By Benedikt Bollig

This publication experiences the connection among automata and monadic second-order common sense, targeting periods of automata that describe the concurrent habit of disbursed structures. It offers a unifying conception of speaking automata and their logical houses. in keeping with Hanf's Theorem and Thomas's graph acceptors, it develops a outcome that permits characterization of many well known types of disbursed computation by way of the existential fragment of monadic second-order logic.

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30. 22). This fact is witnessed by the set Ln of n-supergrids for some n ≥ 4 (cf. [92]). From a grid, we obtain the corresponding n-supergrid if any edge is replaced by a sequence of n new edges. Such a sequence is called a superedge. Relative to the set of graphs over (−, {1, 2}), Ln can be recognized by some graph acceptor equipped with 2n-spheres. But now suppose there is a graph acceptor B with radius 1 such that LDG(−,{1,2}) (B) = Ln and consider ρ to be an accepting run of B on some G ∈ Ln .

Part (c) illustrates a Σ-dag that is not a ﬁfo-dag. Finally, the remaining dag, from Fig. 3d, is a lo-dag over Σ but not a Σ-dag. 8. Any class K ⊆ DAG(Σ, C) has bounded degree. 9. Depending on Σ and C, determine the lowest natural number B such that the degree of DAG(Σ, C) is bounded by B. Is DAGlo (Σ, C) bounded, too? 1 (Σ, 47 Given (V, { } ∈C , λ) ∈ DAGlo (Σ, C), let us introduce the following abbreviations: For i ∈ Ag, Vi shall denote the set of nodes u ∈ V such that λ(u) ∈ Σi . Accordingly, given a ∈ Σ, Va is the set of nodes u ∈ V such that λ(u) = a.

5b. a a a b a b a (a) (b) Fig. 5. 6 Pictures and Grids An important class of graphs is provided by pictures. Many results on pictures will be used to achieve the corresponding results in the framework of dags and message sequence charts. Once more, pictures are a special case of graphs. However, while the node labeling is arbitrary, an edge of a picture is labeled with either 1 or 2. So let Σ be an alphabet in the following and, given n ∈ IN≥1 , let [n] denote the set {1, . . , n}. 26 (Picture).