PART 4: Extended Petri Nets Motivation u Computational power of Petri nets < Turing machines. u In...
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Transcript of PART 4: Extended Petri Nets Motivation u Computational power of Petri nets < Turing machines. u In...
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!