Defining relations on graphs: how hard is it in the presence of node partitions?

Designing query languages for graph structured data is an active field of research. Evaluating a query on a graph results in a relation on the set of its nodes. In other words, a query is a mechanism for defining relations on a graph. Some relations may not be definable by any query in a given language. This leads to the following question: given a graph, a query language and a relation on the graph, does there exist a query in the language that defines the relation? This is called the definability problem. When the given query language is standard regular expressions, the definability problem is known to be PSPACE-complete. The model of graphs can be extended by labeling nodes with values from an infinite domain. These labels induce a partition on the set of nodes: two nodes are equivalent if they are labeled by the same value. Query languages can also be extended to make use of this equivalence. Two such extensions are Regular Expressions with Memory (REM) and Regular Expressions with Equality (REE). In this paper, we study the complexity of the definability problem in this extended model when the query language is either REM or REE. We show that the definability problem is EXPSPACE-complete when the query language is REM, and it is PSPACE-complete when the query language is REE. In addition, when the query language is a union of conjunctive queries based on REM or REE, we show coNP-completeness.