I.S. Behaviour Using Petri Nets

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I.S. Behaviour Using Petri Nets

• Some Problems with statecharts
• They do not represent the resources (data,
  objects or message instances).
• They lack of mathematical semantics for computing
  on them (analysis, verification, ..)
• They cope more on state changes and not on object
  dynamics.

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Petri Nets (PNs)

• Model introduced by C.A. Petri in 1962
• Ph.D. Thesis Communication with Automata
• Applications distributed computing,
  manufacturing, control, information systems,
  communication networks, transportation
• PNs describe explicitly and graphically
• sequencing/causality
• conflict/non-deterministic choice
• concurrency

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Simple Petri Nets Introduction

• General phylosophy under Petri nets
• In any software (information) system, we have
  ressources (e.g. data, objects, events, real
  ressources printers, buffers, ...) .
• There is generally, a number of instances for
  each kind of ressource.
• Actions (behaviour, dynamics) in the system
  allows - consuming (using) some ressources and
  producing new ones or changing existing ones
  (under some conditions).

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Simple Petri Nets Introduction

• With each kind of ressource, a PLACE graphically
  as circle is associated.
• Ressource instances are included as BLACK DOTS
  (called TOKENS) within the corresponding
  ressource kind place.
• Actions called TRANSITIONS are graphically
  represented as a BOX or RECTANGLE.
• Ressources consumption are captured by (INPUT)
  ARCS---valuted or inscribed by the numbers of
  required ressouces--- From the corresponding
  ressource place to the associated action box.

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Simple Petri Nets Introduction

• Ressources production or changes are captured by
  (OUTPUT) ARCS---valuted or inscribed by the
  numbers of produced ressouces--- From the
  corresponding ressource place to the associated
  action box.
• THAT IS, PETRI NETS ARE BIPARTITE GRAPHS.

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Simple Petri Nets Introduction
p1
p2
3
1
t1
p1 Budgetary items p2 personne to be
recruted p3 employees.
1
p3
214
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Simple Petri Nets Basic definitions

• A Petri nets in a five tuples PN (P, T, W, M0)
  
• P p1,p2, ....,pn is a finite sets of places
  (as circles)
• T t1, t2, ...,tq is a finite set of
  transitions (as boxes)
• F ? (P x T) ? (T X P) is a finite set of arcs.
  An input (resp. output) ) arc joins a place to
  transition (resp. transition to place). NEVER
  PLACE TO PLACE or TRANSITION TO TRANSITION.
• W P ? 1,2,... is a weight function attached
  to the arcs. If the weight in missing it is
  implicitly one.
• M0 P ? 0,1,2,.... is the intial marking.
  M0(p), p ? P is the marking (i.e. numbers of
  tokens) of the place p.

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Graphical representation of a classical Petri net.
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Simple Petri Nets Basic definitions

• Notations
• ? t the set of input places of transition t.
  that is the set of places p such that (p,t) ? F.
• t? the set of output places of transition t.
  That is, p such that (t,p) ? F.
• ?p the set of input transitions of place p.
  that is the set of transitions such that (t,p) ?
  F.
• p? the set of output places of transition t.
  That is, t such that (p,t) ? F.

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Simple Petri Nets Basic definitions

• Dynamics of Petri nets
• A transtion t is said to be enabled if, whatever
  p ? ? t , p contains a number of tokens greater
  of equal to W(p,t) (i.e. M(p) ? W(p,t))
• Firing an enabled transition t, consists in
• Removing W(p,t) tokens from each p ? ? t
• Adding W(t, p) tokens from each p ?t?
• That is, M(p) M(p) W(p,t) W(t,p)
• A sequence of transitions firing is generally
  represented by ? ltti1, ti2 , ....., tikgt, and
  the corresponding resulting marking by M0 --
  ?--Mk

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Modeling with Petri Nets
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Modeling with Petri Nets
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Communication Protocol
Send msg
Receive msg
P1
P2
Send Ack
Receive Ack
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Producer-Consumer Problem
Produce
Buffer
Consume
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Simple Petri Nets Basic definitions

• Incidence matrix
• A aij
• is an n ? m matrix of integers and its typical
• entry is given by
• aij aij - aij-
• Where aij w(i,j) is the weight of the arc from
  transition
• i to its output place j and aij- w(j.i) is the
  weight of the
• arc to transition i from its input place j.

