A Foundational View on Integration Problems

The integration of reasoning and computation services across system and language boundaries is a challenging problem of computer science. In this paper, we use integration for the scenario where we have two systems that we integrate by moving problems and solutions between them. While this scenario is often approached from an engineering perspective, we take a foundational view. Based on the generic declarative language MMT, we develop a theoretical framework for system integration using theories and partial theory morphisms. Because MMT permits representations of the meta-logical foundations themselves, this includes integration across logics. We discuss safe and unsafe integration schemes and devise a general form of safe integration.

💡 Research Summary

The paper addresses the longstanding challenge of integrating reasoning and computation services across heterogeneous systems and logical foundations. Rather than focusing on engineering details such as data exchange protocols or syntax conversion, the authors adopt a foundational perspective based on the generic declarative language MMT (Meta‑Meta‑Theory). MMT treats both object-level mathematical theories and meta‑level logical foundations uniformly as theories, and it uses theory morphisms to translate symbols and statements between them. By extending MMT with partial theory morphisms, the framework can represent translations that are incomplete or only partially defined, which is essential when integrating systems that do not share a common foundation.
The authors first formalize the notion of a specification (Spec) as an MMT theory and a concrete system as another theory S together with a morphism v : Spec → S. Soundness of a system corresponds to the morphism preserving logical consequence from Spec to S, while completeness corresponds to the converse preservation. In practice, many computer algebra systems (CAS) violate soundness because they perform unchecked simplifications, and many theorem provers are incomplete relative to richer specifications. Moreover, implementations often commit to concrete definitions that go beyond the abstract specification, leading to underspecification or overspecification problems.
To capture these realities, the paper distinguishes two integration regimes. Safe integration requires a total morphism that respects both typing and equality judgments of MMT, guaranteeing that every theorem proved in the source system remains a theorem in the target system. Unsafe integration allows partial morphisms where some symbols are left unmapped or mapped to ill‑typed objects; in such cases the framework recommends attaching proof sketches or certificates that can be re‑checked in the target system. This approach isolates the “unsafe” part of the translation and provides a systematic way to regain trust through additional verification steps.
The authors illustrate the framework with a detailed example involving natural numbers. The specification Nat is defined in a second‑order logic theory, while two implementations are built on top of ZF set theory: one follows the traditional set‑theoretic encoding of numbers, the other uses a different morphism that maps the same specification symbols to alternative definitions. Both implementations are represented as MMT theories with explicit morphisms from Spec to each implementation, demonstrating how the same abstract specification can admit multiple concrete realizations. The paper also shows how borrowing (using one system to prove theorems in the language of another) and computation (delegating evaluation tasks) can be modeled as morphism applications, with partial morphisms handling cases where the target system lacks the necessary logical machinery.
A significant contribution of the work is the formal definition of safe integration as a “general form” that can be instantiated for any pair of systems, regardless of their underlying logics. By making the semantics of MMT parametric in the chosen foundation, the framework remains agnostic to specific type systems (e.g., LF, CIC) and can therefore support cross‑foundation integration. The authors argue that this meta‑theoretic flexibility overcomes the limitations of existing OpenMath/MathML based content exchange, which typically only aligns syntactic representations and cannot guarantee preservation of logical consequence.
In the related‑work discussion, the paper positions its approach alongside earlier efforts on proof sketches, proof certificates, and heterogeneous tool integration, emphasizing that MMT’s unified treatment of theories, morphisms, and meta‑theories provides a more principled and extensible foundation. The conclusion outlines future directions, including tool support for automatic generation of partial morphisms, integration with existing proof assistants, and empirical evaluation on large‑scale CAS‑theorem‑prover collaborations. Overall, the paper presents a rigorous, foundation‑independent framework for safe and controlled integration of mathematical software, opening a path toward reliable, interoperable computational mathematics ecosystems.