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

Raster pass first — operate on a 2D grid of characters
Locate matrix corner anchors ⎡⎤⎣⎦ — these are highly selective so candidate detection is cheap
Scan right for ⎤, down for ⎣, verify ⎦ at intersection
Use ⎢/⎥ to identify row boundaries within the region
Collapse the identified rectangle into a single matrix token
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.

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