Treehouse of Horror: The LaTeX Massacre

Segment 1: The Formatting

Homer works as a LaTeX typesetter at the nuclear plant. After Mr. Burns demands perfectly aligned equations, Homer goes insane trying to format complex mathematical expressions, eventually snapping when his equations run off the page. In a parody of "The Shinning," Homer chases his family around with a mechanical keyboard while screaming "All work and no proper alignment makes Homer go crazy!"

Segment 2: Time and Compilation

In a nod to "Time and Punishment", Homer accidentally breaks his LaTeX compiler and tries to fix it, but ends up creating a time paradox where every document compiles differently in parallel universes. He desperately tries to find his way back to a reality where his equations render properly.

Segment 3: The Cursed Code

Bart discovers an ancient LaTeX document that contains forbidden mathematics. When he compiles it, it summons an eldrich horror made entirely of misaligned integrals and malformed matrices. Lisa must save Springfield by finding the one perfect alignment that will banish the mathematical monster back to its dimension.

The episode ends with a meta-joke about how even the credits won't compile properly.

A Modest Proposal: Statistical Token Prediction Is No Replacement for Syntactic Construction

A Modest Proposal: Statistical Token Prediction Is No Replacement for Syntactic Construction by Stephen Crowley October 25, 2024

1Current Generative-Pretrained-Transformer Architecture

Given vocabulary $V$, $|V| = v$, current models map token sequences to vectors:

$\displaystyle (t_1, \ldots, t_n) \mapsto X \in \mathbb{R}^{n \times d}$

Through layers of transformations:

$\displaystyle \text{softmax} (QK^T / \sqrt{d}) V$

where $Q = XW_Q$, $K = XW_K$, $V = XW_V$

Optimizing:

$\displaystyle \max_{\theta} \sum \log P (t_{n + 1} |t_1, \ldots, t_n ; \theta)$

2Required Reformulation

Instead, construct Abstract Syntax Trees where each node $\eta$ must satisfy:

$\displaystyle \eta \in \{ \text{Noun}, \text{Verb}, \text{Adjective}, \text{Conjunction}, \ldots\}$

With composition rules $R$ such that for nodes $\eta_1, \eta_2$:

$\displaystyle R (\eta_1, \eta_2) = \left\{ \begin{array}{ll} \text{valid\_subtree} & \text{if grammatically valid}\\ \emptyset & \text{otherwise} \end{array} \right.$

And logical constraints $L$ such that for any subtree $T$:

$\displaystyle L (T) = \left\{ \begin{array}{ll} T & \text{if logically consistent}\\ \emptyset & \text{if contradictory} \end{array} \right.$

3Parsing and Generation

Input text $s$ maps to valid AST $T$ or error $E$:

$\displaystyle \text{parse} (s) = \left\{ \begin{array}{ll} T & \text{if } \exists \text{valid AST}\\ E (\text{closest\_valid}, \text{violation}) & \text{otherwise} \end{array} \right.$

Generation must traverse only valid AST constructions:

$\displaystyle \text{generate} (c) = \{T|R (T) \neq \emptyset \wedge L (T) \neq \emptyset\}$

where $c$ is the context/prompt.

4Why Current GPT Fails

The statistical model:

$\displaystyle \text{softmax} (QK^T / \sqrt{d}) V$

Has no inherent conception of:

• Syntactic validity

• Logical consistency

• Conceptual preservation

It merely maximizes:

$\displaystyle P (t_{n + 1} |t_1, \ldots, t_n)$

Based on training patterns, with no guaranteed constraints on:

$\displaystyle \prod_{i = 1}^n P (t_i |t_1, \ldots, t_{i - 1})$

This allows generation of:

• Grammatically invalid sequences

• Logically contradictory statements

• Conceptually inconsistent responses

5Conclusion

The fundamental flaw is attempting to learn syntax and logic from data rather than building them into the architecture. An AST-based approach with formal grammar rules and logical constraints must replace unconstrained statistical token prediction.