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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:
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Syntactic validity
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Logical consistency
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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:
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Grammatically invalid sequences
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Logically contradictory statements
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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.
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