It is well known from [2, 3] that in general processes in Petri nets are not recoverable from firing sequences. However, firing sequences in Petri nets say something about concurrency. The paper presents a way to define concurrency from firing sequences of nets. It turns out that the information of concurrency in a firing sequences characterizes all its processes

Ban, Đoàn Văn

Hung, Dang Van

Vietnam Academy of Science and Technology: Journals Online

HOW M UCH INFORMATION OF CO NC UR RENCY  
CAN BE GOT FROM FIRING SEQUENCES  
IN PETRI NETS
i
D O A N  VAN B A N  and D A N G  VAN H U N G
A b s tr a c t . It is well known from [2], [3| that in general processes in Petri nets are not 
recoverable from firing sequences. However, firing sequences in Petri nets say som ething  
about concurrency. The paper presents a way to define concurrency from firing sequences 
of nets. It turns out that the information of concurrency in a firing sequences characterizes 
all its processes. - i
1. INTRODUCTION " U lJ  i
• . .¿5$ i
In concerning the concurrent and distributed systems, the way: in which the 
tem poral/causal ordering events is described is a problem  being under discussion.
In the interleaving approach, the fact th a t a set of events may occur in paral­
lel is described by saying th a t they may occur in any order. Models based on 
true  concurrency use instead partial orderings to explicitly describe the tem po­
ral/causal relations among events [4, 6, 8, 9], In [2, 3], a comparison between 
the two approaches has been treated . These authors proved th a t in P /T  nets 
processes (corresponding to the latter) are not recoverable from firing sequences 
(corresponding) to the former), while in C /E  systems they are. This m eans th a t
in general in P /T  nets true  concurrency cannot be obtained from firing sequences.
As firing sequences play an im portant role in studying the behaviours of P /T  nets, 
and as a p a rt of true  concurrency is carried in them , it is worth studying the ways 
to decide w hat we can say about concurrency from firing sequence of P /T  nets. 
By following the approach of Mazurkiewicz [7]. Best [2] and Degano [3] to the 
behaviours of concurrent systems and developing some results in [5], the paper 
presents a way to study concurrency from firing sequences. We show th a t in order 
to obtain  information of true concurrency from firing sequences, only the s ta tis­
tical structures of nets comes into play. We also give a necessary and sufficient 
condition to  a net for which processes are recoverable from firing sequences.
2. LIKE-DEPENDENCY
9
We follow M azurkiewicz/s approach to the behaviours of C /E  system s [l], [7]
in studying firing aquences of P /T  nets. _ ___—
] V i L4B* :0MGTWOJivST lN j
O ur starting  point is the notion of so-called like - dependency. Intuitively 
speaking, when there may be causal dependencies among occurrences of two ac­
tions, we consider them  to be in like - dependence. Formally like - dependency is 
defined below.
Let A  be a finite set whose members are referred to as actions. Let A* (A u 
respectively) denote the set of all finite (infinite) sequences (or words) over A, 
A°° := A* U A w. The em pty sequence is denoted by e.
For u> 6 A°° and a £  A, # aw will denote the number of the occurrences of a 
in w and 0 (w )  denotes the set { (a ,i ) \ # aw >  0 AO <  i <  jf^aw +  l}-
D e f in it io n  1. A like-dependency on A  is a reflexive binary relation on. A.
Since in general an action can depend on other action while the la tte r is 
independent of the former, a like-dependency is not required to be sym m etrical.
Let D  be a like-dependency on A.  Each w G A *, w may represent a"compu­
tation . Then the partial ordering of causal dependency relation < w is defined as 
follows.
D e f in it io n  2. The partia l ordering generated by w over D  is (0(u>),< iu), where 
< w is the reflexive and transitive closure F£ of the relation Fw defined by 
(a ,t)  Fw(b , j ) iff the j’-th  occurrence of b precedes the ¿-th occurrence of a in 
w and (a, 6) £  D.
E x a m p le  1. Let A  = {a, 6}, D  =  {(a, a), (6,6), (6, a)}, w = (ab)w = abab... E A w.