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Simple Petri Nets Basic definitions

• Given an initial marking M0, let ? be a firiable
  sequence of transitions which applies to M0.
• The counting vector V? of is the vector V?
  v1,v2,..,vq is the numbers of times tj is
  included in ?.
• If M is the marking obtained by firing ?, then
  Mt M0t AV?t . With t denotes the transpose.

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Simple Petri Nets Basic definitions

• Example

t4
t1
t2
p2
p1
4
2
t3
p3
2
2
p4
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Simple Petri Nets Basic definitions

• 1 -2 -1 0
• A 0 4 0 -1
• 0 0 1 -1
• 0 0 -2 2
• Let M0 3,0,0,2 and ? ltt2,t3,t4gt, that is
• V? 0,1,1,1 . Then M 0, 3, 0, 2 using the
  formulas.

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PN properties

• Behavioral depend on the initial marking (most
  interesting)
• Reachability
• Boundedness
• Schedulability
• Liveness
• Conservation
• Structural do not depend on the initial marking
  (often too restrictive)
• Consistency
• Structural boundedness

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Reachability

• Marking M is reachable from marking M0 if there
  exists a sequence of firings s M0 t1 M1 t2 M2
  M that transforms M0 to M.
• The reachability problem is decidable.

p2
t2
M0 (1,0,1,0) t3 M1 (1,0,0,1)
t2 M (1,1,0,0)
t1
p1
p4
t3
p3
M0 (1,0,1,0) M (1,1,0,0)
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Liveness

• Liveness from any marking any transition can
  become fireable
• Liveness implies deadlock freedom, not viceversa

Not live
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Boundedness

• Boundedness the number of tokens in any place
  cannot grow indefinitely
• (1-bounded also called safe)
• Application places represent buffers and
  registers (check there is no overflow)

Unbounded
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Conservation

• Conservation the total number of tokens in the
  net is constant

Not conservative
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Analysis techniques

• Structural analysis techniques
• Incidence matrix
• T- and S- Invariants
• State Space Analysis techniques
• Coverability Tree
• Reachability Graph

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Incidence Matrix
t2
p1
p2
p3
t1
t1
t2
t3
p1
t3
p2
p3

• Necessary condition for marking M to be reachable
  from initial marking M0
• there exists firing vector v s.t.
• M M0 A v

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State equations

• E.g. reachability of M 0 0 1T from M0 1 0
  0T

t2
but also v2 1 1 2 T or any vk 1 (k)
(k1) T
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State equations and invariants

• Solutions of Ax 0 (in M M0 Ax, M M0)
• T-invariants
• sequences of transitions that (if fireable) bring
  back to original marking
• periodic schedule in SDF
• e.g. x 0 1 1 T

t2
p1
p2
p3
t1
t3
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Application of T-invariants

• Scheduling
• Cyclic schedules need to return to the initial
  state

i
o

k2
k1
T-invariant (1,1,1,1,1)
Schedule i k2 k1 o
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State equations and invariants

• Solutions of yA 0
• S-invariants
• sets of places whose weighted total token count
  does not change after the firing of any
  transition (y M y M)
• e.g. y 1 1 1 T

t2
p1
p2
p3
t1
-1 1 0 0 1 -1 0 -1 1
AT
t3
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Application of S-invariants

• Structural Boundedness bounded for any finite
  initial marking M0
• Existence of a positive S-invariant is CS for
  structural boundedness
• initial marking is finite
• weighted token count does not change

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Reachability graph
t2
p1
p2
p3
t1
100
t1
010
t3
t2
t3
001

• For bounded nets the Coverability Tree is called
  Reachability Tree since it contains all possible
  reachable markings