Universal Conditional Logic: A Formal Language for Prompt Engineering

TL;DR

Universal Conditional Logic (UCL) offers a formal, systematic approach to prompt optimization in language models, reducing token usage and explaining performance variations through a mathematical framework.

Contribution

This paper introduces UCL, a novel formal language for prompt engineering that enables systematic optimization and explains model performance differences.

Findings

Significant token reduction (29.8%) with cost savings.

UCL's structural overhead explains performance degradation beyond a threshold.

Optimal configurations vary across model architectures.

Abstract

We present Universal Conditional Logic (UCL), a mathematical framework for prompt optimization that transforms prompt engineering from heuristic practice into systematic optimization. Through systematic evaluation (N=305, 11 models, 4 iterations), we demonstrate significant token reduction (29.8%, t(10)=6.36, p < 0.001, Cohen's d = 2.01) with corresponding cost savings. UCL's structural overhead function O_s(A) explains version-specific performance differences through the Over-Specification Paradox: beyond threshold S* = 0.509, additional specification degrades performance quadratically. Core mechanisms -- indicator functions (I_i in {0,1}), structural overhead (O_s = gamma * sum(ln C_k)), early binding -- are validated. Notably, optimal UCL configuration varies by model architecture -- certain models (e.g., Llama 4 Scout) require version-specific adaptations (V4.1). This work establishes UCL as a calibratable framework for efficient LLM interaction, with model-family-specific optimization as a key research direction.