The wonderland of reflections

TL;DR

This paper extends the algebraic framework for classifying the complexity of constraint satisfaction problems (CSPs) over -categorical structures by introducing new algebraic and syntactic tools, leading to a refined dichotomy conjecture.

Contribution

It introduces the concept of reflection and h1 clone homomorphisms, broadening the algebraic approach to CSPs and proposing a new dichotomy conjecture for reducts of finitely bounded homogeneous structures.

Findings

Complexity depends on identities of height 1 in polymorphism clones.

Introduces reflection as a new semantic construction.

Establishes a connection between h1 clone homomorphisms and lattice interpretability.

Abstract

A fundamental fact for the algebraic theory of constraint satisfaction problems (CSPs) over a fixed template is that pp-interpretations between at most countable \omega-categorical relational structures have two algebraic counterparts for their polymorphism clones: a semantic one via the standard algebraic operators H, S, P, and a syntactic one via clone homomorphisms (capturing identities). We provide a similar characterization which incorporates all relational constructions relevant for CSPs, that is, homomorphic equivalence and adding singletons to cores in addition to pp-interpretations. For the semantic part we introduce a new construction, called reflection, and for the syntactic part we find an appropriate weakening of clone homomorphisms, called h1 clone homomorphisms (capturing identities of height 1). As a consequence, the complexity of the CSP of an at most countable $\omega$-categorical structure depends only on the identities of height 1 satisfied in its polymorphism clone as well as the the natural uniformity thereon. This allows us in turn to formulate a new elegant dichotomy conjecture for the CSPs of reducts of finitely bounded homogeneous structures. Finally, we reveal a close connection between h1 clone homomorphisms and the notion of compatibility with projections used in the study of the lattice of interpretability types of varieties.