By Benedikt Bollig

This booklet reports the connection among automata and monadic second-order good judgment, targeting sessions of automata that describe the concurrent habit of disbursed platforms. It presents a unifying conception of speaking automata and their logical houses. in accordance with Hanf's Theorem and Thomas's graph acceptors, it develops a consequence that permits characterization of many well known versions of disbursed computation when it comes to the existential fragment of monadic second-order logic.

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Note that a word over Σ is just a graph over Σ and a singleton, which is, to some extent, extraneous. Actually, W(Σ) is simply DAGH (Σ, −) restricted to graphs (V, , λ) such that ∗ forms a total order. Recall that we do not distinguish isomorphic structures. We can therefore identify a word ({1, . . , n}, , λ) ∈ W with the sequence a1 . . an ∈ Σ ∗ where ai = λ(i) for each i ∈ {1, . . , n}. For example, ({1, 2, 3}, {(1, 2), (2, 3)}, {1 → a, 2 → b, 3 → a}) ∈ W({a, b}) is equated with the string aba ∈ {a, b}∗ .

N}}, and the ui are labeled by λn with a, whereas the vi are labeled with b. For example, Fig. 7a illustrates D4 . An ACAT A over Σ with L(A) = L is (Q, ∆, T, F ) where Q = {q1 , q2 } and ∆ contains the following transitions (recall that there are no edge labelings): ∅ −→ (b, q2 ) {(b, q2 )} −→ (b, q2 ) {(b, q2 )} −→ (a, q1 ) {(a, q1 ), (b, q2 )} −→ (a, q1 ) Moreover, we set T (a, q1 ) T (b, q1 ) T (a, q2 ) T (b, q2 ) =∅ =∅ =∅ = {a} and F = {(q1 , q2 )}. The transitions and the type function of A can be depicted as shown in Fig.

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