Complexity of the Uniform Membership Problem for Hyperedge Replacement Grammars
Tikhon Pshenitsyn
TL;DR
This work analyzes the uniform membership problem for hyperedge replacement grammars (HRGs), revealing that complexity hinges on whether hyperedges may attach to repeated nodes. By connecting HRGs to transformation monoids and rational subset membership, the authors establish an $EXPTIME$ upper bound for general HRGs and an $NP$ upper bound for repetition-free HRGs, while proving $EXPTIME$-hardness for string-generating HRGs and $EXPTIME$-completeness for determining string-generating status; repetition-free string-generating HRGs are $NP$-hard for uniform membership. The results situate HRGs among mildly context-sensitive formalisms and offer insight into how hypergraph definitions and empty/chain productions shape parsing complexity. The findings have implications for fusion grammars and the broader landscape of uniform versus non-uniform membership, and they identify open questions about the existence of a polynomially tractable formalism for certain NC languages.
Abstract
We investigate 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 the problem considered depends heavily 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, if repetitions are allowed, then the problem under consideration is EXPTIME-complete even for string-generating hyperedge replacement grammars while it is NP-complete if repetitions are disallowed. We also prove that checking whether a hyperedge replacement grammar is string-generating is EXPTIME-complete.