On Decidability and Expressive Power of Fusion Grammars

TL;DR

This paper investigates the computational complexity and expressive capabilities of fusion grammars, establishing decidability results for key problems and demonstrating that certain generated languages are semilinear.

Contribution

It proves decidability of non-emptiness and membership problems for fusion grammars, introduces bounded variants, and extends Parikh's theorem to connection-preserving fusion grammars.

Findings

Non-emptiness and membership problems are decidable in NEXPTIME.

Decidability holds for fusion grammars with bounded markers and connectors.

Languages generated by connection-preserving fusion grammars are semilinear.

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.