T hen {0 ( w ) , < w) is represented by the following graph (the transitive arcs are 
om itted). v_____
(a, 1)
I
(a, 2) (M )
i I
* •?
(a, n) (6, n  — 1)
1 1
(6, n)
I
NoWj we introduce a partial order among members of A °° .
D e fin itio n  3 . For w, w'  £  A °° , w C w'  iff 0 (w )  =  0(w ')  and < WC.<W> (w C w'  
iff the partia l ordering generated by w over D  is coarser th an  the one generated 
by to').
We consider the behaviour of a com putation system  as a pair of
• A like-dependency, which approximately represents dependency in the 
system. ,(
• subset C  of A °° , which represents possible com putation of the system  (the 
interleaving behaviours of the system).
Then, for each w £  C,  (0(itf),<u>) represents uncertainty the causal dependen­
cies among occurrences of actions in w, some causal dependencies of which are 
introduced by going to extremes.
Now, we consider w hat the relation C means.
In the sequel, let name : A  X { l, 2 , . . .  } — ► A  be defined as
name ((a, *)) =  a for all integers i >  0,
and let pref : A°° — > 2A be a mapping which returns all prefixes of its argum ent. 
The m apping name  is extended to  a homomorphism from (A x { l, 2 , . . .  })°° to  A°°
in the obvious way. Furtherm ore, for w , w'  £  A*, we w rite w — w'  iff there is a
D
derivation from w to w'  in the rewriting system (A , P)  w ith P  = {ab —> 6a|(a, 6) ^  
£>}•
T h e o re m  1.
(i) Let w, w' £  A* , w Q w '  if  and only if w w ' .
(ii) Let w, w' £ A w  Q w' if and only if (0 ( w ) =  0(w ')
A(Vt> £  pref(iy') 3u £ pref(u>) 3x  £ A* : (u vx)).
Proof.
(i) Only the ’only if’ part is not obvious and can be shown by induction on 
the length |to| of w, and we leave it to the readers.
(ii) (*i=): Let e i, e-i £  0(w )  = 0(w ')  and e\ < w e2 - There m ust be v in 
pref(u/) such th a t ex, e2 £ 0{v).  Let u and x  be such th a t u £ pref(w) and 
u vx.  From (i) it follows el5 e2 £ 0(u) .  By the definition of < w we have 
e i < u  ^2 , and thus e\ <„ e2 by (i). Hence, e\ < w> e2 by the definition of .
(=>): Let v £  pref(u /). Then 0(v)  C 0(w) .  Let u £  pref(u>) such th a t 
0 (u )  D 0 (v ) .  It follows th a t < u=  (< w) n (0 (u ) x 0(u) )  C (< w,) n (0 (u )  x 0(u)) .  
Let a  be a topology sorting of (0 (u ) \  0 (v )) by < w> and x  =  nam e(a). It can be 
seen from the definition of < wi th a t (<„,< n(0(u)x0 (u) )  C.<vx ■ Hence, < UC <  VI . 
By (1) we get u — vx.  □
Theorem  1 says th a t for w, w'  E A°°, w C w'  if and if they have the same set 
of action occurrences and w'  is derived from w by applying a (finite or infinite) 
num ber of rew riting rules ab —> ba w ith (a, 6) ^  D.
Since independent events can occur in any order, it follows from  Theorem  1 
th a t if w E C,  for each w'  such th a t w C t v \  w'  E C as well.
C o ro lla ry  1. Let =  be defined as w = w' iff then w = w' if  and only if
w - w ' , where D s is the symmetrical closure of D.
3. INFORM ATION OF TRUE CONCURRENCY IN FIRING 
SEQUENCES OF P /T  NETS
In this section we investigate how much information of true concurrency can 
be got from firing sequences of P /T  nets. We shall compare the partia l orderings 
am ong events introduced by processes in P /T  nets to the partia l orderings gener­
a ted  by firing sequences w ith respect to the natural like-dependencies defined by 
the s tru c tu re  of nets.
. «=-
A nets is a triple {S,T ' ,F),  where
• 5 f l T  =  oo;
• F  C (S  x r) U (T  x S).
Let, as usual, t* =  {s 6  5 ^ 5 } ,  *t =  { 5  6  S \sF t}  for a net (S , T ; F ).
An occurrence net is a net =  (S, T ; F)  such th a t
• The transitive closure of F,  defined by F + , is irreflexive;
•  Vs E S, Tsl <  1 A Is* I <  1.' 1 *. ' 1
Furtherm ore,
• S  U T  is considered as ordered by < , defined as F +\
• The slices of K  are maximal subsets of S  which do not contain elements 
related by < .
A m arked p lace/transition  net (P /T  net) is a quintiple N  = (S , T ] F , W , M ), 
where y
• {S , T \ F )  is a net, w ith S  and T  finite;
• W  : F  N  assigns a positive weight to each arc;
• M  : S  —* N  is the initial marking of N .
Given a P /T  net N  = (S , T \ F , W , M ) ,  a firing sequence of N  is { M o t o M i t i M 2 -.. 
where for i =  1, 2 ,...
