Constraint Satisfaction Problems over Finitely Bounded Homogeneous Structures: a Dichotomy between FO and L-hard
Leonid Dorochko, Michał Wrona
TL;DR
This work tackles the Bodirsky-Pinsker program by proving a general dichotomy for CSPs over first-order expansions of finitely bounded homogeneous model-complete cores: each such CSP is either first-order definable (and hence in non-uniform $AC^0$) or $L$-hard under first-order reductions. The authors introduce a two-phase strategy, first reproving the finite Larose–Tesson dichotomy with a new argument and then extending it to infinite structures via $k$-homogeneous and $\ell$-bounded cores, balanced and stable implications, and a tree $\mathbb A$-formula framework. Central to the approach are notions of finite duality, $\mathbb A$-definitions, and a $(k,\max(k,l))$-minimality procedure that either yields a finite obstruction set or an implication that encodes a reduction from Graph Unreachability, yielding $L$-hardness. The result is the most general dichotomy known for the class of structures within Bodirsky-Pinsker, with concrete examples including expansions of $(\mathbb{Q};<)$ and equality-free, unbalanced cases. Together, these insights illuminate when infinite-domain CSPs can be solved by local-consistency methods or are inherently as hard as $L$-computations, and pave the way for identifying NL vs. L boundaries in further work.
Abstract
Feder-Vardi conjecture, which proposed that every finite-domain Constraint Satisfaction Problem (CSP) is either in P or it is NP-complete, has been solved independently by Bulatov and Zhuk almost ten years ago. Bodirsky-Pinsker conjecture which states a similar dichotomy for countably infinite first-order reducts of finitely bounded homogeneous structures is wide open. In this paper, we prove that CSPs over first-order expansions of finitely bounded homogeneous model-complete cores are either first-order definable (and hence in non-uniform AC$^0$) or L-hard under first-order reduction. It is arguably the most general complexity dichotomy when it comes to the scope of structures within Bodirsky-Pinsker conjecture. Our strategy is that we first give a new proof of Larose-Tesson theorem, which provides a similar dichotomy over finite structures, and then generalize that new proof to infinite structures.