Uniform Membership for Hyperedge Replacement Grammars and Related Decision Problems

Abstract

This paper investigates complexity of the uniform membership problem for hyperedge replacement grammars in comparison with other mildly context-sensitive grammar formalisms. It turns out that the complexity of this problem depends on how one defines a hypergraph. There are two commonly used definitions in the field, which differ in whether repetitions of attachment nodes of a hyperedge are allowed in a hypergraph or not. We show that, in general, the problem under consideration is EXPTIME-complete, even for string-generating hyperedge replacement grammars, but it is NP-complete if repetitions are not allowed. We extend the developed proof techniques in order to prove a general meta-theorem: checking whether a given hyperedge replacement grammar generates a hypergraph satisfying a non-Parikh property is EXPTIME-hard. Non-Parikh properties are those that are not preimages of properties on Parikh vectors of hypergraphs. This includes any graph property relying significantly on structure of graphs, e.g. connectivity, Eulerianity, Hamiltonianity, acyclicity. A tight upper bound is established for EXPTIME-compatible properties via Filter Theorem.