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Event structures Mauro Piccolo

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Interleaving Models Trace Languages: computation described through a non-deterministic choice between all sequential order of actions HO games: A play: a trace of computation Strategy: A set of play

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Example P = a1. a2 || b Traces Labelled Transition System

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Causal models Ordering, concurrency and conflict between actions is explicitly represented Order between action which are causally related Choice is modeled by a conflict relation Two action are concurrent if they are neither in conflict nor causally related Example: Event Structures

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Plan Event structures: Definitions A category of Event Structures Domain of configurations Event structure semantics of CCS

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EVENT STRUCTURES: DEFINITIONS

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(Prime) Event Structure: definition An event structure is a triple E = where E is a countable set of events is a partially ordered set the set [e) = {e' | e'

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Some notation parents(e): set of maximal events of [e) [e] = [e) U {e} e1 e2 is inherited if there exists e3

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Labelled Event Structures Events are occurrence of actions A labelled event structure is an event structure together with a labelling function λ : E --> L (where L is a set of labels)

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Configurations A configuration is a downward closed conflict free set of events We denote with D(E) the set of configurations of E L.T.S. of Labelled Event Structure: State: configuration Transitions:

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A CATEGORY OF EVENT STRUCTURES

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Morphisms on event structures Let and two event structures: a morphism is a map f : E1 --> E2 satifsfying f(e) = e' can be interpreted as the fact that e' is a component of the event e

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Morphism on event structures Prop: A morphism between event structures is a partial function f: E1 --> E2 such that [f(e)] ⊆ f([e]) Products and co-products are always defined

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Co-product (Sum) Let two event structure. The co-product is the event structure where and

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Product (Synchronous Parallel Composition) e1 e2 e3 E1 E2 (e1,*, ∅ )(e1,e3, ∅ )(*,e3, ∅ ) e f g (e2,*,{e}) (e2,e3,{e})(e2,*,{f}) E1 x E2

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DOMAIN OF CONFIGURATIONS

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Let a poset (we denote l.u.b. of a subset X with ⊔ X D is bounded complete if all subsets X that have an upper bound, have a ⊔ X in D D is coherent if all subsets X which are pairwise bounded have a l.u.b. ⊔ X in D A complete prime of D is an element p such that for all X that have l.u.b. we have that D is prime algebric iff for all x in D x = ⊔ {p≤x|p is complete prime} D is finitary iff for all q complete prime the set {p≤q|p is complete prime} is finite

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Prime algebric domains and Event Structures Let E an event structure then is a finitary prime algebric domain where the complete primes are the set {[e] | e in E} Let a finitary prime algebric domain and let P the set of complete primes then is an event structure where p p' if they do not have an upper bound in D The finitary prime algebric domains are precisely the dI-domains

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EVENT STRUCTURE SEMANTICS OF CCS

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Synchronization algebra A synchronization algebra is a triple where L is a set of labels that contains * . is a partial commutative associative operator with * as neutral element. Synchronization algebra of CCS L = N U N U {τ,*} for all α in N, α.α=α.α=τ and for all α, α. * = *. α = α

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The language Proc_L Syntax Operational semantics (LTS) S is an endomorphism of L

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Constructions on Event Structures Prefixing where Sum E1 + E2 (categorical product) Restriction where X is a set of labels Relabelling where f : E --> L Parallel Composition E1 || E2 = E1 x E2 is the categorical product X is the set of pair of labels where. is undefined f(l1,l2) = l1. l2

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Example of parallel composition

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Semantics of Proc_L ρ is the environment function mapping process variables into event structures

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Properties [[ ]] is well defined Prefix order We say that an event structure E is a prefix of E' (written E ≤ E') if there exists an event structure E'' isomorph to E' such that E ⊆ E'' and no event of E''\E is below any other event of E. It is possible to show that the class of event structures with the prefix order form a cpo all the constructions above are continuos

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