Read e-book online Formal Models of Communicating Systems. Languages, Automata, PDF

By Benedikt Bollig

ISBN-10: 3540329226

ISBN-13: 9783540329220

This publication reports the connection among automata and monadic second-order common sense, concentrating on periods of automata that describe the concurrent habit of disbursed structures. It offers a unifying thought of speaking automata and their logical homes. in response to Hanf's Theorem and Thomas's graph acceptors, it develops a end result that enables characterization of many renowned versions of dispensed computation when it comes to the existential fragment of monadic second-order logic.

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So let i ∈ {1, . . , l} and j ∈ {1, . . , li } and suppose G ∈ Lij . , we may choose an arbitrary G ∈ Lij . ,li } Clearly, we have L = {G ∈ K | G |= α}, which concludes the proof. 4 Graph Acceptors 27 The next step towards a logically founded automata theory over graphs is to establish a connection between EMSO-definable and locally threshold testable languages. So let us introduce some further notions and let n ≥ 1. Given an extended graph G = (V, { } ∈C , λ) ∈ DG(Σ × {0, 1}n , C) over (Σ, C), the projection of G is defined to be h(G) := (V, { } ∈C , λ ) ∈ DG(Σ, C) where, for any u ∈ V , we have λ (u) = a if λ(u) = (a, (b1 , .

26 3 Graphs, Logics, and Graph Acceptors An occurrence formula α ∈ Cond (O, t) is canonically interpreted over a mapping m : O → IN. Precisely, • • • • m |= o ≥ n m |= ¬α m |= α ∨ β m |= α ∧ β if if if if m(o) ≥ n, not m |= α, m |= α or m |= β, and m |= α and m |= β. Let S be a set of R-spheres over (Σ, C). An occurrence formula from Cond (Σ, t) is satisfied by the graph G ∈ DG(Σ, C), written G |= α, if α is satisfied by the mapping that assigns |G|a to any a ∈ Σ. Accordingly, an occurrence formula α ∈ Cond (S, t) is satisfied by the graph G if it is satisfied by the mapping that assigns to each R-sphere H ∈ S the natural number |G|H .

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.

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Formal Models of Communicating Systems. Languages, Automata, and Monadic Second-order Logic by Benedikt Bollig


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