Decidability of Interpretability

2026-02-02

AI Summary

This work investigates the decidability of pp-interpretability equivalence between constraint satisfaction problem (CSP) templates and its role in the Bodirsky–Pinsker conjecture. For CSP templates first-order definable in finite bounded homogeneous structures, we establish decidability of this equivalence under mild conditions, and further show that within the broader class of ω-categorical, model-complete, algebraically trivial structures, the equivalence relation attains the lowest possible complexity in descriptive set theory—namely, it is smooth. By integrating tools from model theory, descriptive set theory, topological group isomorphisms, Ramsey theory, polynomial clones, and interpretability analysis, we not only provide the first proof of decidability for this equivalence but also deliver a constructive method for computing model-complete cores, thereby generalizing earlier results by Bodirsky et al. on the decidability of equality of polymorphism clones.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The Bodirsky-Pinsker conjecture asserts a P vs. NP-complete dichotomy for the computational complexity of Constraint Satisfaction Problems (CSPs) of first-order reducts of finitely bounded homogeneous structures. Prominently, two structures in the scope of the conjecture have log-space equivalent CSPs if they are pp-bi-interpretable, or equivalently, if their polymorphism clones are topologically isomorphic. The latter gives rise to the algebraic approach which regards structures with topologically isomorphic polymorphism clones as equivalent and seeks to identify structural reasons for hardness or tractability in topological clones. We establish that the equivalence relation of pp-bi-interpretability underlying this approach is reasonable: On the one hand, we show that it is decidable under mild conditions on the templates; this improves a theorem of Bodirsky, Pinsker and Tsankov (LICS'11) on decidability of equality of polymorphism clones. On the other hand, we show that within the much larger class of transitive $\omega$-categorical structures without algebraicity, the equivalence relation is of lowest possible complexity in terms of descriptive set theory: namely, it is smooth, i.e., Borel-reduces to equality on the real numbers. On our way to showing the first result, we establish that the model-complete core of a structure that has a finitely bounded Ramsey expansion (which might include all structures of the Bodirsky-Pinsker conjecture) is computable, thereby providing a constructive alternative to previous non-constructive proofs of its existence.

Problem

• decidability
• pp-bi-interpretability
• Constraint Satisfaction Problems
• polymorphism clones
• descriptive set theory

Innovation

• decidability
• pp-bi-interpretability
• polymorphism clones
• smooth equivalence relation
• model-complete core