• M i  are markings of S  and M q = M , t {  E T;
.  Mi[t i > M t+ 1 , where M[t > M '  implies th a t Vs £ s ,  M (s )  > w ( s , t )  and 
M '(s )  = M (s )  -  W{s,  t) +  W(t ,  s).
We shall call the sequences obtained from firing sequences by dropping the m ark­
ings also firing sequences w ithout fear of confusions.
Given a P /T  net N  = (s ,  T ,F ]W ,  M )  and an occurrence net K  =  { s ' , T ' - F r) , 
a P /T  process of Ẹ  is a function
p : K  -> N
such th a t
• p(S ')  c  5 , p ( r ')  c  T ;
• {S'  u T 1, < ')  is finitely preceded. Let ° K  be the set of its minima;
.  Vs E s ,  M { s ) =  {p_1(5) n °K\\
• Vt' e  T ’. V s e s
(i) w(i,i>_(t#)) =  |p-1W n V | ,
(ii) W(p(t'),s) =  \p x(s) m '* | .
D e fin itio n  4. The labeled partia l ordering generated by a process p : K  —y N  of 
a P /T  net N  (denoted as above) is < p ), where <p is F ' * \ t ' x T ‘-
From the results in [2], [3] it follows:
For a P /T  net N ,  a  is a firing sequence of N  if and only if there exists a
process p : K  —► N  of N  such th a t a = p(ß),  where ß  is a topology sorting of T'
by p is extended to a homomorphism on sequences in obvious way.
It can be seen easily th a t if p : K  —»■ N  is a process of N  w ith K  =  ( s ' ,  T'] F ' ), 
S 'u T ' is countable. Furtherm ore, since isomorphic processes are not distinguished, 
in the  sequel T '  is usually considered as a subset o f T  X {1, 2 ,...}  satisfying:
(i) t' =  (a, n) G T '  implies (a, i) £ T '  for 0 <  i < n,
(ii) (a, n ), (a , n f) 6 T '  and n < n'  implies (a , n ')
(iii) p(t) =  nam e(i) for all t E T ' .
As in [4] processes are considered to be equivalent iff the partia l order of event 
occurrences agrees in them . We give the following definition.
D e fin itio n  5. Let K  and N  be denoted as above, p : K  —> N  is a process of N . 
{T', <p) is called a concurrency characteristic of p (characteristic of p for short).
As in [2], let us denote for a firing sequence a  and for a process p : K  —> N  of 
N  w ith K  = (S ' , T ' ] F '):
Lin(p) :=  {a|o! =  name(/?) w ith ß  being a topology sorting of T '  by <p}.
P roc(a) :=  {pỊô G Lin(p)}.
From the resup in [2], [3], we have
T h e o re m  2. Let N  be a P / T  net, a  is a firing sequence of N  if and only if there 
exists a process p : K  —* N  of N  with K  = (S' ,  0 ( a ) ,  F ’) such that a  =  name(/3) 
with (3 being a topology sorting of 0 ( a )  by <p (p is said to correspond to a).
This is the  first result on the relationship between firing sequences and pro­
cesses. Now we give some anothers.
Definition 6. Let N  — ( S , T ] F , W , M )  be a P /T  net.
D  = {(t, í ') j í , €  T A ( f n * t '  0Ví =  t 1)} is called like-dependency generated
by N .
Let, in the sequel, N  be a P /T  net, D  its like dependency, a  a firing sequence 
of N  w ith  ( 0 ( a ) ,  < a ) being its partial ordering on D , and let <  be defined in as 
Definition 3 w .r.t D.
T h e o re m  3 . Let K  =  (5 ',  0 ( a ) ,  F 1) and p : K  —* N  be a process of N  corre­
sponding to a ,  < p =  F 1*. Then ( 0 ( a ) ,< p) is coarser than ( 0 ( a ) , < a ).
Proof. It is sufficient to  prove:
Since a  G Lin(p), the ¿-th occurrence of a precedes the j - th  occurrence of 6 in a . 
By Definition 2 (a, 6) G D  implies (a, t) < Q
Theorem  3 says th a t if (a, t) and (b,j) are not related by < a , neither are they 
by any process corresponding to a . T h a t means we can get some inform ation of 
concurrency from  firing sequences of the net by its like-dependency.
V(a,¿), (&J) G 0(a) : (a, i) F n (b,j) => (a , i ) < a (b , j ) .
we have
{a , i ) F ,2(b , j ) => 3s E S '  : (a , t )F 'sF ' (b , j )
=>• a* n  *b D {p(,s)} =>• (a, b) G D  .
E x a m p le  2.
Let N  =
a b
Then, D  =  { (a ,a ), (6,6), (6 ,a)}.
a  =  (a6)w is a firing sequence of N ,  and (0 ( a ) ,  < a ) is the same as in Exam ple 
1. A process corresponding to a  is the following
In th is case, P ro c(a) contains one process, and (0 ( a ) ,  < a ) is its characteristic.
T h eorem  4 . I f  the parallel occurrence of the same transitions is impossible in N, 
(this means that i f  p : (S ' ,T ' ;  F') —> N  is a process of N ,  and if  ( f ,t) , ( t , j )  G T '  
