2011 seminar talk: Positive Horn definability in \aleph_0-categorical structures
Talk held by Moritz Müller (KGRC)
at the KGRC seminar on 2011-10-27.
We show that in an \(\aleph_0\)-categorical structure \(A\) a relation is positive
Horn definable if and only if it is preserved by the surjective
periomorphisms of \(A\). These are homomorphisms from the periodic power of \(A\)
into \(A\), and the periodic power of \(A\) is the structure induced on all
periodic sequences in \(A^\omega\).
It follows that the complexity (up to polynomial time reducibility) of the
problem to decide the positive Horn theory of some \(\aleph_0\)-categorical
structure is determined by the structures set of surjective
Such problems are known as quantified constraint satisfaction problems and
have been studied in depth for finite structures - there a preservation
theorem as above has been established via surjective polymorphisms.
The talk gives some background information concerning the so-called
algebraic approach to constraint complexity which is based on such
This is joint work with Hubie Chen.