1 Formal Models for Transactions: Petri Nets Roberto Bruni Dipartimento di Informatica Università...

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Formal Models for Transactions: Petri Nets

Roberto BruniDipartimento di Informatica Università di Pisa

Models and Languages for Coordination and Orchestration

IMT- Institutions Markets Technologies - Alti Studi Lucca

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Roberto Bruni @ IMT Lucca 12 April 2005


InstitutionsMarketsTechnologies

IMT

Contents Orchestration with Petri Nets Transactions with Zero Safe Nets From BizTalk to Zero Safe Nets

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Contents Orchestration with Petri Nets Transactions with Zero Safe Nets From BizTalk to Zero Safe Nets

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Petri Nets Place/Transition Petri nets [Petri 1962]

Well-known model of concurrency Theory / Applications

Well-supported by a large community Academy / Industry

Well-developed tools http://www.daimi.au.dk/PetriNets

A media for conveying ideas to non-expert Suggestive graphical presentation

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Foundational Model of Concurrency

Basic monoidal structure of states A graph whose set of nodes is a monoid Precursor of most modern models and calculi of

concurrency Monoidal structure of computations

Essentially multiset rewriting Framework for studying issues of

Causality, concurrency, conflict Event structures and domain Deadlock, liveness, boundedness Reachability, coverability, invariants

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General-purpose Model of (Concurrent) Computation

Algebraic representations of data, states, transitions, steps, computations

Basic rewriting system later generalized by Term rewriting, graph rewriting, term-graph rewriting,

concurrent constraint models A framework where to mix concurrency with other

features Data types, time, probability, dynamic reconfiguration,

read-without-consuming, negative preconditions, objects, Semantic framework for encoding other models

and languages Useful in studying and comparing expressiveness issues

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Practical Specification Language

Standard theory and notation Simple and natural graphical

representation Exploited in many heterogeneous areas Supported by tools of “industrial quality” Suitable for concurrent orchestration!

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Terminology and Notation Places a,b,c,… classes of resources Transitions t,t’,… basic activities Tokens instances of a resource class Markings u,v,w,… multisets of resources

Multiset union uw Empty marking Places are unary markings a Multiset inclusion (e.g. 2a3c 3ab3c)

Pre-sets t resources necessary to execute t multiset fetched by the execution of t

Post-sets t resources produced by t We write t:uv for t=u and t=v

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Formal Definition A P/T Petri net is a graph N=(S,T,pre,post,u0)

S is the set of markings (is the free monoid over the set of places S) Nodes of the graph

T is the set of transitions Arcs of the graph

pre:T S assigns pre-sets to transitions pre(t) = t Source map of the graph

post:T S assigns post-sets to transitions post(t) = t

Target map of the graph u0 is the initial marking

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Graphically

a2a1

a5 a4

a3

2

2

places are circlestransitions are boxesweighted arcs model pre-/post-sets

t1 t2

t3

t4

t5

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Enabling A transition t is enabled in the

marking u if t u

Meaning that there exists u’ such that u=u’t

If t is enabled in u, it means that the system has enough resources to

execute t

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Firing

The execution of t is called a firing t can

first fetch the resources in its pre-set and then release fresh resources according to its

post-set the system moves

from the state u = u’ t to the state v = u’ t

the firing is usually written u [t v

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Steps A multiset of transition i niti is enabled in

the marking u if there exists u’ such that u = u’ i ni

ti Several transitions that are concurrently

enabled can fire concurrently their concurrent execution is called a step

from u = u’ i niti

to v = u’ i niti

Usually written u [i niti v

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Example: Step Sequencesa2a1

a5 a4

a3

2

2t1 t2

t3

t4

t5

t1:a5a1

t2:2a1a4 a22a5

t3:a2a3

t4:a3a2

t5:a3a4

4a12a2

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a5 a4

a3

2

2t1 t2

t3

t4

t5

t1:a5a1

t2:2a1a4 a22a5

t3:a2a3

t4:a3a2

t5:a3a4

4a12a2 [2t3 4a12a3

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a5 a4

a3

2

2t1 t2

t3

t4

t5

t1:a5a1

t2:2a1a4 a22a5

t3:a2a3

t4:a3a2

t5:a3a4

4a12a2 [2t3 4a12a3 [t4t5 4a1a2a4

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a5 a4

a3

2

2t1 t2

t3

t4

t5

t1:a5a1

t2:2a1a4 a22a5

t3:a2a3

t4:a3a2

t5:a3a4

4a12a2 [2t3 4a12a3 [t4t5 4a1a2a4 [t2 2a12a2 2a5

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Operational Semantics We can describe concurrent computations

by means of three simple inference rules:

aS

aNa[reflexivity]

t:uvT

uNv[firing]

uNv u’Nv’

uu’Nvv’[parallel composition]

uNv vNw

uNw[sequential composition]uNw*

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Basic Properties Proposition

