2010 | OriginalPaper | Buchkapitel
The Lattice Structure of Sets of Surjective Hyper-Operations
verfasst von : Barnaby Martin
Erschienen in: Principles and Practice of Constraint Programming – CP 2010
Verlag: Springer Berlin Heidelberg
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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)
$\mathcal{B}$
, the set of relations definable over
$\mathcal{B}$
in positive fo logic without equality consists of exactly those relations that are invariant under the surjective hyper-endomorphisms (shes) of
$\mathcal{B}$
. 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
$\mathcal{B}$
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.