8 research outputs found
An Algebraic Preservation Theorem for Aleph-Zero Categorical Quantified Constraint Satisfaction
We prove an algebraic preservation theorem for positive Horn definability in
aleph-zero categorical structures. In particular, we define and study a
construction which we call the periodic power of a structure, and define a
periomorphism of a structure to be a homomorphism from the periodic power of
the structure to the structure itself. Our preservation theorem states that,
over an aleph-zero categorical structure, a relation is positive Horn definable
if and only if it is preserved by all periomorphisms of the structure. We give
applications of this theorem, including a new proof of the known complexity
classification of quantified constraint satisfaction on equality templates
The complexity of positive first-order logic without equality
We study the complexity of evaluating positive equality-free sentences of first-order (FO) logic over a fixed, finite structure B. This may be seen as a natural generalisation of the nonuniform quantified constraint satisfaction problem QCSP(B). We introduce surjective hyper-endomorphisms and use them in proving a Galois connection that characterizes definability in positive equality-free FO. Through an algebraic method, we derive a complete complexity classification for our problems as B ranges over structures of size at most three. Specifically, each problem either is in L, is NP-complete, is co-NP-complete, or is Pspace-complete
The complexity of positive first-order logic without equality
We study the complexity of evaluating positive equality-free sentences of first-order (FO) logic over a fixed, finite structure B. This may be seen as a natural generalisation of the nonuniform quantified constraint satisfaction problem QCSP(B). We introduce surjective hyper-endomorphisms and use them in proving a Galois connection that characterizes definability in positive equality-free FO. Through an algebraic method, we derive a complete complexity classification for our problems as B ranges over structures of size at most three. Specifically, each problem either is in L, is NP-complete, is co-NP-complete, or is Pspace-complete
The complexity of positive first-order logic without equality
We study the complexity of evaluating positive equality-free sentences of first-order (FO) logic over a fixed, finite structure B. This may be seen as a natural generalisation of the nonuniform quantified constraint satisfaction problem QCSP(B). We introduce surjective hyper-endomorphisms and use them in proving a Galois connection that characterizes definability in positive equality-free FO. Through an algebraic method, we derive a complete complexity classification for our problems as B ranges over structures of size at most three. Specifically, each problem either is in L, is NP-complete, is co-NP-complete, or is Pspace-complete
The complexity of positive first-order logic without equality II: the four-element case
We study the complexity of evaluating positive equality-free sentences of first-order logic over fixed, finite structures B . This may be seen as a natural generalisation of the non-uniform quantified constraint satisfaction problem QCSP (B) . Extending the algebraic methods of a previous paper, we derive a complete complexity classification for these problems as B ranges over structures of domain size 4. Specifically, each problem is either in L, is NP-complete, is co-NP-complete or is Pspace-complete
The complexity of positive first-order logic without equality II: the four-element case
We study the complexity of evaluating positive equality-free sentences of first-order logic over fixed, finite structures B . This may be seen as a natural generalisation of the non-uniform quantified constraint satisfaction problem QCSP (B) . Extending the algebraic methods of a previous paper, we derive a complete complexity classification for these problems as B ranges over structures of domain size 4. Specifically, each problem is either in L, is NP-complete, is co-NP-complete or is Pspace-complete