On Infinite–finite Duality Pairs of Directed Graphs

Abstract

The (A, D) duality pairs play crucial role in the theory of general relational structures and in the Constraint Satisfaction Problem. The case where both classes are finite is fully characterized. The case when both side are infinite seems to be very complex. It is also known that no finite-infinite duality pair is possible if we make the additional restriction that both classes are antichains. In this paper (which is the first one of a series) we start the detailed study of the infinite-finite case.Here we concentrate on directed graphs. We prove some elementary properties of the infinite-finite duality pairs, including lower and upper bounds on the size of D, and show that the elements of A must be equivalent to forests if A is an antichain. Then we construct instructive examples, where the elements of A are paths or trees. Note that the existence of infinite-finite antichain dualities was not previously known.

Regular families of forests, antichains and duality pairs of relational structures

“…Given a set O ⊆ F, we can define a graph G whose vertex-set is F r and whose edges join pairs (A, a), In [6] and [7], the natural description of paths and caterpillars in terms of words over an alphabet allowed a more direct correspondence between regular languages and regular families of paths or caterpillars. Definition 3.1 is in the spirit of the Myhill-Nerode theorem, which states that a language L over an alphabet Σ is regular if and only if the infinitely many words a ∈ Σ * define only finitely many distinct extension sets L − a = {b ∈ Σ * | ab ∈ L}.…”

“…In the present paper we extend the context of [6] from digraphs to general relational structures, and the context of [7] from caterpillar dualities to general forest dualities. The criterion of regularity remains relevant, but in the context of forest dualities it is necessary to generalize it.…”

Universal Structures with Forbidden Homomorphisms

“…In special cases we relate lifts to the concept of homomorphism dualities. Motivated by a recent characterization of infinite-finite dualities by P. L. Erdős, Pálvölgyi, Tardif, Tardos [8], we introduce a notion of regular families of relational structures. These (possibly infinite) families of structures generalize regular forests, used in [8] to characterize infinitefinite dualities.…”

“…Motivated by a recent characterization of infinite-finite dualities by P. L. Erdős, Pálvölgyi, Tardif, Tardos [8], we introduce a notion of regular families of relational structures. These (possibly infinite) families of structures generalize regular forests, used in [8] to characterize infinitefinite dualities. In fact the regular families of trees (and forests) corresponds to well established notion of recognizable tree languages, see [5].…”

Caterpillar Dualities and Regular Languages

“…in a σ-structure B recursively, by a repeated application of the rules that apply, until a certain "goal" is achieved. Note that all the rules can be rewritten in terms of the type σ 2 : The rule 1 can be written (4) a ∈ ρ i ← b ∈ ρ j and (b, a) ∈ R (n,m) .…”