[Paper Review] Dedukti: a Logical Framework based on the $λ$$Π$-Calculus Modulo Theory

Ali Assaf, Guillaume Burel+8 more|arXiv (Cornell University)|Nov 13, 2023

Logic, programming, and type systems4 citations

TL;DR

This paper presents Dedukti, a logical framework based on the $\lambda\Pi$-calculus modulo theory, which enables the formalization and verification of diverse logical systems—including constructive and classical logic, type theories, programming languages, and proof assistants like Coq, Matita, and HOL Light. It demonstrates that large proof libraries from these systems can be translated into and checked within Dedukti, establishing it as a unifying platform for interoperable formal proofs.

ABSTRACT

Dedukti is a Logical Framework based on the $λ$$Π$-Calculus Modulo Theory. We show that many theories can be expressed in Dedukti: constructive and classical predicate logic, Simple type theory, programming languages, Pure type systems, the Calculus of inductive constructions with universes, etc. and that permits to used it to check large libraries of proofs developed in other proof systems: Zenon, iProver, FoCaLiZe, HOL Light, and Matita.

Motivation & Objective

• To unify diverse formal systems—such as constructive logic, classical logic, type theories, and programming languages—within a single logical framework.
• To overcome the limitations of predicate logic as a unifying framework, including inability to express dependent types, bound variables beyond quantifiers, and computational reductions.
• To enable interoperability between proof assistants by translating proofs from external systems into a common, checkable format in Dedukti.
• To demonstrate the scalability and expressivity of Dedukti through end-to-end translation and verification of large proof libraries.
• To lay the foundation for a unified proof ecosystem by enabling combination of lemmas from different systems and reverse engineering of proofs for cross-framework reuse.

Proposed method

• Leveraging the $\lambda\Pi$-calculus modulo theory as the foundational logical framework, which combines dependent types with equational reasoning via rewrite rules.
• Expressing various logical systems (e.g., Simple type theory, Calculus of Inductive Constructions, Pure Type Systems) as theories within this framework using definitional encodings.
• Designing and implementing Dedukti as a concrete system that supports type checking, normalization, and proof verification under the $\lambda\Pi$-calculus modulo theory.
• Developing translators to convert proof artifacts from external systems (e.g., Zenon, iProver, FoCaLiZe, HOL Light, Matita) into Dedukti’s syntax and semantics.
• Validating translations by checking the resulting Dedukti libraries for correctness, with benchmarks on file sizes and verification times.
• Extending the framework to support features like universe polymorphism and proof irrelevance, though these remain partially unimplemented in current versions.

Experimental results

Research questions

• RQ1 Can the $\lambda\Pi$-calculus modulo theory serve as a sufficiently expressive and practical foundation for encoding a wide range of logical systems and programming languages?
• RQ2 To what extent can proof libraries from major proof assistants like Coq, Matita, and HOL Light be faithfully translated into Dedukti without loss of expressivity?
• RQ3 Can Dedukti effectively unify formal proofs across heterogeneous systems, enabling cross-library reasoning and lemma combination?
• RQ4 How scalable is Dedukti in handling large-scale proof corpora, such as the TPTP library or the Matita arithmetic library?
• RQ5 What are the practical challenges in reverse-engineering proofs from classical systems (e.g., HOL Light) to extract constructive content usable in Dedukti?

Key findings

• Dedukti successfully expresses a broad spectrum of logical systems, including constructive and classical predicate logic, Simple type theory, Pure Type Systems, and the Calculus of Inductive Constructions with universes.
• The translation of the Matita arithmetic library into Dedukti produced a 1.11 MB gzipped file that was successfully verified, demonstrating scalability to non-trivial proof corpora.
• Proofs from Zenon Modulo, iProverModulo, FoCaLiZe, and HOL Light were successfully translated and checked in Dedukti, confirming cross-framework interoperability.
• The system supports the combination of lemmas from different theories when they share a common language fragment, enabling combined reasoning across systems.
• Many proofs in classical systems like HOL Light are inherently constructive, suggesting that reverse engineering can unlock reuse potential across formal systems.
• Although full support for features like universe polymorphism and proof irrelevance remains pending, preliminary results show feasibility, and ongoing work aims to close these gaps.