Decidability of Interpretability

TL;DR

The paper addresses decidability and classification of interpretability notions for CSP templates arising from ω-categorical, finitely bounded homogeneous structures. It develops a dual descriptive-set-theoretic and computational framework around pp-bi-interpretability, bi-definability, and model-complete cores, proving decidability and smoothness under Ramsey and no-algebraicity assumptions and providing a constructive method to compute model-complete cores. It then shows how bi-definability lifts to bi-interpretability (via group and endomorphism-moniod analyses) and extends these ideas to a broader interpretability program, including answering a question of Bodirsky and Junker in the no-algebraicity regime. The results justify the “topological clone isomorphism” and pp-bi-interpretability perspective as a reasonable and tractable foundation for a general complexity dichotomy framework for infinite-domain CSPs. Overall, the work links computability, model theory, and descriptive set theory to advance the structured understanding of CSPs in the Bodirsky–Pinsker program and provides constructive tools for obtaining model-complete cores.

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 $ω$-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.

Theorems & Definitions (66)

• Conjecture 1: Bodirsky and Pinsker $\sim$2011, see BPP-projective-homomorphisms
• Theorem 1: Bodirsky, Pinsker, and Tsankov BPT-decidability-of-definability
• Example 2
• Theorem 3: Mottet and Pinsker MottetPinskerCores
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Definition 9