On Decidability and Expressive Power of Fusion Grammars
Tikhon Pshenitsyn
TL;DR
This work studies fusion grammars, extending hyperedge replacement grammars with a fusion operation and optional markers/connectors. It establishes decidability and $NEXPTIME$ bounds for the membership and non-emptiness problems in fusion grammars without markers/connectors, and shows membership remains decidable under bounded usage of markers/connectors. The paper also generalises Parikh's theorem to connection-preserving fusion grammars, proving their generated languages have semilinear Parikh images. The methods combine vertex colourings encoded in hyperedge labels with evidence paths to manage identifications during fusion, and introduce normal forms and a linearisation framework to connect fusion grammars to context-free grammars. Overall, the results delineate the boundaries of decidability and expressive power for fusion grammars and highlight practical implications for DNA computing and hypergraph languages, while outlining key open questions for the unbounded case and broader logical characterisations.
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.