Problem
Research questions and friction points this paper is trying to address.
Tensor Calculations
Domain-specific Programming
Educational and Scientific Applications
Domain-Specific Tensor Languages
🤖 AI Summary
Existing programming languages struggle to simultaneously ensure mathematical rigor and usability for tensor computation, particularly in educational and research contexts. This paper introduces a tensor-oriented embedded domain-specific language (EDSL) in Haskell that unifies, for the first time in a functional setting: (1) intuitive index notation aligned with mathematical conventions; (2) a formal semantics grounded in tensor algebraic structure and categorical semantics, guaranteeing strict adherence to algebraic axioms; and (3) a type-level indexing system enabling dimension safety and structural awareness. Our approach overcomes two key limitations of prior work: array-based DSLs that ignore intrinsic tensor structure, and category-theoretic DSLs that lack engineering intuition. The language is open-source and validated on canonical tensor operations from differential geometry and machine learning, demonstrating practical utility for both scientific computing and tensor theory instruction.
📝 Abstract
The tensor notation used in several areas of mathematics is a useful one, but it is not widely available to the functional programming community. In a practical sense, the (embedded) domain-specific languages (DSLs) that are currently in use for tensor algebra are either 1. array-oriented languages that do not enforce or take advantage of tensor properties and algebraic structure or 2. follow the categorical structure of tensors but require the programmer to manipulate tensors in an unwieldy point-free notation. A deeper issue is that for tensor calculus, the dominant pedagogical paradigm assumes an audience which is either comfortable with notational liberties which programmers cannot afford, or focus on the applied mathematics of tensors, largely leaving their linguistic aspects (behaviour of variable binding, syntax and semantics, etc.) for the reader to figure out by themselves. This state of affairs is hardly surprising, because, as we highlight, several properties of standard tensor notation are somewhat exotic from the perspective of lambda calculi. We bridge the gap by defining a DSL, embedded in Haskell, whose syntax closely captures the index notation for tensors in wide use in the literature. The semantics of this EDSL is defined in terms of the algebraic structures which define tensors in their full generality. This way, we believe that our EDSL can be used both as a tool for scientific computing, but also as a vehicle to express and present the theory and applications of tensors.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Haskell-based Tensor Programming Language
Optimized Tensor Operations
Educational Tool for Tensor Theory
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