On Families of Graphs Having a Decidable First Order Theory with Reachability
We consider a new class of infinite graphs defined as the smallest solution of equational systems with vertex replacement operators and unsynchronised product. We show that those graphs have an equivalent internal representation as graphs of recognizable ground term rewriting systems. Furthermore, we show that, when restricted to bounded tree-width, those graphs are isomorphic to hyperedge replacement equational graphs. Finally, we prove that on a wider family of graphs — interpretations of trees having a decidable monadic theory — the first order theory with reachability is decidable.
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