MMV-RAM: Accelerated Vector & Matrix Computation

• MMV-RAM model is a computational framework integrating vector and small matrix-multiply operations for accelerated AI/ML primitives and formal complexity analysis.
• It employs a Matrix-Multiply Unit (MMU) that computes dense matrix products in one parallel step, bridging AC⁰ and TC⁰ circuit complexity classes.
• The model underpins O(logₛ n) segmented scan algorithms and demonstrates tangible speedups over vector-only methods in high-performance and irregular computations.

The MMV-RAM (Matrix-Multiply and Vector Random Access Machine) model is a formal computational paradigm devised to reflect the architecture and algorithmic possibilities of modern AI/ML accelerators equipped with both vector compute units and dedicated small matrix-multiply hardware. The model captures the interaction between vector-level parallelism and small dense-matrix multiplications, and enables rigorous theoretical analysis of the complexity of parallel primitives, particularly those that underpin high-performance numerical and irregular data operations. The MMV-RAM model was introduced and analyzed in (Sobczyk et al., 30 Jun 2025), providing formal cost metrics, circuit-theoretic power separation, and corresponding algorithmic results.

1. Machine Architecture and Model Definition

MMV-RAM augments the traditional Vector-RAM by incorporating a Matrix-Multiply Unit (MMU) capable of computing an $(n\times s)$ by $(s\times s)$ product in a single parallel step, where $s$ is a global model parameter ($s\ge 2$). The architecture comprises:

• Memory organization: A single shared main memory with unbounded capacity, storing words of length $O(\log n)$ bits. Two I/O interfaces are provided:
  • Vector port: Reads/writes any $n$ consecutive words in one step.
  • Matrix port: Reads/writes any $n\cdot s$ consecutive words in one step.
• Processing units per step:

1. Scalar unit: Handles address computation, scalar arithmetic, and control flow.
2. Vector Compute Unit (VCU): Executes one vector instruction per step, each realized as an unbounded fan-in, constant-depth (AC⁰) circuit family.
3. Matrix-Multiply Unit (MMU): In one step, computes $C \leftarrow \text{matmul}(A,B)$ for $A$ as an $n\times s$ matrix (in row-major layout) and $B$ as $s\times s$. The MMU corresponds to a uniform TC⁰ circuit of depth $O(1)$ and size $O(n s^2)$.

The model's core innovation is the explicit demarcation between the vector (AC⁰) and matrix-multiply (TC⁰) primitives, mirroring architectural constraints and circuit-complexity separations—specifically, the well-known result that parity does not belong to AC⁰.

2. Instruction Set and Computational Cost Model

Matrix Operations

• Matrix multiplication: $C \leftarrow \text{matmul}(A, B)$ computes $C = AB$ for $A$ ($n\times s$), $B$ ($s\times s$); takes one step, with a work cost $M(n) = \Theta(n s^2)$.

Vector Operations

VCU primitives execute in one step, realized as AC⁰ circuits. Representative instructions include:

• Bitwise operations: AND, OR, NOT on $n$-bit vectors ($O(n)$ size).
• Addition/Subtraction: On $n$ $B$-bit integers ($O(nB^2)$ size).
• mask, ISZERO, FILLS, gathers, scatters: Each with circuit size and fan-in/fan-out as in Table 3 of (Sobczyk et al., 30 Jun 2025).
• revertspecs: AC⁰ circuit to correct speculative unsegmented scans during block-wise segmented scan computation.

Memory access: Vector ($n$ words) or matrix ($n s$ words) reads/writes take no additional steps, folded into the cost of the respective compute units.

3. Complexity Bounds and Theoretical Separation

MMV-RAM is designed to expose the algorithmic impact of hardware acceleration for matrix multiplication while preserving the limits of vector-level parallelism.

