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Large language models (LLMs) are powerful AI tools that can generate andcomprehend natural language text and other complex information. However, thefield lacks a mathematical framework to systematically describe, compare andimprove LLMs. We propose Hex a framework that clarifies key terms and conceptsin LLM research, such as hallucinations, alignment, self-verification andchain-of-thought reasoning. The Hex framework offers a precise and consistentway to characterize LLMs, identify their strengths and weaknesses, andintegrate new findings. Using Hex, we differentiate chain-of-thought reasoningfrom chain-of-thought prompting and establish the conditions under which theyare equivalent. This distinction clarifies the basic assumptions behindchain-of-thought prompting and its implications for methods that use it, suchas self-verification and prompt programming.  Our goal is to provide a formal framework for LLMs that can help bothresearchers and practitioners explore new possibilities for generative AI. Wedo not claim to have a definitive solution, but rather a tool for opening upnew research avenues. We argue that our formal definitions and results arecrucial for advancing the discussion on how to build generative AI systems thatare safe, reliable, fair and robust, especially in domains like healthcare andsoftware engineering.

Beyond Words: A Mathematical Framework for Interpreting Large Language Models