A Dichotomy Theorem for Automatic Structures

The field of constraint satisfaction problems (CSPs) studies homomorphism problems between relational structures where the target structure is fixed. Classifying the complexity of these problems has been a central quest of the field, notably when both sides are finite structures. In this paper, we study the generalization where the input is an automatic structure – potentially infinite, but describable by finite automata. We prove a striking dichotomy: homomorphism problems over automatic structures are either decidable in non-deterministic logarithmic space (NL), or undecidable. We show that structures for which the problem is decidable are exactly those with finite duality, which is a classical property of target structures asserting that the existence of a homomorphism into them can be characterized by the absence of a finite set of obstructions in the source structure. Notably, this property precisely characterizes target structures whose homomorphism problem is definable in first-order logic, which is well-known to be decidable over automatic structures. We also consider a natural variant of the problem. While automatic structures are finitely describable, homomorphisms from them into finite structures need not be. This leads to the notion of regular homomorphism, where the homomorphism itself must be describable by finite automata. Remarkably, we prove that this variant exhibits the same dichotomy, with the same characterization for decidability.

💡 Research Summary

The paper investigates the computational complexity of constraint satisfaction problems (CSPs) when the source structure is an automatic structure—potentially infinite but finitely representable by synchronous finite automata—while the target structure B is a fixed finite relational structure. The authors establish a sharp dichotomy: for any such B, the homomorphism problem Hom(Aut, B) (does there exist a homomorphism from a given automatic structure A to B?) and its regular variant Hom_reg(Aut, B) (does there exist a homomorphism that itself can be described by a finite automaton?) are either decidable in nondeterministic logarithmic space (NL) or undecidable.

The decisive property of B is finite duality: B has finite duality if there exists a finite set of finite “obstructions” such that a structure maps homomorphically into B precisely when none of these obstructions embed into the source. This classical notion is known to be equivalent to the existence of a first‑order definition of the homomorphism problem. The authors prove that finite duality is exactly the condition that makes both Hom(Aut, B) and Hom_reg(Aut, B) NL‑decidable.

When B enjoys finite duality, the authors give two complementary algorithmic approaches. The first is a logical reduction: because every automatic structure admits an NL‑complete model‑checking procedure for existential‑positive first‑order formulas, the existence of a homomorphism can be expressed as such a formula and decided in NL. The second, more constructive method, adapts the hyper‑edge consistency algorithm (originally for finite structures) to the automatic setting. They show that for structures with finite duality the algorithm terminates after a bounded number of steps, yielding an NL decision procedure. The same algorithm works for regular homomorphisms because the regularity constraint can be enforced by synchronously checking the automaton that would describe the homomorphism.

If B lacks finite duality, the paper proves undecidability for both variants. For the ordinary homomorphism problem, the authors adapt the L‑hardness proof of Larose and Tesson (originally for finite structures) and combine it with the known undecidability of reachability in automatic graphs. They construct, from an arbitrary automatic graph, a target structure B without finite duality such that a homomorphism exists exactly when a certain reachability condition holds, thereby transferring the undecidability.

The regular homomorphism case is more intricate. Starting from the regular reachability problem for linear Turing machines (known Σ₁⁰‑complete), they reduce it to a problem called regular unconnectivity in automatic graphs, and then, using a refined version of the Larose‑Tesson reduction, to Hom_reg(Aut, B) for a B without finite duality. Throughout, the regularity requirement is preserved by encoding the Turing‑machine computation into the synchronous automata that would describe the homomorphism. Consequently, Hom_reg(Aut, B) is also undecidable when B does not have finite duality.

The main theorem (Theorem 3.1) lists five equivalent conditions for a finite target B: (i) B has finite duality; (ii) Hom(Aut, B) is decidable; (iii) Hom_reg(Aut, B) is decidable; (iv) the two problems always have the same answer; (v) homomorphisms from any automatic structure to B are uniformly first‑order definable. Moreover, when the conditions fail, both problems are recursively enumerable‑complete, while when they hold they are NL‑complete.

The paper also surveys related work on automatic structures, regular homomorphisms, and homomorphism problems with infinite targets, and discusses extensions to broader classes of definable structures, conjectures about the boundary between decidable and undecidable cases, and connections to automatic isomorphisms.

In summary, the authors provide a clean, robust dichotomy for CSPs over automatic structures: finite duality of the target is the exact line separating NL‑tractable homomorphism problems from undecidable ones, and this line remains unchanged even when the homomorphism itself must be regular. This result deepens our understanding of the interplay between logical definability, automata‑based representations, and computational complexity in infinite‑but‑finitely‑presentable domains.