🤖 AI Summary
This work addresses the prevailing lack of systematic understanding of foundational formal theories in current AI compiler design, which hinders rigorous evaluation of the completeness and desirability of intermediate representations and compilation abstractions. For the first time, it systematically establishes precise correspondences between core mechanisms of MLIR—such as term rewriting systems, refinement calculi, and abstract interpretation—and classical formal theories. By grounding compiler abstractions in formal semantics, the paper clarifies the theoretical underpinnings of these constructs, articulates a precise notion of “design completeness,” and provides assessable criteria and guiding principles to navigate trade-offs between engineering pragmatism and theoretical ideals.
📝 Abstract
Compiler infrastructures such as MLIR rest on a set of design principles: IR abstractions, interfaces, match-and-rewrite, flow analysis, type conversion, staged lowering, and so on. These concepts have proven themselves in practice. Good designs typically arrive through engineering knowledge, intuition and experience. Many of them, however, have correspondences in formal theory. MLIR's match-and-rewrite engine has correspondence to a \emph{term-rewriting-system}~\cite{baadernipkow1998}; staged lowering has the structure of \emph{refinement calculus}~\cite{back1998}; and range analysis is grounded in \emph{abstract interpretation}~\cite{cousot1977,cousot1979}. Highlighting these correspondences is useful because each theory supplies vocabulary precise enough to discuss structural questions. Moreover, as coding agents lower the cost of implementation, good design and abstractions become the main concern~\cite{Lattner2026ClaudeCCompiler}. A coding agent can generate a pass, but it can only reason over the semantics the representation exposes. When essential structure is missing, the limitation is one of abstraction, not of implementation. The natural next question is how to design that substrate well. Well-chosen abstractions emerge from experience and intuition, but they often mirror concepts given a more precise treatment in formal theory. We argue that knowledge of these formal concepts clarifies what completeness means for a given abstraction, what the ideal design would be, and where practical trade-offs depart from it.