Event structuresMauro 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 = <E,≤, > where

E is a countable set of events <E,≤> 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

E1E2

(e1,*,∅) (e1,e3,∅) (*,e3,∅)e f g

(e2,*,{e}) (e2,e3,{e}) (e2,*,{f})

E1 x E2

DOMAIN OF CONFIGURATIONS

Let <D,≤> 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 Dx = ⊔{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 <D(E),⊆> is a finitary prime algebric

domain where the complete primes are the set {[e] | e in E}

Let <D,≤> a finitary prime algebric domain and let P the set of complete primes then <P,≤, > 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 <L,*,.> 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 functionmapping process variables intoevent 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