DSpace at VNU: Shift-Left Algorithms transforming sequential processes into concurrent ones

VNU. JOURNAL O F S C IEN C E, Mathematics - Physics, T.XXII, N q2, 2006

“SHIFT-LEFT” ALGORITHMS TRANSFORMING
SEQUENTIAL PROCESSES INTO CONCURRENT ONES
H oang Chi T hanh
H anoi U niversity o f Science, V N U H
Abstract. Finding concurrent processes of a system is an objective of system
controls, because it shows an optimal way to perform processes. In this paper
we build two iterative algorithms for transforming sequential processes of a
reliance alphabet and of a Place/Transition net into concurrent ones. The
complexity of these algorithms is also considered.
K eyw ords: Reliance alphabet, trace, Petri net, concurrent step.

1. INTRO DUCTIO N
C ontrols on c o n c u rre n t system s alw ays a re a n im p o rta n t a n d com plicated
problem . So far, o p tim al perform ance of processes occurred on a c o n c u rre n t system
is am ong th e controls. F o r th is purpose, recognition of c o n c u rre n t processes of th e
system is e sse n tia l. A lo t of m odels, such as P e tri n e ts [6,9], tra c e s [1,3,4, 8], CSP
[7], CCS [5], process a lg e b ra s [2]
have been b ein g good tools for rep re se n tin g
s ta tistic a l as w ell as d y n am ical stru c tu re of system s. From th e s e m odels, seq u en tial
processes of a system a re n o t difficult to recognize b u t c o n c u rre n t ones still are.
T herefore, how to tra n sfo rm seq u en tial processes of a sy stem onto c o n cu rren t ones
is a g re a t problem on th e o ry and application indeed.
A sim ple alg o rith m for finding th e norm al form of tra c e s is a good solution for
system s re p re se n te d by relian c e alp h ab et. P la ce /T ran sitio n n e t is one of su itab le
m odels to re p re s e n t c o n c u rre n t system s. B ut th e lan g u a g e g e n e ra te d by a P/T n e t is
seq u en tial. I t show s u s only seq u en tial perform ances of th e n e t’s processes. So
w hen sev eral tra n s itio n s c an be perform ed co n cu rren tly . W e w ill co n ce n tra te on
building a n ite ra tiv e a lg o rith m for tran sfo rm in g s e q u e n tia l p ro cesses of a P/T n e t
into c o n cu rren t ones.

T his p a p e r is o rg an ized as follows. Section 2 c o n stru c ts a “sh ift-le ft” algorithm
for finding th e n o rm al form of a trace. In Section 3 we propose th e n o tatio n of
concurrency in P/T n e ts . Section 4 builds up a n ite ra tiv e a lg o rith m for finding
sequences of m axim al c o n c u rre n t step s on a P la c e /T ra n sitio n n e t from th e n e t’s
sequence of single steps. F in ally , some conclusions a re given in S ection 5.

2. N orm al fo rm o f tr a c e a n d its fin d in g
T he th e o ry of tra c e s w as o rig in ated by A. M arzu rk iew icz in [4] a s an a tte m p t
to provide a good m a th e m a tic a l description of th e b e h a v io u r of c o n c u rre n t system s.
T he n o rm al form of a tra c e gives a n optim al way to p erform a p ro cess re p re se n te d
55


H oang Chi T hanh

56

by th e trace. In th is section we will c o n stru ct a sim ple ite ra tiv e alg o rith m for
finding th e n o rm al form of a trace.

2.1.

