Lindenmayer graph languages, first-order theories and expanders

Speaker: Teodor Knapik, Université de la Nouvelle-Calédonie, Nouméa

Tuesday 14 May 2024, 14:00, Room 1Z56

Abstract: Imagined by Kolmogorov in the middle of past century, expanders form remarkable graph families with applications in areas as diverse as robust communication networks and probabilistically checkable proofs, to name just two. Since the proof of the existence of expanders, it took several years to come up with an explicit algebraic construction [Margulis 1973] of some expander families. Their first elementary (combinatorial) construction has been published in 2002 and awarded Gödel Prize in 2009.

In this talk, we introduce a framework that captures most of the known combinatorial constructions of expanders. It is based on a generalisation of Lindenmayer systems to the domain of graphs. We call this formalism Lindenmayer graph grammars. We identify a few essential properties which make decidable the language checking problem with respect to first-order sentences. This result is obtained by encompassing a graph language into an automatic structure. By language checking in this specific context, we mean the following problem.

Instance: a Lindenmayer graph grammar and a first-order sentence. Question: Does there exist a graph in the language for which the sentence holds?