Homomorphism Counts, Database Queries, and Modal Logics
Jesse A. Comer
Abstract:
We study applications of restrictions of homomorphism vectors for finite relational models in database theory and modal logic. Assume some fixed enumeration (Mi )i∈ω of all finite relational models (up to isomorphism), and let M be some finite relational model. The left homomorphism vector of M is the countably infinite vector whose ith entry is the number of homomorphisms from Mi to M . Similarly, the right homomorphism vector of M is the countably infinite vector whose ith entry is the number of homomorphisms from M to Mi . A restriction of a homomorphism vector is a vector obtained by removing some of the entries.
In particular, we study (right) finite characterizations, which are restrictions of the right homomorphism vector of a model M to a finite number of entries which characterize M up to isomorphism. Interpreting the characterized model as a canonical model of a conjunctive query, a finite characterization can equivalently be seen as a collection of database instances, where the answers to the query in each instance of the collection under the bag semantics determine the query up to isomorphism. Given an arbitrary finite model M , we construct a finite characterization of M containing at most exponentially-many examples, each of which has domain size bounded by the domain size of M . We also construct polynomially-large finite characterizations for certain special classes of models.
Additionally, we study which relations preserving different modal languages can be captured by restricting the left homomorphism vector of labeled transition systems. In this vein, we show that simulation and graded bisimulation are captured by the restriction of the left homomorphism vector to the class of directed tree-shaped labeled transition systems under the Boolean and counting semantics, respectively. We then lift these results to show that modal languages with backward and global modalities are captured by the restriction of the left homomorphism vector to appropriate classes of tree-shaped and directed forest models, respectively. We also show that no relation finer than directed simulation and coarser than bisimulation can be captured by a restriction of the left homomorphism vector under the Boolean or the counting semantics.