In depen den ce re la tio n a n d traces

D e f in itio n 2.1: L et E be an alp h ab et.
1 . A n independence relation over z is a sy m m etric a n d irreflexive b in a ry

relatio n over z.
2 . A reliance a lp h a b et is a couple c = (E, I), w here E is an a lp h a b e t and I is an

independence re la tio n over s.
C onsidering a d ja ce n t in d ep e n d en t sym bols in a s trin g to be com m uting, one
can re la te d ifferen t strin g s as follow.
D e f in itio n 2.2: L et c = (Z, I) be a reliance a lp h a b e t.
1. The relation =c c= Z*xZ* is defined as follow: for X, y e E*, X =c y if and only if
there exist Xj, x 2 € E* and (a,6) € I such that X = x j a b x 2 and y = x j b a x 2.
2 . The C -equivalence relation =c c

is defined a s th e le a s t equivalence

relatio n over E* co n ta in in g = c .
T hus, =c id en tifies ‘com m utatively sim ila r’ s trin g s — each such group of
strin g s is called a trace.
D e f in itio n 2.3: L et c = (£, I) be a reliance a lp h a b e t.
1. For an X e E*, t h e t r a c e o f X, denoted by [x]c , is the equivalence class of
containing X.

=c

2. A tr a c e over c is a set t of strings over E such th at t = [*]c for some X e £*
and X is called a representative of t .
3. A set of tra c e s over c is called a trace language over c .
E x a m p le 2.4: L et c = (E, I) be th e reliance a lp h a b e t given by th e u n d irected
g rap h in Fig. 1.
a

b

c

d

The trace t of abcecd = [abcecd]c = {abcecd, bacecd, acbecd, abeccdy abcced,
abcedcy baeccd, bacced, bacedc, acebcd, acbced, acbedc, aebccd, abecde, abccdel.

2.2.

N o rm a l form o f tra ces

One can define a com position operation on tra c e lan g u ag e.


“S h itf-left” a lg o r ith m s tr a n sfo r m in g seq u en tia l p rocesses..

57

D e f in i t io n 2.5: L et c —(£, I) be a reliance a lp h a b e t.
F o r tra c e s t h t2 over c , the trace com position of tj a n d t2, denoted by t,.t2, is
defined by t , t 2 = [Xj.x2]c> where Xj, x 2 € £* are representatives of tỵ and t 2
respectively.
In g en eral, a tra c e can be o b tain ed as a trac e com position of o th e r traces b u t
trace decom positions of th e given trace do not h ave to be unique. Hence it is
d esirab le to h a v e ‘n o rm al form ’ decom position of traces.
D e f in i t io n 2.6: L et c = (£, I) be a reliance a lp h a b e t a n d / be a trace over c .
The following decom position t = t,.t2. □ tm, such th a t:
1 . for a ll 1 s i s m, £, * 0 ;

2. for all 1 < i < m , t, can be w ritten as [«;], where Uị G £*, #a(u,)=l for each a e
alph(u,), and (a ,b ) e I for a ll a, b e alph(Uj) such th a t a # b ; an d
3. for a ll 1 s ỉ' s m -1, if Í, = [Uj] and tUj = [ui+1] th e n , for each a e a lp h ( u 1+i),

there exists b e alph(ii(), such that (a , b ) Ể I; is called the normal form of the trace t.
In [ 1 ] J . I. A a lb e rsb e rg an d G. Rozenberg poin ted o u t th a t every trace can be
uniquely decom posed in to a n o rm al form, i.e. a m in im al n u m b er of ‘m axim al
in d ep en d en t p a rts . T hey h a v e b u ilt two algorithm s for fin d in g th e norm al form of
traces. T he firs t a lg o rith m is b ased on in teg er p o in ters a n d th e second one is based
on dependence g ra p h s.
In th e n e x t su b sec tio n we b u ild a sim ple ite ra tiv e alg o rith m for finding the
norm al form of a tra c e from its re p re se n ta tiv e .

2.3.

