PART 4:

Extended Petri Nets

Motivation

Computational power of Petri nets < Turing machines.

In many real-time applications, it is often desirable to give certain jobs higher priorities over others, so that critical actions can be finished within their time constraints. For example, one way to do so is to assign each transition of a process a priority which indicates the degree of importance or urgency.

Inhibitory arcs “Colored” tokens Program variables Enabling predicates on transitions Adding time Maximum firing rule …

Generalizing Petri Nets

2-counter Machines

Turing machines ≡ 2-counter Machines

Finite-StateControl

Counters

(1) Add one to a counter(2) Subtract one from a counter(3) Test a counter for zero

inhibitory arc

Can simulate zero-test

2-counter machines

Turing machines

Inhibitor arc

Simulating a counter by Inhibitor Arcs

ctr.

=0? -1

Simulating a Counter by Priority

ctr.

=0? -1

Higher priority

Lower priority

Adding Time to Petri Nets

Variation 1: Transitions have a delay time; firing takes a non-zero time from enabling. Time may be bounded from above or below.

Variation 2: Places have a delay time: A token must dwell on a place a certain amount of time (determined by the place) before becoming usable in firing.

Variation 3: Like 2, but tokens have a delay time.

– A Time Petri Net is like a Petri Net with a time interval on each transition:

[t1, t2] or [t1, )

– From the time the transition is enabled, it cannot fire before t1 and must fire by t2 (unless disabled by firing another transition).

Simulating a Counter by Timed PN

ctr.

=0? -1

Delay = 0

Delay = 1

Simulating a Counter by Maximum Parallelism

ctr.

ctr 0Ctr = 0

The above extensions render the new PN models

Turing-equivalent

?

Reset / Transfer nets

Reset nets: set of reset arcs FR T x P

when t is fired with (t, p) FR

place p is reset to zero.

Transfer nets: set of transfer arcs FT P x T x P

when t is fired with (p, t, q) FT

1. Removing the enabling tokens

2. Transferring all tokens from p to q

3. Adding the usual output tokens

Reset nets

An unbounded Reset Petri net with no iterated sequence

2 21 2 3 4 1 2 3 4 1 2 3 4(1,1,0,0) (1,2,0,0) (1,3,0,0)...(1, ,0,0) (1, 1,0,0)...

i it t t t t t t t t t t ti i

p2p1

p3 p4t3

t2

t1

t4

Reset arc (t2, p2), resetting p2 when t2 is fired

(Un)decidability results

Reset nets Transfer nets P/T nets

Σ1-complete Decidable Decidable

Σ1-complete Σ1-complete Decidable

Π2-complete Π1-complete Decidable (PTIME !)

Decidable Decidable Decidable

Π1-complete Π1-complete Decidable (PTIME !)

boundedness

place-boundedness

structural boundedness

termination

structural termination

Turing-equivalent extended PNs

Priority PNs PNs with inhibitor arcs

timed PNs

colored PNs

extended Petri nets

transfer net

reset net

general Petri nets

restricted Petri nets

conflict-free

normal

sinkless

BPP

free-choice

persistent More complexity analysis

Upper bound for reachability

Boundary between decidability/undecidability

What is next?

PART 5:

Petri Net Languages

Formal Language Approach

View the behavior of a system by the set of all executable transition sequences – a language.

Allow us to take advantage of known results in formal language theory.

Let the alphabet of the system be

the system’s behaviors can be captured by the following sequential language:

•

•

•

As for the comcurrent language of the system, the following is a possible concurrent sequence:

, 1 ,i ip s i m r c

i i# # p # S 1 for 1 iS i m

# # # c 1c

i1# # #

i mS

1, 2, , 1, 2, ,' ..., ..., ... '

m

m mp p p s s s c c

p1

sm

pm

Various Types of PN LanguagesGiven a PN (P,T,I,O) with initial marking , a set of final

markings F and a labeling function is T {} – Free-labeled PN: (t)= (t’) t=t’ -free-labeled PN: (t) t

L-type: L={()* | T*, - ’ ( F)}G-type: L={()* | T*, - ’ ( ’’ F)}P-type: L={()* | T*, - ’ is defined}

T-type: L={()* | T*, - ’ (’ is dead)}

Containment Relationship

Arbitrary labeling

-free labeling

Free labeling

T-type T T T f

L-type L L L f

G-type G G G f

P-type P P P f

Known Results: AB means A contains B

Relationship of Petri Net Languages to Chomsky Hierarchy

Context-Sensitive

PN Lang Context-free

BCF

R

Regularity

What makes regularity important?

