Recognizable languages of arrows and cospans

In this article, we generalize Courcelle's recognizable graph languages and results on monadic second-order logic to more general structures.First, we give a category-theoretical characterization of recognizability. A recognizable subset of arrows in a category is defined via a functor into the category of relations on finite sets. This can be seen as a straightforward generalization of finite automata. We show that our notion corresponds to recognizable graph languages if we apply the theory to the category of cospans of graphs.In the second part of the paper, we introduce a simple logic that allows to quantify over the subobjects of a categorical object. Again, we show that, for the category of graphs, this logic is equally expressive as monadic second-order graph logic (msogl). Furthermore, we show that in the more general setting of hereditary pushout categories, a class of categories closely related to adhesive categories, we can recover Courcelle's result that everymsogl-expressible property is recognizable. This is done by giving an inductive translation of formulas of our logic into automaton functors.

Random graph languages

There tend to be no related researches regarding the relationships between graph theory and languages ever since the concept of graph-semigroup was first proposed in 1991. In 2011, after finding out the inner co-relations among digraphs, undirected graphs and languages, we proposed certain concepts including undirected graph language and digraph language; moreover, in 2014, we proposed a broaden concept–(V,R)-language and proved: (1) both undirected graph language and digraph language are (V,R)-languages; (2) both undirected graph language and digraph language are regular languages; (3) natural languages are regular languages. In this paper, we propose a new concept–Random Graph Language and build the relationships between random graph and language, which provides researchers with the possibility to do research about languages by using random graph theory.