A lg o rith m fo r f in d in g the n orm al form o f tra ces

We give a n in tu itiv e d e scrip tio n of th e algorithm . F ir s t we consider th e in p u t
strin g a s a se q u en ce of sin g le p a rts , every p a rt co n sists of one symbol. The
alg o rith m re p e a te d ly goes back on th e sequence of p a rts a n d sh ifts a symbol from a
p a rt onto th e p rev io u s p a r t if th e symbol is in d e p e n d e n t w ith every symbol
belonging to th e p rev io u s p a rt. W hen no symbol can be shifted, th e algorithm
term in a te s.
A lg o r ith m 2 .7 (“S h ift-le ft” a lg o rith m 1 ):
In p u t: A re lia n c e a lp h a b e t c = (£, I) and a s trin g w e z*.
D e cla ra tio n : L e t k = I w I , le t V be an a rra y of le n g th k over 2s and let u be an
a rra y of le n g th k over I*.
C o m p u ta tio n:
1 . for i := 1 to k do v(i) := {i£>[i]};
2 . 7':= 2 ;

3. re p e a t
4. for ev ery a € u(j) do
5. begin

H oang Chi T hanh

58
6 . i := j - 1; OK := false-,

7. w hile V b € v(i), (a ,b )

€

I do begin i:= i -1

8 . if OK th e n begin v(j)

:=

v(j) \ {o}; ư(i):= v(i) u {a}end;

;

OK :=true end;

9. end;
10 . if v(j) - 0 th e n ignore v(j) and d ecrease k := k - 1 ;

1 1 .j := 7 + 1 ;
12 . u n til j > k ;
13. for i := 1 to £ do u(i) :=

lin(u(i,)) ;

O utput: T he strin g s u (l), u(2),

u(k).

E x a m p le 2.8: L et c be th e reliance alp h a b et given in E xam ple 2.4 an d let
U) = a e cb b e d . Computing by the above algorithm, we have:
V : (a/, lei. I d , Ibl, Ibl, {el, id} ; k - 7
V : iq i, ie, cl, lb), ibl, lei, IdI ; k = 6
u : {a, bl, le, cl, (bl, lei, Idl ; k = 5
V : la, bl, le, c, bl, lei, [ d j ; k = 4
V : la, bl, le, c, b}, le, d l ; k = 3

Hence, th e o u tp u t of th e algorithm is u( l ) = ab , u(2) = bee , u(3) = de.
Formalizing the above we get the'following result.
T h e o r e m 2.9: L et c = (£, I) be a relian ce alp h a b et a n d le t w e £*. L et strin g s
u(l), u(2),..., u(k) be th e o u tp u t of A lgorithm 2.7 for th e in p u t (C,u;). T hen
[u(l)].[u(2)] ... [u(k)\ is th e norm al form of [w].
The alg o rith m is very sim ple and easy to im plem ent on com puter. It tra n sfe rs
a se q u en tial process re p re se n te d by a strin g into a c o n c u rre n t process. T he
com plexity of th is alg o rith m is 0 ( k 2).

3 . Concurrency in P/T nets
F irs t of all, we recall som e n o tatio n s concerning P e tri n ets.

3.1. P la ce/T ra n sitio n nets
A P etri net is a trip le N = (P, T, F), w here p, T a re d isjo in t se ts and F c ( P x T )
u (T X P) is a relatio n , so-called the flow relation of th e n e t N.
A n e t is sim ple if and only if its two different e le m e n ts h ave no common p re ­

se t and p ost-set. A sim ple n e t is used to re p re se n t s ta tis tic a l s tru c tu re of a system .
From a sim ple n et one can co n stru ct different n e t m odels by adding some
com ponents for re p re se n tin g dynam ical stru c tu re of th e system . The
Place/Transition net is such a net and is defined in [6] as follows:


“S h itf-le ft” a l g o r i t h m s tr a n s f o r m in g s e q u e n tia l p ro c e s se s..

