# Mathematical Expression Subsystem

> [!NOTE]
> This document is written by Claude by Anthropic using Sonnet 4.6 and has yet to be vetted by Mikael Lövqvist

## Overview

A math-like expression language built on top of the reduction scanner, supporting
operator notation, matrix literals, subscripts, superscripts, and symbolic operators.

## Operator Notation

Operators are identified by their symbol name rather than semantic meaning, since the
same symbol can mean different things depending on operand types:

- `*` (ASTERISK) — could be scalar multiplication, Hadamard product, or scale depending on types
- `·` (DOT) — dot product
- `×` (CROSS) — cross product
- `⊕` (OPLUS) — direct sum or XOR

Semantic resolution (e.g. `ASTERISK(matrix, matrix)` → Hadamard) is a separate
type-inference pass, not part of the structural reduction.

## ASCII Input for Special Symbols

LaTeX-inspired escape sequences for entering special symbols in plain ASCII:

- `\oplus` → ⊕
- `\times` → ×
- `\cdot` → ·
- `\otimes` → ⊗

`^` is reserved for superscript (not XOR), `_` for subscript. `S_12` reads as S₁₂.

## Matrix Literals

Single-line input using nested brackets:

```
[[1, 0, 0], [0, 1, 0], [0, 0, 1]]
```

Pretty-printed output using Unicode bracket characters:

```
⎡1 0 0⎤
⎢0 1 0⎥
⎣0 0 1⎦
```

## 2D Raster Reduction Scanner

For parsing pretty-printed multi-line matrix literals within larger expressions like
`M + 2 * N` where M and N are written in 2D notation, a raster-based reduction pass
is needed before the standard 1D reduction pass.

### Approach

1. **Raster pass first** — operate on a 2D grid of characters
2. Locate matrix corner anchors `⎡⎤⎣⎦` — these are highly selective so candidate
   detection is cheap
3. Scan right for `⎤`, down for `⎣`, verify `⎦` at intersection
4. Use `⎢`/`⎥` to identify row boundaries within the region
5. Collapse the identified rectangle into a single matrix token
6. **1D pass second** — the surrounding expression now contains ordinary tokens and
   the collapsed matrix nodes, reducible by standard rules

### Scope Boundaries

Fraction bars define containment — a matrix appearing in a numerator or denominator
is only part of that sub-expression. The horizontal extent of the fraction bar bounds
the operand scan. Containment must be resolved outside-in: find outermost structure
first, recurse into sub-regions.

### Generalization

A 2D reduction scanner is a natural generalization of the 1D scanner — the "sequence"
becomes a 2D array and conditions match spatial patterns rather than linear ones.
The same anchor-point and backtracking concepts apply.