Axial Grammar Framework Overview

• Axial Grammar Framework is a formal system that encodes multi-dimensional structured data into linear token sequences using fixed-arity coordinate systems.
• It employs deterministic, streaming parsing with single-pass coordinate assignment to simplify intermediate representation generation for language models.
• The Memelang instantiation demonstrates its practical utility by translating compact linear syntax into robust SQL and vector-relational queries.

The Axial Grammar Framework is a formal system for encoding multi-dimensional, structured data representations within linear token sequences, designed to facilitate deterministic, streaming parsing for LLM intermediate representations (IRs), especially in the context of tool-oriented and hybrid vector-relational query generation. The framework was introduced and instantiated in the Memelang query language, which provides a compact, LLM-emittable IR that maps directly to structured query constructs, such as those required for parameterized SQL and vector similarity operations without reliance on clause order, parentheses, or context-free parsing mechanisms (Holt, 18 Dec 2025).

1. Formal Structure of Axial Grammar

Axial grammar employs a fixed arity $n > 0$, interpreted as the number of axes, or "ranks," in an $n$-dimensional coordinate space. The input alphabet $\Sigma$ is partitioned into separator tokens $S$ and all other tokens $A$ ("atoms"):

$\Sigma = S \,\dot{\cup}\, A$

Each separator $s \in S$ has a rank $\rho(s) \in \{0,1,\ldots,n-1\}$. The framework processes a token stream $\tau = (t_1, \ldots, t_T) \in \Sigma^T$ with a left-to-right scan, maintaining an $n$-vector of nonnegative integer indices, $$i^{(k)} = (i^{(k)}_{n-1}, ..., i^{(k)}_0) \in \mathbb{N}^n$$. The coordinate update function $U_r$ for separator rank $r$ is defined as:

$$(U_r(i))_m = \begin{cases} i_m & \text{if } m > r \ i_r + 1 & \text{if } m = r \ 0 & \text{if } m < r \ \end{cases}$$

Non-separator (atom) tokens are assigned the current coordinate. Tokens are grouped into "cells" $C_\tau(x)$, each indexed by $x \in \mathbb{N}^n$. Optionally, a bijective reindexing $\pi : \mathbb{N}^n \to \mathbb{N}^n$ transforms the coordinate space (e.g., for alignment). Semantic interpretation then proceeds by applying a partial decoder $g: A^* \rightharpoonup \mathcal{X}$ to nonempty cells, yielding $E(x) = g(C^\pi_\tau(x))$.

Additional mechanisms—coordinate-stable relative references, variable binding, and implicit inheritance (carry-forward)—are layered over this core representation.

2. Deterministic, Single-Pass Coordinate Assignment

Parsing in axial grammar is realized by a streaming, single-pass algorithm. As the token sequence is traversed:

• On encountering a separator of rank $r$, the $r$-th axis coordinate is incremented, and all lower axes are reset to zero.
• Atom tokens inherit the current coordinate.
• After the scan, all atoms are grouped by their coordinates into cells, preserving stream order within each cell.

Pseudocode for this process:

Input: token stream τ[1..T], rank-map ρ:S→{0..n−1} i ← [0, 0, ..., 0] # n-vector for k in 1..T: if τ[k] ∈ S: r ← ρ(τ[k]) for m in 0..n−1: if m > r: i[m] ← i[m] elif m = r: i[m] ← i[m] + 1 else: i[m] ← 0 else: coords[k] ← copy(i) emit (τ[k], coords[k])

This process yields $O(T \cdot n)$ complexity with no need for backtracking or context-free stack management, and deterministic coordinate assignments for semantic parsing (Holt, 18 Dec 2025).

3. Concrete Grammar Specification and Memelang Instantiation

Memelang is a practical implementation of an axial grammar with $n=3$ axes (Matrix, Vector, Limit), each delineated by a unique separator:

• Axis 2: ;; (double semicolon)
• Axis 1: ; (single semicolon)
• Axis 0: whitespace (one or more spaces or tabs)

The effective EBNF for Memelang’s query surface is:

query := ( matrix ";;" )+ ; matrix := vector ( ";" vector )* ; vector := limit ( WS limit )* ; limit := left | right | ( cmp right ) | ( left cmp right ) ; left := [ term ] ( ":" func )* ; right := term ( "," term )* ; term := atom | ( mod atom ) | ( atom mod atom ) ; atom := ALNUM | QUOT | INT | DEC | "_" | "@" | "$" VAR | EMB ; cmp := "=" | "!=" | "<" | "<=" | ">" | ">=" | "~" | "!~" ; mod := "<->" | "<#>" | "<=>" | "+" | "-" | "*" | "/" | "%" | "**" ; func := "grp" | "sum" | "cnt" | "min" | "max" | "avg" | "last" | "asc" | "des" | "$" VAR ; WS := [ \t\r\n ]+ ;

Each rank’s separator enforces scope boundaries (tables, fields, values) in queries, supporting projection, selection, grouping, and ordering as inline, tag-based annotations.

