Technology Category
Knowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesCognitive Modeling & Cognitive Systems: Symbolic Representations
Uniform Membership for Hyperedge Replacement Grammars and Related Decision Problems
🤖 AI Summary
This paper investigates the computational complexity of the uniform membership problem for hyperedge replacement grammars (HRGs), distinguishing two hypergraph semantics: attachment nodes allowed to repeat versus disallowed to repeat. Using carefully constructed polynomial-time reductions and deterministic Turing machine simulation techniques, we establish that the problem is EXPTIME-complete under the repeat-allowed semantics and NP-complete under the repeat-forbidden semantics. Furthermore, we prove that deciding HRG string generability is also EXPTIME-complete. This work provides the first precise complexity classification of the HRG uniform membership problem, revealing how subtle semantic differences—specifically, the treatment of node attachments—fundamentally affect computational hardness. By resolving long-standing ambiguities tied to semantic definitions, our results establish a key theoretical boundary for lightweight grammar-based modeling of context-sensitive languages.
Technology Category
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
Hyperedge Replacement Grammar
Membership Problem
Complexity Classification
Innovation
Methods, ideas, or system contributions that make the work stand out.
Hyperedge Replacement Grammars
Uniform Membership Problem
Complexity Analysis
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