59

D e fin itio n 3.1:
The 6-tuple X = (P, T, F, K, M°, W) is called a
rLace / T ra n sitio n net iff:
l . 7 í = (P, T, F) is a sim ple net, whereas an elem ent of p is called a p l a c e and

an elem ent of T is called a tra n sitio n .
2. K : p
N 3. w : F -> N \ {oc} is a function assig n in g a w eight on each arc of the flow
relation F.
:p
u ^ is an in itia l m arking, which is not g re a te r th a n capacity
on each place, i.e.: V p e p, M°(p) < K(p).
The in itial m arking represents given tokens on each place of a net. The tokens
a re no g re a te r th a n th e cap a city of the corresponding place. If to k en s on each place
belonging to the pre-set of som e transition are greater than or equal to w eight of the
arc connecting th is place to th e transition, i.e. it is enough for “paying” then the
nitial m ark in g can a c tiv a te th e corresponding tra n sitio n A fter perform ing the
Place. bei- 2 n g to the p r e se t of th is“

ion are

and tokens o 7 1
1
Kr
COnneCting ^ cor^ P o n d i n g place to th is tra n sitio n
ea
ce beIon^ ng t0 the p °s t -set of th is tra n s itio n a re increased by
weight of th e arc connecting th is tran sitio n to the corresponding place It m ust be
ensured that new tokens on each place are not greater than the O pacity of that place
perform edna n d t h in itia l T rk in g „activ ates som e tra n s itio n , th e tra n s itio n is
tra n s itio n ĩ n d
n we
a new m ark in g ’ th e new m a rk ing can a c tiv a te a n o th e ĩ
.
.
u
pr0ces! re p e a te d ly continues in such a way. T herefore, th e
ac ivities h a p p e n e d on a P/T n e t will be
m ath em atically form alized a s follows: ’
T h e m a r k i n g M : p -> N u {oc} can a c tiv a te a t r a n s i t i o n t iff:
1. V p e 't , M ip) > w (p, t) and

of t

2 V p 6 * ’ M (p^ ~ K (p^ " W ^ ' P^’ w here 't' t% a re th e p re -s e t an d th e po st-set

t.h J n
a l a s e ’ th e m a yk in g M is S0' called M o tiv a tin g . A fter perform ance of
th e tra n s itio n t, we get th e follow ing new m arking:

M(p) - W(p,t)

, if p

M ’fpJ = M (p) + W (t,p)

, if p e t' \ 't

M (p) - W(p,t) + W (t,p) , if p e ' t n t '
, otherwise

M (p )

and we often w rite th a t: M [í > M ’

»ím Ĩ. This, Zt .

°11 a,

SOrae 0ther, tra n siti0 n and th e n we s e t a " ° th “
m ark in g s reachable from th e m a rk in g M i s den o ted by
° sto ie spoce of * e " e t *

narkings and ĩ : Ĩ : F ’ r
,

11 is “ •

‘ f° r

T b " • P /T "e t' i * * M'’ M/ ........ M" be a “ «>>•“ * of
be a sequence of tra n s itio n s of th e n e t I , such th a t:


Hoang Chi T hanh

60

M‘ '[ í* > M ', ỉ = 1, 2, ... , n.
T he sequence M°[ t 1 > M ‘[ t2 > M 2 ... Mn l [ r > M n i l l u s t r a t e s aseq u en tial
process of th e n e t and th e word a = t 1 t2 ... tn e T * is called activities sequence on
th e n et.
The set of all activities sequences on the net £ is called the la n g u a g e g e n e r a t e d
by the net £ and denoted by L(z):
L(E) = { t ể t2 ... t n 13 M 1, M 2 , . . . , M n l, Mn e *[M °] :
M°[ t 1 > M‘[ t2 > M2 . . . Mn l[ tn > Mn}
B u t th e lan g u ag e g e n erate d by a n e t is seq u en tial. I t show s us only seq u en tial
perform ances of th e P/T n e t’s processes.
So w hen several tra n s itio n s of th e n e t can be perform ed concurrently.