There is a step sequence leading from u to v iff uNv

Decidable properties termination reachability coverability

*

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An Algebra of Computations

We can use proof terms to denote computations:

aS

a:aNa[reflexivity]

t:uvT

t:uNv[firing]

:uNv :u’Nv’

:uu’Nvv’[parallel composition]

:uNv :vNw

;:uNw[sequential composition]

associativecommutativeunit :N

monoid homomorphismidentities u:uNu

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a5 a4

a3

2

2t1 t2

t3

t4

t5

t1:a5a1

t2:2a1a4 a22a5

t3:a2a3

t4:a3a2

t5:a3a4

4a12a2 [2t3 4a12a3 [t4t5 4a1a2a4 [t2 2a12a2 2a5

(4a12t3);(4a1t4t5);(2a1a2t2)

idle resources

activities

: 4a12a2 2a12a2 2a5

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Basic Facts About Concurrency

idling monoid homomorphism

Suppose t:uv and t’:u’v’ t;v = t = u;t

Idle steps are immaterial tt’: uu’Nvv’ // concurrent execution tt’ = (t;v)(u’;t’) = (tu’);(vt’) // t precedes t’ tt’ = (u;t)(t’;v) = (ut’);(tv’) // t’ precedes t

If two activities can be executed concurrently, they can be executed in any order The vice versa is not true

Take t:aa and t’:aa t;t’ and t’;t are very different from tt’

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Token Philosophies This semantics follows the so-called

Collective Token Philosophy (CTPh) Any two tokens in the same place are

indistinguishable one from the other computationally equivalent

Other semantics follow the Individual Token Philosophy (ITPh) Any token carries its own history

tokens have unique origins fetching one token makes an activity causally

dependent from the activities that produced it such analysis can be important for recovery purposes,

detecting intrusions, increase parallelism, …

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Process Semantics Non-sequential behaviour of nets

Causality and concurrency within a single run Runs are described by Processes = (P,)

A process net P acyclic net pre-/post-sets are just sets, not multisets transitions have disjoint pre-sets transition have disjoint post-sets

A net morphism :PN places to places transitions to transitions

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Example

a2a1

a5 a4

a3

2

2t1 t2

t3

t4

t5

Graphically is rendered by a suitable labeling (herewe use colours for simplicity)

a1 a1 a1 a1 a4 a2

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Example

a2a1

a5 a4

a3

2

2t1 t2

t3

t4

t5

t3t2

a3a5a5 a2

a1 a1 a1 a1 a4 a2

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Example

t5t1

a4a1

a2a1

a5 a4

a3

2

2t1 t2

t3

t4

t5

t3t2

a3a5a5 a2

a1 a1 a1 a1 a4 a2

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Example: Process 1

t2

a5a5 a2

a5 a4

a3

2

2t1 t2

t3

t4

t5

a2a1

t5t1

a4a1

t3t2

a3a5a5 a2

a1 a1 a1 a1 a4 a2 the last firing of t2 causally depends on all other events

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Example: Process 2

Now there are two disjoint activities!

a5 a4

a3

2

2t1 t2

t3

t4

t5

a2a1

t2

a5a5 a2

t5t1

a4a1

t3t2

a3a5a5 a2

a1 a1 a1 a1 a4 a2

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FOURTH HOMEWORK

Using the algebra of computations that we have seen: write the expression associated with 1

( 2a1t2t3 ) ; ...

write the expression associated with 2

show that the two expressions can be proved equal to

( t2 ; a2a5t1 ) ( 2a1t3 ; 2a1t5 ; t2 )

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Concatenable Processes Each process defines

an initial marking u -image of the set of places with no antecedents – minimal places

and a final marking v -image of the set of places with no successors – maximal places

:uv Can processes be composed analogously to CTPh runs?

