Caterpillar Dualities and Regular Languages

Abstract

Abstract. We characterize obstruction sets in caterpillar dualities in terms of regular languages, and give a construction of the dual of a regular family of caterpillars. We show that these duals correspond to the constraint satisfaction problems definable by a monadic linear Datalog program with at most one EDB per rule.

On Infinite–finite Duality Pairs of Directed Graphs

“…Indeed, while the family of words {+ s (−+ s−1 ) k + | k ≥ 0} is a regular language for any s, the family {+(+−) k ) k + + | k ≥ 1} or any of its infinite subfamilies are not regular. This connection was the basis of our upcoming paper [3] that establishes regularity as necessary and sufficient condition for having a finite dual in this case. We state the following easy observation regarding regularity to further motivate this connection.…”

“…For simplicity we drop the term "oriented" when referring to (oriented) paths, (oriented) trees and (oriented) forests, these are 1 Research supported in part by the Hungarian NSF, under contract NK 78439 and K 68262 2 Research supported by grants from NSERC and ARP. 3 Research supported in part by the NSERC grant 329527 and by the Hungarian NSF grants T-046234, AT048826 and NK-62321 (directed) graphs whose underlying undirected graphs are path, trees, respectively forests in the traditional sense. In particular, paths, trees and forests have no loops and no pair of vertices is connected in both directions.…”

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 [7], it is shown that regular languages can be used to characterize the "caterpillar dualities" in general relational structures. Caterpillars are generalizations of paths, and the caterpillar dualities (O, {D}) are of interest in the constraint satisfaction community as the dualities for which O can be described in the "smallest natural recursive fragment of Datalog" (see [2]).…”

On Ramsey properties of classes with forbidden trees

“…Recall that two pieces (M, m), (M , m ) are ≈-equivalent if their incompatible sets are equal, that is, if I(M, m) = I(M , m ). By Definition 4.4, the signature τ is finite if and only if ≈ has finitely many equivalence classes on the pieces of the trees contained in F. In this case, we call F a regular class of σ-trees; the term is motivated by a connection to regular languages, highlighted in [9]. This definition of regularity coincides with the one from [16].…”