The lattice structure of sets of surjective hyper-operations

Conference paper

Martin, B. 2010. The lattice structure of sets of surjective hyper-operations. The 16th International Conference on Principles and Practice of Constraint Programming (CP 2010). St Andrews, Scotland 06 - 10 Sep 2010 Springer Verlag. https://doi.org/10.1007/978-3-642-15396-9_31
TypeConference paper
TitleThe lattice structure of sets of surjective hyper-operations
AuthorsMartin, B.
Abstract

[NB some mathematical symbols may have been omitted from this abstract due to software limitations]. 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 under inverse. These problems turn out either to be in L or to be Pspace-complete.
We go on to apply our results to certain digraphs. We prove that the evaluation of positive fo without equality on a semicomplete digraph is always Pspace-complete. We go on to prove that this problem is NP-hard for any graph of diameter at least 3. Finally, we prove a tetrachotomy for antireflexive and reflexive graphs, modulo a known conjecture as to the complexity of the QCSP on connected non-bipartite graphs. Specifically, these problems are either in L, NP-complete, co-NP-complete or Pspace-complete.

Research GroupFoundations of Computing group
ConferenceThe 16th International Conference on Principles and Practice of Constraint Programming (CP 2010)
SeriesLecture Notes in Computer Science
ISBN
Hardcover9783642153952
PublisherSpringer Verlag
Publication dates
Print2010
Publication process dates
Deposited01 Feb 2013
Output statusPublished
Additional information

Online ISBN: 9783642153969

Digital Object Identifier (DOI)https://doi.org/10.1007/978-3-642-15396-9_31
LanguageEnglish
Book titlePrinciples and practice of constraint programming – CP 2010
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