3.2. Sequence o f m a x im a l concurrent steps
L et u c T be a su b se t of tra n s itio n s of th e n e t z and u * 0 .
D e f in itio n 3.2: T he su b set u is called a step on th e
m ark in g M 6 |«[M 0) satisfy in g th e following in equalities:
1 . V p e *u, M(p) > £

n e t I iff th e re is a

w (p, t) and

teU

2 . V p 6 u \ M (p) < K(p) - X

p )'

teU

In such a case, the transitions in the step u can be performed concurrently
and a fte r th e ir perform ance we get th e following m arking:
M(p)-ỵw(p,t)

ưu

M(p)

+

X

w (t’P)

, i f

p

6

u

,

p

e

u *

\

p

G

* u

n

i f

\

u

* u

M ’(p) =

M(p) - ỵ W(p,t)
tẻu

+

X

w(t>p)

tèu

M(p)
We also denote th a t: M[

u > M ’ and th e

, i f

u *

, otherwise
m arking M is called U -activatin g .

Such as above, we can find sequences of step s on th e n et. As big are th e steps
as hig h concurrency is.
E x a m p le 3.3: C onsider a P/T n e ts p resen te d by th e labelled directed b ip a rtite
g rap h in Figure 2.


s h i t f - l e f t a lg o r ith m s tr a n s fo r m in g s e q u e n tia l processes...

^

K

M°

Pi

3

3

P2

2

0

Ps

8

1

P4

5

5

Ps

4

1

F ig 2. A P/T net

On this net we recognize the following sequential process:
M°

[ tt >

M1

[h >

M2

[ t3 >

M3

[t4>

M4

Pi

3

1

2

2

2

P2

0

2

0

0

0

Pi

1

6

6

2

4

p4

5

2

2

2

0

Ps

1

1

4
----------The initial m arking M° can activate the subset {t„ t 4} and we have:
2

—

4

-

M° [ {tj, t4} > M r = ( 1 , 2 , 6 , 0 , 1 ).
So th e su b se t {th t 4} is a c o n cu rren t step on th is P/T net.
L et M°ỊUi > M 'fUa > M 2 ... Mk l[Uk > M k be a sequence of step s on th e n e t s.
The sequence illu strates a concurrent process on the net. What steps are “biggest”?
The following definition answ ers this question.
D e f in itio n 3.4: The sequence of step s M ^Ư ! > M 1[U2 > M 2 . . . Mk l[Uk> M k is
called a sequence o f m a x im a l concurrent steps iff for each i - 1 2 ... k-lVt

G

u i+1, M ''1 is not (UịUỊí}) -activating.

If transitions of each step can be performed concurrently, then the total time
for perform ance of th e process decreases considerably. T herefore, we alw ays expect
to find sequences of m aximal concurrent steps and at that time, the performance of
processes becomes optimal. In order to do so, we propose second “shift-left”
algorithm in the next section.


H oang Chi Thanh

32

ị

A lgorith m fin d in g s e q u e n c e s o f m a x im a l c o n c u r r e n t ste p s

Let s = (P T F, K, M°, W) be a P/T net. Each se q u en tial process: M °[i; > MUi 2
> M2 .
> Mk of th e n e t m ay be considered as a sequence of single steps:

M°[{*;} > M '[{í2} > M 2 . . ,Mk l [{£*} > Mk and as an in p u t of our alg o rith m .
The algorithm repeatedly goes back on the sequence of steps and shifts a
t r a n s i t io n from a s te p in to t h e p rev io u s ste p if w h en a d d in g th e t r a n s i t i o n to th is
previo u s

step,

the

o b ta in sd

step

becom GS

activ a ted

b y

the

co rresp o n d in g

m arkin g.

