  Large language models (LLMs) are powerful AI tools that can generate and
comprehend natural language text and other complex information. However, the
field lacks a mathematical framework to systematically describe, compare and
improve LLMs. We propose Hex a framework that clarifies key terms and concepts
in LLM research, such as hallucinations, alignment, self-verification and
chain-of-thought reasoning. The Hex framework offers a precise and consistent
way to characterize LLMs, identify their strengths and weaknesses, and
integrate new findings. Using Hex, we differentiate chain-of-thought reasoning
from chain-of-thought prompting and establish the conditions under which they
are equivalent. This distinction clarifies the basic assumptions behind
chain-of-thought prompting and its implications for methods that use it, such
as self-verification and prompt programming.
  Our goal is to provide a formal framework for LLMs that can help both
researchers and practitioners explore new possibilities for generative AI. We
do not claim to have a definitive solution, but rather a tool for opening up
new research avenues. We argue that our formal definitions and results are
crucial for advancing the discussion on how to build generative AI systems that
are safe, reliable, fair and robust, especially in domains like healthcare and
software engineering.
