Theoretical Foundations of Prompt Engineering: From Heuristics to Expressivity

• URL: http://arxiv.org/abs/2512.12688v1
• Date: Sun, 14 Dec 2025 13:42:20 GMT
• Title: Theoretical Foundations of Prompt Engineering: From Heuristics to Expressivity
• Authors: Dongseok Kim, Hyoungsun Choi, Mohamed Jismy Aashik Rasool, Gisung Oh,
• Abstract summary: We study the family of functions obtainable by holding a Transformer backbone fixed as an executor and varying only the prompt.<n>We prove a constructive existential result showing that a single fixed backbone can approximate a broad class of target behaviors via prompts alone.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Prompts can switch a model's behavior even when the weights are fixed, yet this phenomenon is rarely treated as a clean theoretical object rather than a heuristic. We study the family of functions obtainable by holding a Transformer backbone fixed as an executor and varying only the prompt. Our core idea is to view the prompt as an externally injected program and to construct a simplified Transformer that interprets it to implement different computations. The construction exposes a mechanism-level decomposition: attention performs selective routing from prompt memory, the FFN performs local arithmetic conditioned on retrieved fragments, and depth-wise stacking composes these local updates into a multi-step computation. Under this viewpoint, we prove a constructive existential result showing that a single fixed backbone can approximate a broad class of target behaviors via prompts alone. The framework provides a unified starting point for formalizing trade-offs under prompt length/precision constraints and for studying structural limits of prompt-based switching, while remaining distinct from empirical claims about pretrained LLMs.

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