arXiv subjects

Lenore Mullin

Publications and source records attributed to Lenore Mullin.

4 recordsLinked to original sources

arXiv2026

The Mathematical Evolution of Addressing: From Yushchenko's Address Programming Language to A Mathematics of Arrays How Addressing Evolved into a Mathematical Theory of Indexing, Array Computation, and Machine Realization

This article argues that a central development in the history of computing is the mathematical evolution of addressing. During the three decades separating Kateryna Yushchenko's Address Programming Language (1958) from Lenore Mullin's A Mathematics of Arrays (1988), addressing evolved from a machine-level mechanism for locating data into a formal theory of indexing, reduction, and machine realization, passing through Kenneth Iverson's algebra of arrays and Philip Abrams' recognition of shape as an architectural resource. Mullin's theory establishes indexing itself, via the function ψ, as the primitive from which array operations compose, reduce to a Denotational Normal Form, transform into an Operational Normal Form, and realize on hardware through dimension lifting. Historical claims are sourced against primary material, with inferences flagged as such; mathematical claims are stated as identities; performance claims are marked validated, theoretically established, or provisional. The result links programming languages, array mathematics, and computer architecture in one account.

cs.GL↗

arXiv2026

Attention at the Theoretical Minimum: A Mathematics of Arrays Framework for Memory-Optimal Transformer Kernels

The attention mechanism is the dominant computational bottleneck in modern transformer-based AI. Its standard implementation incurs quadratic memory traffic in the sequence length~$n$, and DRAM accesses cost 100--1000$\times$ more energy than arithmetic operations on contemporary hardware, so any analysis focused solely on FLOP counts fundamentally mischaracterises the bottleneck. We present a Mathematics of Arrays (MoA) reformulation of scaled dot-product attention and its numerically stable softmax, deriving a Denotational Normal Form (DNF) that eliminates all intermediate arrays -- including the implicit transposed-key buffer and every softmax temporary -- by algebraic construction rather than empirical tuning. The DNF achieves $O(n_{dk} + n{_{dv}})$ data movement versus $O(n^2 + n_{dk} + n_{dv})$ for the standard implementation, where $n$ is the sequence length, $dk$ is the key dimensionality and $dv$ the value dimensionality, and is verified numerically against PyTorch at full double-precision floating-point on concrete inputs. Unlike hardware-specific accelerators or empirical tiling schemes such as FlashAttention, MoA simultaneously provides array fusion, shape-transformation correctness, and predictive cost models from a single algebraic framework. Memory minimality is a theorem established before any code is written. A predictive performance model projects $2$--$100\times$ speedup and $2$--$50\times$ energy reduction, with the advantage widening at exascale. The derivation establishes a formally verified pipeline from Python specification through (ONF) Operational Normal Form, and dimension-lifted hardware mapping, providing performance-portable AI kernels of direct relevance to DARPA edge-deployment and DOE exascale priorities.

cs.LG↗

arXiv2023

Parallelizing Optimal Multiple Sequence Alignment by Dynamic Programming

Optimal multiple sequence alignment by dynamic programming, like many highly dimensional scientific computing problems, has failed to benefit from the improvements in computing performance brought about by multi-processor systems, due to the lack of suitable scheme to manage partitioning and dependencies. A scheme for parallel implementation of the dynamic programming multiple sequence alignment is presented, based on a peer to peer design and a multidimensional array indexing method. This design results in up to 5-fold improvement compared to a previously described master/slave design, and scales favourably with the number of processors used. This study demonstrates an approach for parallelising multi-dimensional dynamic programming and similar algorithms utilizing multi-processor architectures.

cs.DC↗

arXiv2019

MoA Interpretation of the Iterative Conjugate Gradient Method with Psi Reduction - A Tutorial to teach the Mathematically literate in Linear and Tensor Algebra: Part I

It is often difficult to learn new mathematics semantically and syntactically, even when there are similarities in the words and meaning when discussed aloud. The goal of this document is to facilitate learning through explanations and definitions relating our common mathematical knowledge and highlighting what is new. It is meant to be a working document that will evolve based on feedback from target audiences, those mathematically literate in linear and tensor algebra, those that want to learn MoA, Psi Calculus, and its uses, those that want and need the ability to prove a design, either in hardware or software through the ONF, Operational Normal Form, and those wanting to exploit all resources optimally, especially when Tensor Algebra, i.e. algorithms foundational to their application,are needed: Knowledge Representation, Machine Learning, Signal Processing, AI, HPC, etc.

cs.OH↗