Decidability of Interpretability

TL;DR

This paper investigates the decidability and complexity of the equivalence relation of pp-bi-interpretability in the context of the Bodirsky-Pinsker conjecture, linking algebraic and computational aspects of CSPs.

Contribution

It proves the decidability of pp-bi-interpretability under mild conditions and shows that, for a broad class of structures, this relation is of minimal complexity (smooth), advancing understanding of the algebraic approach to CSP complexity.

Findings

Decidability of pp-bi-interpretability under mild conditions.

The equivalence relation is smooth (Borel-reducible to equality on reals) for certain structures.

Constructive proof of the computability of the model-complete core of structures with finitely bounded Ramsey expansions.

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.