Algebraic Theory of Processes by Matthew Hennessy

By Matthew Hennessy

Algebraic concept of methods offers the 1st normal and systematic advent to the semantics of concurrent structures, a comparatively new learn zone in machine technological know-how.

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Wir zeigen [b] ⊆ [a]. Bezeichne x ein beliebiges Element aus [b]. Dann gilt (b, x) ∈ R. , x ∈ [a]. Also gilt [b] ⊆ [a]. Analog zeigt man [a] ⊆ [b]. Folglich haben wir [a] = [b]. (3): Angenommen, es existieren a, b ∈ A mit [a] ∩ [b] = ∅ und [a] = [b]. , es gilt (a, c) ∈ R und (b, c) ∈ R. Wegen der Symmetrie von R haben wir (c, b) ∈ R und (wegen der Transitivit¨ at von R) (a, b) ∈ R. Nach (2) gilt dann [a] = [b], im Widerspruch zur Annahme. Definition Sei A eine nichtleere Menge. Nichtleere Mengen Ai (i ∈ I) nennt man eine Zerlegung (Partition, Klasseneinteilung) der Menge A, wenn A= Ai ∧ (∀i, j ∈ I : Ai = Aj ∨ Ai ∩ Aj = ∅) i∈I gilt.

U e(tni )). u e(t) := fiA (e Seien u, u Abbildungen von V ar in A. Wir schreiben u =xk u , ur alle j = k (j, k ∈ N). falls u(xj ) = u (xj ) f¨ Es sei A := (A; (fi )i∈I , (Rj )j∈J ) eine Struktur der Signatur δ und u : V ar −→ A eine Abbildung. , u e(tmj )) ∈ RjA ; ur alle ◦ ∈ J0 ; vA,u (¬α) := ¬(vA,u (α)) und vA,u (α ◦ β) := vA,u (α) ◦ vA,u (β) f¨ v (ϕ) = 1 f¨ u r alle u mit u = xk u vA,u (∀xk ϕ) = 1 genau dann wenn A,u V (wir schreiben: vA,u (∀xk ϕ) = u , u=x u vA,u (ϕ)); k vA,u (∃xk ϕ) = 1 genau dann, wenn W es ein u mit u =xk u und vA,u (ϕ) = 1 gibt (wir schreiben: vA,u (∃xk ϕ) = u , u=x u vA,u (ϕ) ).

Geordnete Liste der Konjunktionen, in der die Konjunktionen aus der disjunktiven Normalform nach der Anzahl der auftretenden Negationszeichen geordnet sind: Nummer der Konjunktionen 1. 2. 3. 4. 5. xyz xyz xyz x yz xyz Zusammenfassungen mittels der Regel xy∨xy = x (unter Beachtung von xy = yx) sind dann nur bei Konjunktionen m¨ oglich, die zu benachbarten Gruppen“ ” in obiger Liste geh¨ oren. Man erh¨ alt dann folgende 1. , 2. 3. 5. 5. xy yz xz x y. Sortiert man nun die erhaltenen Konjunktionen nach gleichen Variablen und anschließend nach der Anzahl der auftretenden Negationszeichen, so erh¨alt man die 2.

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