• Segmented-scan algorithms: The core result is an $O(\log_s n)$-step algorithm for segmented scan on input of length $n$, using block-wise speculative scan via matrix-multiply and AC⁰ correction circuits.
• Vector-only lower bound: Any algorithm relying solely on the VCU (AC⁰) requires $\Omega(\log_2 n / \log_2 \log_2 n)$ steps for prefix sum or segmented scan. This reflects the AC⁰ lower bound for parity (Håstad's result), since prefix sum computation subsumes parity detection.
• Work cost: Full algorithmic work for segmented scan is $O(M(n/s) + n (sB + B^2/s))$ for $n$-length input, with matrix-multiply cost $M(n) = n s^2$.

These complexity separations precisely capture the benefit of hardware tensor and matrix-multiply acceleration on workloads featuring irregular parallel primitives.

4. Principal Algorithms in the MMV-RAM Model

Segmented Scan (SegScan)

Recursive partitioning of an input vector $A$ and flag vector $F$ (segment boundaries) into blocks of size $s$ enables matrix-based speculative scans at each level:

Procedure SegScan(A, F; s): B ← BlockSegScan(A, F; s) return Recurse(A, F; s) Procedure Recurse(A, F; s): if |A| ≤ s then return A T ← matmul(A, U_s) // U_s = upper-triangular s×s all-ones F′ ← gathers(T; s) F′ ← ¬ISZERO(F′) F′ ← gathers(F′; s) BlockSummary ← BlockSegScan(F′, F′; s) BlockPrefix ← Recurse(F′, BlockSummary; s) C ← scatters(BlockPrefix; s) UpdateFirstSegment(A, C; s) // masked broadcast return A Procedure BlockSegScan(X, Flags; s): Y ← matmul(X, U_s) Z ← matmul(Flags, U_s) return revertspecs(Y, Z; s)

The overall depth is $O(\log_s n)$.

Segmented Sum (SCD)

Derived as a composition:

1. SCAN (unsegmented) on $A$,
2. COMPRESS to gather results at segment ends,
3. DIFF to recover per-segment sums (vector-differentiation).

All steps are expressible via MMU-accelerated matmul and VCU primitives, aggregate depth $O(\log_s n)$.

Additional Primitives

• Elementwise integer multiplication: Reduces to segmented scan on $B$-bit integer factors; all $n$ products in $O(\log_s B)$ steps.
• Dense $n\times n$ matrix multiplication: Flattened to an unsegmented scan of $n^3$ elements, segments corresponding to block summations; $O(\log_s (n B))$ depth.

5. Empirical and Practical Observations

Experimental results on platforms such as the Ascend 910B demonstrate practical efficacy:

• Single-core segmented scan via SegScan achieves up to $2\times$ speedup over vector-only baselines for cumsum+mask.
• COMPRESS primitive often dominates runtime ($\sim$50%), while SCAN/DIFF approach memory-bandwidth ceilings.
• Fusion of matmul and vector instructions in multicore settings yields significant gains over purely vectorized approaches.

This validates the MMV-RAM abstraction as both a lower-bound model and a facilitator for pragmatic, architecture-aware algorithm design (Sobczyk et al., 30 Jun 2025).

6. Significance and Impact

MMV-RAM rigorously delineates the algorithmic power granted by small matrix multiplication in hardware, balancing it against vector-level AC⁰ primitives. The separation achieved—rooted in circuit complexity theory—proves both necessary and sufficient for accelerating key irregular data primitives on modern hardware, providing foundational bounds for algorithm designers and a platform for modeling hardware/software co-design. The model offers both a formal complexity-theoretic tool and a guide for practical high-performance implementation, as illustrated by algorithms for segmented scan, segmented sum, and matrix algebra.

7. Relation to Prior and Contemporary Models

By extending the Vector-RAM with MMU primitives and aligning cost models to realistic circuit complexity classes (AC⁰/TC⁰), the MMV-RAM model bridges the substantial gap between abstract theoretical computation (e.g., PRAM, Vector-RAM) and practical accelerator architectures. No prior model captured the computational dichotomy imposed by tight AC⁰ limits combined with matrix-multiply acceleration, nor did any achieve the lower bounds and practical speedups observed for key parallel primitives on contemporary AI accelerators (Sobczyk et al., 30 Jun 2025).