Dynamic Causality in Event Structures (Technical Report)

In [1] we present an extension of Prime Event Structures by a mechanism to express dynamicity in the causal relation. More precisely we add the possibility that the occurrence of an event can add or remove causal dependencies between events and analyse the expressive power of the resulting Event Structures w.r.t. to some well-known Event Structures from the literature. This technical report contains some additional information and the missing proofs of [1]. 1 Event Structures for Resolvable Conflict For a transition based ES with a few additional properties, there is a natural embedding into RCESs. Definition 20. Let μ be an ES with a transition relation → defined on configurations such that X → Y implies X ⊆ Y and X ⊆ X ′ ⊆ Y ′ ⊆ Y implies that X → Y =⇒ X ′ → Y ′ for all configurations X,Y,X , Y ′ of μ. Then rces(μ) = (E, {X ⊢ Z | ∃Y ⊆ E . X→Y ∧ Z ⊆ Y }). Note that the SESs and GESs satisfy these properties. We show that the resulting structure rces(μ) is indeed an RCES and that it is transition equivalent to μ. Lemma 2. Let μ be an ES that satisfies the conditions of Def. 20. Then rces(μ) is a RCES and rces(μ)≃tμ. Proof. By Def. 9 in [1], rces(μ) is a RCES. Assume X→Y . Then, by Def. 20, X ⊆ Y and X ⊢ Z for all Z ⊆ Y .Then, by Def. 10 in [1], X→rcY . Assume X→rcY . Then, by Def. 10 in [1], X ⊆ Y and there is some X ′ ⊆ X such that X ′ ⊢ Y . By Def. 20 for rces(·), then X ′ ⊢ Y ′ for all Y ′ ⊆ Y . So there is a set Ỹ such that Y ⊆ Ỹ and X ′ ⊢ Ỹ ′ for each Ỹ ′ ⊆ Ỹ . Then, by Def. 20 for rces(·), it follows X ′ → Ỹ ′ and X ′ ⊆ X ⊆ Y ⊆ Ỹ . Finally, by the second property of Def. 20, X→Y . ⋆ Supported by the DFG Research Training Group SOAMED. 2 Youssef Arbach, David Karcher, Kirstin Peters, Uwe Nestmann