4. Distinctive Mechanisms and Semantic Apparatus

The axial grammar framework incorporates mechanisms facilitating complex query generation and parsing:

• Rank-Specific Separators: Enforce structural boundaries per axis, eliminating ambiguity in clause and sub-clause demarcation.
• Coordinate-Stable Relative References: Special atoms (e.g., @, ^) encode integer vector offsets; their resolution is always relative to the current coordinate, referencing other cells after optional carry-forward.
• Parse-Time Variable Binding: The `:$v` tag at a coordinate binds$v$to$x$; `$v$` is dereferenced to$E(\beta(v))$$in subsequent occurrences, supporting self-joins and value sharing in flat sequences.</li> <li><strong>Implicit Context Carry-Forward (Inheritance):</strong> For axes declared as inheriting (notably Matrix and Vector in Memelang), undefined cells inherit the value from the previous coordinate along that axis:</li> </ul> <p>$$cf_r(E)(x) = E(x) \text{ if defined, else } cf_r(E)(x - e_r) \text{ if } x_r > 0$
  • Inline Aggregation, Grouping, Ordering: Attached "func" chains (e.g., :min, :grp, :asc) annotate the left term in value cells and are parsed to SQL operations (GROUP BY, aggregates, ORDER BY) in a single pass.

  These features collectively enable the generation of highly compact, semantically rich representations, directly mappable to parameterized SQL and vector-relational query plans (Holt, 18 Dec 2025).

  5. Illustrative Example and Token-to-Coordinate Mapping

  The following example demonstrates Memelang’s axial grammar encoding:

  Token stream:

  movies year <1970 ; title _ ;;

  With axes and separators (;;, ;, and whitespace), and n=3, the coordinate assignments are:

  Token	Axis2 sep seen	Axis1 sep seen	Axis0 sep seen	Coord	Role
  movies	i=(0,0,0)	–	–	(0,0,2)	Table
  year	(0,0,2)	–	–	(0,0,1)	Column
  <1970				(0,0,0)	Value
  ;	(sep r=1) ↑	i=(0,1,0)	resets axis0	–	sep Axis1
  title				(0,1,1)	Column (cf)
  _				(0,1,0)	Value
  ;;	(sep r=2) ↑	...	...	–	sep Axis2

  After cell grouping and inheritance, the sequence is deterministically compiled to SQL:

  SELECT t0.year, t0.title FROM movies AS t0 WHERE t0.year < $1 ;

  Parameters (e.g., $1 = 1970) are externally supplied (Holt, 18 Dec 2025).

  6. Parsing Properties and Theoretical Guarantees

  • Streaming, Deterministic Parsing: All semantics are defined in a single left-to-right pass, with coordinate grouping enabling clause and subclause recognition without recursion or context-free parsing.
  • Absence of Grammar Ambiguity: Fixed axis/coordinate roles remove class and order ambiguity. No need for nested delimiters or backtracking.
  • Low-Entropy, Unambiguous Surface Form: Minimal use of separators, stable syntax, and absence of variant keywords reduce LLM prompt complexity and generation errors.
  • Parse Complexity: $O(T \cdot n)$for initial coordinate assignment and grouping;$O(|\text{cells}|)$ for semantic embodiment.
  • Contextual Carry-Forward: Reduces token repetition and implicit context specification, enabling economical linear encodings.

  A plausible implication is that these properties are especially advantageous for LLM-based IR generation in constrained, streaming, or low-entropy settings (Holt, 18 Dec 2025).

  7. Practical Applications and System Benefits

  Axial grammar’s properties are operationalized in Memelang as an IR for LLM-driven tool use, with the following integrable benefits:

  • Direct Mapping to Parameterized SQL: Deteministic translation to PostgreSQL and vector-relational queries (including pgvector) supports injection resistance and safe plan caching.
  • Reduced Prompt Length: The compact, unambiguous linear encoding reduces input/output token consumption for LLMs.
  • Constrained and Streaming Decoding: Deterministic surface form enables validation and decoding under tight resource and security requirements.
  • Compositional Query Expression: Supports joins, groupings, aggregates, filters, ordering, vector similarity lookups, and variable-based self-joins within a uniform linear grammar.
  • Reference Implementation: While no formal benchmark is reported, the provided open-source implementation validates LLM-to-IR-to-SQL workflows, affirming correctness and efficiency with hybrid query outputs and low prompt overhead (Holt, 18 Dec 2025).

  These attributes position the axial grammar framework and its Memelang instance as a robust methodology for structured IR emission and parsing in neural-assisted data toolchains.