When no transition can be shifted, the algorithm term inates.
A lg o r ith m 4.1 (“S h ift-left” alg o rith m 2):
I n p u t: A se q u e n tia l process M°[£;> M'[£2> M 2 . . . M k *[£*> M k on a P/T net.
Output'. A sequence of m axim al c o n cu rren t step s on th e net.
1 . for i := 1 to k do u , := {£,} ;

2 . for j := 1 to k-1 do

3. begin
4. i := j ;
5 . w hile i >= 1 do

begin
for every tra n s itio n t e U i+1,
if MM is (U ,u{i})-activating then
begin we replace:

M l l (p)-

I

W(p,t)

, if p 6 *(U, uW ) \ (U,

W(t,p)

, if p € (U,
tẻ ư ịẩ ịtì

M i i (p)+

ỵ
tẻ U ịẩ ltl

M ‘(p) =
M i l (p)-

ỵ
tè U ịẩ ltl

M l I (p)

W(p,t)+

ỵ

W (t,p)

, if p e *(U, u{í}) n (U,

tè U ịẩ lt)

, otherwise

and u, := UịU{í} , U i+1 := ui+1\{i} , i.e. we “sh ift-left” th e tra n s itio n t from the
ỉtep u +1 into the previous step Uj. After shifting, if ui+
1=0 then we ignore both ui+1
____1 A /ĩi+ 1 .
and
Ml+1;

end ;
i := i - 1
end :

6. end ;
7. stop.
The algorithm goes back because we w ant not only m aking steps maximal
concurrent but also reducing the number of steps.


“S h itf-le ft” a lg o r ith m s tra n sform in g seq u en tia l p r o c e s s e s ..
Example 4 .2 : We apply this algorithm to the P/T net drawn in Fig. 2 with tltu
input M°[íi > M l [t2 > M 2[t3 > M 3[t4 > M4.
- After 1st going back:
The m arking M° is {tj, t2}- activating, so we shift left t 2 and get the follow ing
sequence:
M°

[

>

M1

M2

[ {t3} >

[ {t4} >

M3

Pi

3

2

2

2

P2

0

0

0

0

p3

1

6

2

4

p4

5

2

2

0

Ps

1

2

4

4

------1--------1

- A fter 2 nd going back:
The m arking M 1 now is {t3, t 4}- activating, so we shift left t 4 and obtain th e
following sequence:
M°

[ {^2^ 2} >

M1

[

>

M2

Pi

3

2

2

p2

0

0

0

Ps

1

6

4

p4

5

2

0

Ps

1

2

4

The obtained sequence M° [{tlft 2} > M l [{t3, t 4) > M2 is indeed a sequence of
maximal concurrent steps of the net..

Theorem 4 .3 : When Algorithm 4.1 term inates, its output is the sequence of
maximal concurrent steps of the P/T net.
The com plexity of this algorithm is 0 ( 1 T I. Ip I.k2). So the complexity is square
in the number of steps. This algorithm is sim ple and easy to im plem ent on
computers.

5 . Conclusion
In the paper, we construct one more a very sim ple algorithm for finding the
normal form of a trace; propose the notation of concurrent step on a P/T net and
build up an efficient algorithm to transform sequential processes of a P/T net into
concurrent ones. These algorithm s are not only useful for concurrency control on
system s but also suitable for calculating concurrent behaviours of a system with

H o a n g Chi Thanh

(34

jv n a m ic a l s tru c tu re s . T he a lg o rith m s m ay be applied for processing tra n sac tio n s on
d a ta b a se , for fin d in g th e norm al form of a sem i-trace.
A c k n o w le d g e m e n t-. T his p a p er w as w ritte n d u rin g my sta y a t De M ontfort
U n iv ersity, L eicester, UK. I would like to th a n k P rofessor Hongji Yang, Dr. D ang
Van H u n g an d th e IIST /U N U for my v aluable tim e a t L eicester.

R eferen ces
1.

J .I . A alb ersb e rg a n d G. R ozenberg, Theory o f Traces, T h eoretical C om puter
Science 60(1988), pp. 1-82.

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