Event structures Mauro Piccolo

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

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

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

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

EVENT STRUCTURES: DEFINITIONS

(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'<e} is finite  is an binary irreflexive symmetric relation on events satifying

Some notation parents(e): set of maximal events of [e) [e] = [e) U {e} e1 e2 is inherited if there exists e3<e1 s.t. e3 e2. It is immediate (written ) otherwise Remark: conflict and causal order are mutually exclusive.

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)

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:

A CATEGORY OF EVENT STRUCTURES

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

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

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

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

DOMAIN OF CONFIGURATIONS

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

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

EVENT STRUCTURE SEMANTICS OF CCS

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

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

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

Example of parallel composition

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

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