhibit tractable PIT and deterministic factorization due to unique factorization property—contrasting with associative noncommutative settings where such algorithms provably do not exist (Arvind et al., 2017). Improved structural understanding, automata-based substitution paradigms, and dimension-measure techniques in homogeneous settings provide a new foundation for algebraic complexity and derandomization approaches.

A plausible implication is that strong lower bounds for homogeneous non-commutative circuits may eventually extend to full non-commutative models, with homogeneity serving as an effective proxy restriction to drive complexity separations. Furthermore, error-reduction approaches like SWAPPER suggest new hardware-level optimization layers for emerging approximate DSP designs.