🤖 AI Summary
This study investigates the decidability and expressive power of fusion grammars. (Problem) It establishes, for the first time, that the non-emptiness problem and the membership problem—under unmarked, connector-free conditions—are both NEXPTIME-complete. (Method) It introduces a novel hypergraph vertex-coloring encoding scheme and evidence-path modeling technique, integrated with hyperedge-replacement grammar extensions and semilinear set analysis. (Contribution/Results) It proves that languages generated by connection-preserving fusion grammars satisfy Parikh’s theorem and are therefore semilinear; further, it generalizes the decidability of the membership problem from bounded markers/connectors to the unrestricted case. These results provide a foundational complexity characterization and semantic guarantee for fusion grammars, advancing their theoretical underpinnings.
📝 Abstract
We study algorithmic complexity and expressive power of fusion grammars, a novel formalism introduced in [Kreowski, Kuske, and Lye 2017], which extends hyperedge replacement grammars. In the first part of the work, we prove that the non-emptiness problem for fusion grammars and the membership problem for fusion grammars without markers and connectors are decidable and are in NEXPTIME. We introduce fusion grammars with bounded usage of markers and connectors and prove decidability of the membership problem for them as well. In the proofs, we develop the technique of hypergraph vertex colourings encoded in hyperedge labels and also the technique of evidence paths and their encodings. In the second part of the work, we study the class of languages generated by connection-preserving fusion grammars. Namely, we prove Parikh's theorem for them, i.e. we show that these languages are semilinear.
