On the Containment of Forbidden Patterns Problems

Abstract

Forbidden patterns problems are a generalisation of (finite) constraint satisfaction problems which are definable in Feder and Vardi’s logic mmsnp [1]. In fact, they are examples of infinite constraint satisfaction problems with nice model theoretic properties introduced by Bodirsky [2]. In previous work [3], we introduced a normal form for these forbidden patterns problems which allowed us to provide an effective characterisation of when a problem is a finite or infinite constraint satisfaction problem. One of the central concepts of this normal form is that of a recolouring. In the presence of a recolouring from a forbidden patterns problem Ω₁ to another forbidden patterns problem Ω₂, containment of Ω₁ in Ω₂ follows. The converse does not hold in general and it remained open whether it did in the case of problems being given in our normal form. In this paper, we prove that this is indeed the case. We also show that the recolouring problem is $\Pi^p_2$-hard and in $\Sigma^p_3$.