Path querying using conjunctive grammars

Graphs are used as a data structure to represent large volumes of information in a compact and convenient for analysis form in many areas: bioinformatics, graph databases, static code analysis, etc. In these areas, it is necessary to evaluate a queries for large graphs in order to determine the dependencies between the nodes. The answer to such queries is usually a set of all triples (A, $m$, $n$) for which there is a path in the graph from the vertex $m$ to the vertex $n$, such that the labels on the edges of this path form a string derivable from the nonterminal A in some context-free grammar. This type of query is calculated using relational query semantics and context-free grammars but there are conjunctive grammars, which form a broader class of grammars than traditional context-free grammars. The use of conjunctive grammars in the path querying allows us to formulate more complex queries to the graph and solve a wider class of problems. It is known that the path querying using relational query semantics and conjunctive grammars is undecidable problem. In this paper, we propose a path querying algorithm which uses relational query semantics and conjunctive grammars, and approximates the exact solution. The proposed algorithm is based on matrix operations, and our evaluation shows that it is possible to significantly improve the performance of the algorithm by using a graphics processing unit.