with i < 3 ,  ( M )  < p  (t , j ), then < Q =  (J <p/ . *
pGProc(a)
Proof. From  Theorem  3 we have
U  < P'C < Q . •
p £ P ro c (a )
Now we have to show the inverse inclusion. It is sufficient to prove th a t V(a, z), (b,j)  G 
0 (a )  : (a,*) < a (b,j)  => 3p G Proc(a) : (a ,t)  <p (b , j ).
Let p be a  process of N  corresponding to a. If (o, 0  <P [b,j) ,  or a =  b, the
theorem  has been proved. Suppose th a t (a,i) (b, j)  and a ^  b. Since (a, 6) G
D => 3s G a* fl*6. Let si  G (a, ¿)* and s 2 G *{b,j) such th a t p (s i) =  p(«2 ) =  s. Of 
course, s i  /  s 2- Now we construct a occurrence net K '  =  (S ' , 0 ( a ) ,  F") ,  where
F "  =  F ' \ { ( 52,( 6 ,y ) ) } u { ( ,  u { b , m ,  
p -  p.
It can be seen th a t F " ) + is acyclic and p' : K '  —+ N  is & process in P roc(a) 
as well. Furtherm ore, (a, i) < p<
The theorem  will not be true in general w ithout the assum ption th a t the
parallel occurrence of the same transitions is impossible. Let us consider the
following example.
E x a m p le  3. Let
N
Then, aa is a firing sequence of N  w ith (0 ( a a ) , < aa) is represented by
while
(3,1)□ (3.2) □
D O A N  VAN B AN ,  D A N G  VAN H U NG
fa,7)
O—
Proc(aa) =  ça 2\
O -n -O
and its characteristics is ({(a, 1), (o ,2)} ,0).
Theorem  4 says, in the case when the parallel occurrence of the same transi­
tions is impossible in N, th a t the information of concurrency in each firing sequency 
is m axim al am ount derived from all processes corresponding to the firing sequence.
T h eo rem  5 . I f  a  is a firing sequence of N  and aÇ. ¡3 then /3 is a firing sequence 
of N  also. Moreover P roc(a) Ç Proc(/?).
Proof. It follows from Theorem  1 th a t if is a firing sequence, then  P roc(a) ^  0. 
Let p E P ro c(a), from Theorem  3 we have < pÇ < a C < ^ , which implies /3 G Lin(p). 
Hence, /? is a firing sequence of TV, and every process in Proc(o;) is a process in' 
Proc(/?), too. □
In the sequel, we assume th a t N  be such net in which the parallel occurrence 
of the  same transitions In impossible. We have the following corollaries.
C orollary  2. For firing sequences a,  (3 of N, P roc(a) =  Proc(/?) if  and only if 
a  =  ¡3, where =  is defined as in Corollary 1.
Proof. The ’only if’ p a rt follows from Theorem  4, and the ’if’ p a rt follows from 
Theorem  5. □
C orollary  3. For a firing sequence a  of N, all processes in P roc(a) are equivalent 
(by = defined in [2]) if and only if ( 0 ( a ) ,< a ) is their characteristic.
Corollary 3 shows th a t the firing sequence approach and the process approach 
to  the behaviour of P /T  nets coincide only for a restricted class of Petri nets 
concluding 1-safe nets.
REFERENCES >-
1. Albersberg I. J., Rozenberg G., Theory of trances, Theoret. Comput. Sci., 60  (1988), 1 -8 2 .
2. Best E., Concurrent behaviour: sequences, processes and axioms, in S .D . Brookes et al., eds., 
Seminar on Concurrency, Lecture Notes in computer Science 197, Springer, Berlin, 1985, 
221-245.
3. Degano P., Montanari U., Concurrent histories: A  basis for observing distributed systems, Jour, of 
Com put. and Sys. Sci., 34 (1987), 422-461.
4. Goltz U. and Reisig W., The nonsequential behavior of Petri nets, Information and control, 57  
(1983), 125- 147.
5. Hung D. V. and Knuth E., Semi-computations andPetri nets, Theoret. Comput. Sei., 6 4  (1989), 
6 7-81 .
6. Loogen R. and Goltz U., A  non-interleaving semantic model for nondeterministic concurrent pro­
cesses, RWTH Aachen Fachgruppe Informatic, Nr. 87-15.
7. Mazrukiewicz A., Concurrent program schemes and their interpretations, DAIMI Report PB-78, 
Aarhus University, 1977.
8. Nielsen M., Plotkin G., and Winskel G., Petri nets, event structures and domains, Theoret. 
Comput. Sei., 13 (1983), 85-108.
9. Starker P. H., Processes in Petri nets, Informationsverarbeitung und Kybernet. EIK, 17 (1981) , 
389-416.
Institute of  Computer Science 
Hanoi, Vietnam.
Nhận bài ngày 1-8-1995


How much information of concurrency can be got from firing sequences in petri nets

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How much information of concurrency can be got from firing sequences in petri nets

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