Containment, equivalence, … properties are decidable for regular languages.

(Not so for context-free languages)

Related Work Petri nets and regular languages, Valk & Vidal-

Naquet, JCSS’81. Vector addition systems and regular languages,

Ginzburg & Yoeli, JCSS’80. On the rationality of Petri net languages, Schwer,

IPL’86. Fine covers of a VAS language, Schwer, TCS’92. The context-freeness of the languages associated

with vector addition systems is decidable, Schwer, TCS’92.

Semilinearity of reachability set is decidable for Petri nets, Hauschild, Ph.D. Dissertation, Univ. Hamburg, 1990.

Necessary & Sufficient Conditions(Valk & Vidal-Naquet, JCSS, 1981)

A Petri net is not regular iff μ0 μ1 μ2

μ3 μ4 such that

1. μ1 μ≦ 2 and μ1 ≠ μ2 ,

2. μ1(p) μ≧ 2(p) implies μ3(p) μ≦ 4(p), for

every p P (where P is the set of places),

3. μ3(p) ＞ μ4(p), for some p P.

σ1 σ2

σ3 σ4

No complexity analysis was given regarding the above particular algorithm (nor in (Ginzburg & Yoeli, JCSS, 1980))

NOTE: A Petri net is regular iff the set of all (finite) firable sequences of transitions defines a regular language (over T).

.

Intuitively, σ4 constitutes a ‘pumpable’ negative loop, provided a sufficient number of non-negative ‘loops’ σ2s is fired in advance.

Complexity of the Regularity Problem

(Yen, info, & Comp. ’96)

Petri net class Complexity result

Conflict-free PTIME-complete

BPP NL-complete

Sinkless NP-complete

Normal NP-complete

Trap-circuit NP-complete

Extended trap-circuit NP-complete

General EXPSPACE-complete

Context-freeness of PN Languages

Decidable for unlabelled Petri nets – `The Context-Freeness of the Languages

Associated with Vector Addition Systems is Decidable ‘ [Schwer’92].

No complexity analysis

Deterministic Petri Nets

Deterministic Petri net:

for every reachable marking of a PN

Known result (Pelz, STACS 1987)

The language equivalence problem for deterministically labeled Petri nets is decidable.

(Notice that the problem is undecidable for general PNs.)

, , ,P

1 2t t

1 2 1 2 , , then if and t t t t

Iterating systems

Given a loop (q, ω)

does there exist an infinite number of paths

q with loop value ω

…

Pumping lemma for VASSs

Iterating system w.r.t. Loop (q, ω):

11. || || = ,

2. || || 1 || || 1 , and

3. || || 1 || || .

i j

j

j i i p

j p

μ0 μ1 μ2 μ3 μp q

σ1 σ2 σ3 σp ω

ω1 ω2 ω3 ωp+1

(↑↑-↑-↑) (↑↓-↑↑-) (↑↑-↓↓↑) (↑↓-↑↓↑) (↓↓-↓↓↓)

PART 6:

Applications of

Petri Net Theory

Applications of Petri Net Theory

Membrane Computing Supervisory Control Computer-Aided Verification / Formal

Methods Asynchronous Circuit Design/Analysis …

Application to Membrane Computing

Membrane Computing (MC) -- branch of molecular computing initiated by Gheorghe P aun in [TUCS Research Report 1998], [JCSS 2000]

MC identifies an unconventional computing model, called P system -- abstracts from the way living cells process chemical compounds in their `compartmental' (membrane) structure.

A P system consists of a finite number of membranes, each of which contain multisets of objects (symbols) which change during the computation.

Membrane Computing

Membranes organized as Venn diagram/tree structure, where one membrane may contain other membranes.

Dynamics of the system is governed by a set of rules associated with each membrane. Rules specifies how objects evolve/move into neighboring membranes; how membranes can be dissolved/divided/created.

Rules used in nondeterministic, maximally parallel manner define transitions between configurations.

A P system can be used as: acceptor of configurations, or generator of configurations (from a fixed initial configuration).

1-Membrane Symport/Antiport System

A Maximally Parallel Move:R1 applied twice; R2 once a, c, c, d, or R1 applied once; R2 once; R3 once a, a, a, c, d

FACT: Both deterministic and nondeterministic SA areuniversal [Freund & Paun'03]

1-Membrane Communicating P System

FACT: Both deterministic and nondeterministic communicating P systems are universal [Sosik'02]

1-Membrane Catalytic System

A Maximally Parallel Move:R1 applied three times; R3 once c2 d8 e2 or

R1 applied twice; R2 once; R3 twice b c3 d6 e2 or ...

