Symbolical Index Reduction and Completion Rules for Importing Tensor Index Notation into Programming Languages

TL;DR

This paper introduces a programming language framework that supports symbolical tensor index notation, enabling flexible and concise tensor operations with user-defined functions, including differential forms, through index reduction and completion rules.

Contribution

It proposes a novel set of tensor index rules compatible with scalar and tensor parameters, allowing user-defined tensor operators and index notation in programming languages.

Findings

Supports arbitrary user-defined tensor functions with index notation

Enables concise definition of differential form operators

Allows tensor operators to be passed as high-order function arguments

Abstract

In mathematics, many notations have been invented for the concise representation of mathematical formulae. Tensor index notation is one of such notations and has been playing a crucial role in describing formulae in mathematical physics. This paper shows a programming language that can deal with symbolical tensor indices by introducing a set of tensor index rules that is compatible with two types of parameters, i.e., scalar and tensor parameters. When a tensor parameter obtains a tensor as an argument, the function treats the tensor argument as a whole. In contrast, when a scalar parameter obtains a tensor as an argument, the function is applied to each component of the tensor. On a language with scalar and tensor parameters, we can design a set of index reduction rules that allows users to use tensor index notation for arbitrary user-defined functions without requiring additional description. Furthermore, we can also design index completion rules that allow users to define the operators concisely for differential forms such as the wedge product, exterior derivative, and Hodge star operator. In our proposal, all these tensor operators are user-defined functions and can be passed as arguments of high-order functions.