In general there is some ambiguity The correspondence between final places of the first process and

initial places of the second process must be fixed Concatenable processes come equipped with suitable

orders on minimal / maximal places of P The orders concern places that are mapped to the same place of N

unique source of ambiguity

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Concatenable Processes: Graphically

a’1 a’2 a’n’ b’1 b’2 b’m’

superscripts denote order on minimal placessubscripts denote order on maximal places

d’1 d’2 d’k’

a1 a2 an b1 b2 bm d1 d2 dk

: namb…kd n’a’m’b’…k’d’

a,b,…dare places of N

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Composing Concatenable Processes

Idle computations any place is both minimal and maximal (no

transitions) minimal and maximal orders coincide

Parallel composition 12 juxtaposition (NOT COMMUTATIVE) the orders in the result are obtained by assuming that

places of the first process precede places of the second process

Sequential composition 1;2 maximal places of 1 are merged with minimal places

of 2 according to their orders

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Symmetries Special concatenable processes allow to rearrange

the orders of minimal and maximal places called symmetries : uv vu

(note that uv = vu) no real computation the order of minimal places differs from that of maximal

places symmetries are important to generate all possible

causal dependencies arising from different combination of minimal and maximal places during composition

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Unfolding Semantics Instrumental in giving denotational semantics to nets

a unique prime event structure that faithfully represent causality, concurrency and conflict between all possible events that can be generated from the net

unfolding approximations can be used for verification Unfolding combines all processes in a unique

structure non-deterministic exploration of computation space define a nondeterministic net U(N) together with a net

morphism from U(N) to N acyclic, no backward conflicts, pre-/post-sets are sets places are tokens, transitions are events

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a

Example: Three Processes

b

t s

t

a

r

a

c

s

r

c

b

t

t

a

a

s

r

c

b

a

a

s

r

c

b

1 2 3 N

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c

Example: Unfolding I

b

t s

r

a

c

a

s

r

b

t

a

ra

t

r

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a

c

Example: Unfolding II

b

t s

r

a

c

t

t

a

a

s

r

b

#

co

r

r

Three relations: : Causalityco : Concurrency# : Conflict

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Unfolding Construction Immediate precedence

<0 = { (a,t) | a•t} {(t,a) | at•} Causal dependence

is the transitive closure of <0 Binary Conflict

# is the minimal symmetric relation that is hereditary w.r.t. and contains #0 defined by: s#0t iff st •s•t

Concurrency co(x,y) iff (x<y y<x x#y) we also write co(X) iff for all x,yX we have

co(x,y)

Note that and # have empty intersection

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Unfolding Construction:Notation

Places of U(N) a,n,H

a is the corresponding place in N n is a positive natural number introduced to distinguish

tokens with the same history H is the history of the place

either the empty set or a single event (the transition that generated the token)

Transitions of U(N) (events) t,H

t is the corresponding transition in N H is the history of the event (the set of fetched tokens)

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b

t s

r

a

c

x1 = a,1, x2 = b,1,

e1 = t,{x1}

x3 = a,1,{e1} e3 = r,{x1,x4}

e2 = s,{x2}

x4 = c,1,{e2}

re4 = r,{x3,x4}

re6 = r,{x5,x4}

Unfolding Construction:Example

e5 = t,{x3}

x5 = a,1,{e5}

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Unfolding Construction:Inference Rules

The net U(N) is the minimal net generated by the two rules below

ka u0

a,k, SU(N) initial marking of U(N)

t:iai jnjbj T ={ai,ki,Hi}i SU(N) co()

e=t,TU(N) ={bj,m,{e} | 1 m nj}j SU(N)

pre(e)= post(e)=

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Unfolding Construction:Remarks

The condition co() depends exclusively on the histories Hi and cannot be altered by successive firings

Histories can be completely cabled inside the tokens so that it is not necessary to re-compute them at every firing (as in memoizing or dynamic programming)

Histories retain concurrent information, not just sequential Each token / event is generated exactly once

It can be referred several times successively

Several occurrences of the second rule can be applied concurrently

The unfolding can be implemented as a distributed algorithm

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References I Kommunikation mit automaten (PhD Thesis,

Institut fur Instrumentelle Mathematik, Bonn 1962) C.A. Petri

Petri nets: an introduction (EATCS Monograph on TCS, Springer Verlag 1985)

W. Reisig Petri nets are monoids (Information and

Computation 88(2):105-155, Academic Press 1990) J. Meseguer, U. Montanari

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References II The non-sequential behaviour of Petri nets

(Information and Computation 57:125-147, Academic Press 1983)

U. Goltz, W. Reisig Petri nets, event structures and domains, part I

(Theoretical Computer Science 13:85-108, 1981) M. Nielsn, G. Plotkin, G. Winskel

Configuration structures (Proc. LICS’95, IEEE, pp.199-209)

R.J. van Glabbeek, G.D. Plotkin

1 Formal Models for Transactions: Petri Nets Roberto Bruni Dipartimento di Informatica Università...

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