FACT: Nondeterministic CS is universal; [Freund, Paun, Oswald, Sosik, TCS 2005] Determinisdtic CS is not universal [Ibarra, Yen, CIAA 05]

Sequential 1-Membrane Communicating P System

The set of reachable configurations of a sequential 1-Membrane communicating P system can be computed as a finite union of upper-closed sets -- only ``generating" a proper subclass of the semilinear sets.

Extended CPS: (R4) ab axbyccomedcome– equivalent to VAS (Petri nets)

[Dang, Ibarra, DCFS'04 ]

Sequential 1-Membrane Symport/Antiport System

The radius of an antiport rule (x, out; y, in) is (|x|, |y|). For a symport rule (x, out) or (x, in), the radius is |x|.

Sequential 1-Membrane Symport/Antiport Systems are equivalent to VAS (Petri nets)

Every VAS can be simulated by a sequential 1-membrane SA all of whose rules are antiport with radius (1,2) or (2,1).

– [Dang, Ibarra, DCFS'04 ]

Sequential Multi-Membrane Catalytic Systems

Sequential Multi-Membrane CSs are equivalent to communication-free Petri nets (i.e., BPP-nets)– [Ibarra, Yen, Dang, DLT 2005]

In contrast, 1-Membrane CSs under 3-MAX-Parallel Mode are universal– [Freund, Kari, Osward, Sosik, 2003]

Universality Proof using CF-Petri Nets under 3-Max-Parallel Mode

In [Ibarra, Yen, Dang, DLT2004], a simpler proof using 3-max-parallel communication-free PNs is shown:

Set oif addition vectors (W1, W2, W3) – applying at most one vector from each group at any time (nondeterministically)

Proof (Cont’d)

Controllability

Given a plant L (over alphabet =uc) and a desired behavior K

(prefix-closed), a control policy F with is feasible if and only if

Where is the behavior of L under and denotes the set of uncontrollable events of the plant.(Remadge & Wonham, SIAM J. Control & Optimization, 1987)A language K satisfying the above conditions is said to controllable with respect to L.

)

1. , and

2.( L Ku

K L

K

( )L f u f

K

…….controllable

uncontrollableplant L

( )L f K

Controlled Petri Nets CtlPNs – PNs with external enabling conditions called

control places that allows an external controller to influence the transition firing in the net

A control place has the value 0 or 1. A set of transitions is state enabled using the normal PN definition. A set of transitions is control enabled if the control places associated with the transitions all have the value 1

Control Places

A control for a CtlPN is a function u : C -> {0,1} associating a binary value to each control place.

A control u U is said to be as permissive as control u’ if u(c) > u’(c) for all c C

Control u is said to be more permissive than control u’ , if u is as permissive as u’ and u(c) > u’(c) for some c C

The most permissive control is uone :=1. The least permissive control is uzero :=0.

Controllability issues of Petri nets

Given a controlled Petri net P, does there exist a control policy f such that Pf (i.e., P under f) is – bounded– fair– live– safe– self-stabilizing– …

Conclusions In this short course, we have looked at

various Petri net models and their related problems.

Various analytical techniques have been examined for analyzing Petri net problems. A number of complexity results have also been discussed.

We briefly discussed the applications of Petri nets to membrane computing and supervisory control.

Conclusions (cont’d)

In summary, Petri nets are not only of importance practically, they also pose many interesting and challenging mathematical questions.

Future research directions

Solving open problems reported in the literature

e.g. (Exact) complexity of the general reachability problem problems related to persistent Petri nets deciding whether the reachability set of a Petri net

is semilinear (known to be decidable). Petri nets with one inhibitor arc (known to have

decidable reachability problem) …

Future research directions

Pose/solve new Petri net modes/problems motivated by new theoretical/practical research areas

e.g. Membrane computing (in which

maximal parallelism is key)– Complexity/decidability for subclasses of

Petri nets under maximal parallelism

(such as BPP-nets, conflict-free Petri nets under max parallelism)

Future research directions– Relationships between various types of P

systems and classes of (restricted/extended) Petri nets

New problems motivated from, e.g.,

• fault-tolerance: self-stabilization• new programming languages:

-- model of non-blocking Petri nets• …

Future research directions

Expressive power of various extended Petri nets (which are weaker than Turing machines)

…

Thank you!