The Lattice Structure of Sets of Surjective Hyper-Operations

Abstract. We study the lattice structure of sets (monoids) of surjective hyper-operations on an n-element domain. Through a Galois connection, these monoids form the algebraic counterparts to sets of relations closed under definability in positive first-order (fo) logic without equality. Specifically, for a countable set of relations (forming the finite-domain structure) B, the set of relations definable over B in positive fo logic without equality consists of exactly those relations that are invariant under the surjective hyper-endomorphisms (shes) of B. The evaluation problem for this logic on a fixed finite structure is a close relative of the quantified constraint satisfaction problem (QCSP). We study in particular an inverse operation that specifies an automorphism of our lattice. We use our results to give a dichotomy theorem for the evaluation problem of positive fo logic without equality on structures that are she-complementative, i.e. structures B whose set of shes is closed u...

• Type: Journal
• Year: 2010
• Where: CP
• Authors: Barnaby Martin
• Keywords: Artificial Intelligence, CP 2010, Evaluation Problem, Positive Fo, Positive Fo Logic