By Benedikt Bollig

This booklet experiences the connection among automata and monadic second-order good judgment, targeting sessions of automata that describe the concurrent habit of dispensed structures. It offers a unifying idea of speaking automata and their logical homes. according to Hanf's Theorem and Thomas's graph acceptors, it develops a end result that permits characterization of many well known types of allotted computation by way of the existential fragment of monadic second-order logic.

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**Extra info for Formal Models of Communicating Systems. Languages, Automata, and Monadic Second-order Logic**

**Example text**

Xm , xm+1 ) ∈ FOk (Σ, C). By the induction hypothesis, we have that {L DG(Σ,C),{0,1}m+1,0 (ψ ) | ψ (x1 , . . , xm , xm+1 ) ∈ FOk (Σ, C)} is ﬁnite and, thus, so is {L DG(Σ,C),{0,1}m,0 (ϕ) | ϕ(x1 , . . , xm ) ∈ FOk+1 (Σ, C)}. 11. Let L ⊆ DG(Σ, C). We have L ∈ FO(Σ, C)DG iﬀ there is some k ∈ IN such that L is the union of ≡k -equivalence classes. 12. 11. 13 (Threshold Equivalence). Let R, t ∈ IN. Given graphs G1 , G2 ∈ DG(Σ, C), we write G1 R,t G2 if, for any R-sphere H over (Σ, C), either • |G1 |H = |G2 |H or • both |G1 |H ≥ t and |G2 |H ≥ t.

Describe each of the following word languages over Σ = {a, b} by an MSO(Σ, −)-sentence relative to W(Σ): (a) {awa | w ∈ W(Σ)}, (b) {w ∈ W(Σ) | |w |b > 3}, (c) {w ∈ W(Σ) | |w | is even}. 2 Finite Automata We now recall a well-known automata model, which is tailored to words. 5 (Finite Automaton). A ﬁnite automaton over Σ is a structure (S, ∆, sin , F ) where • • • • S is its nonempty ﬁnite set of states, ∆ ⊆ S × Σ × S is the set of transitions, sin ∈ S is the initial state, and F ⊆ S is the set of ﬁnal states.

The dag from Fig. 4b is clearly not an M+ -trace. Apart from the edge labelings, it lacks the mandatory edge between the a and the b-labeled node. However, the latter dag is just a diﬀerent view of the former. Its edge relation allows us to access the history of a node only with respect to (cf. 11). 11 (Mazurkiewicz Traces (2)). An M− -trace over Σ is a v imstructure (V, , λ) ∈ DAGH (Σ, −) such that, for any u, v ∈ V , u plies λ(u) DΣe λ(v). Note that, as we consider a subclass of DAGH (Σ, −), and coincide.