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“SHIFT-LEFT” ALGORITHMS TRANSFORMING SEQUENTIAL PROCESSES INTO CONCURRENT ONES

H oang Chi T hanh

H a n o i 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 IN TR O D U C TIO N

C ontrols on c o n c u rre n t sy stem s alw ays a re a n im p o r ta n t a n d com plicated problem So far, o p tim a l p erfo rm an ce of processes o ccu rred on a c o n c u rre n t system

is am ong th e controls F o r th is pu rp o se, recognition of c o n c u rre n t p ro cesses of th e system is e ss e n tia l A lo t of m odels, such as P e tri n e ts [6,9], tr a c e s [1,3,4,8], CSP [7], CCS [5], process a lg e b ra s [2] have been b e in g good to o ls for re p re s e n tin g

s ta tis tic a l as w ell as d y n a m ic a l s tru c tu re of sy stem s F rom th e s e m odels, se q u e n tia l processes of a sy stem 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 s fo rm se q u e n tia l processes of a sy ste m o n to c o n c u rre n t ones

is a g re a t pro b lem on th e o ry a n d ap p licatio n indeed

A sim p le a lg o rith m for fin d in g th e n o rm al form of tr a c e s is a good so lu tio n for

sy stem s re p re s e n te d by re lia n c e a lp h a b et P la c e /T ra n s itio n n e t is one of su ita b le

m odels to r e p r e s e n t c o n c u rre n t sy stem s B u t th e la n g u a g e g e n e r a te d by a P/T n e t is

se q u en tial I t show s u s only se q u e n tia l p erfo rm an ces of th e n e t ’s p rocesses So

w hen se v era l tr a n s itio n s c a n be perform ed c o n cu rren tly W e w ill c o n c e n tra te on

b u ild in g a n ite ra tiv e a lg o rith m for tra n sfo rm in g s e q u e n tia l p ro c e sse s of a P/T n e t

in to c o n c u rre n t ones

T his p a p e r is o rg a n iz e d as follows Section 2 c o n s tru c ts a “sh ift-le ft” alg o rith m for finding th e n o rm al form of a trace In Section 3 we p ro p o se th e n o ta tio n of concurrency in P/T n e ts Section 4 b u ild s up a n ite r a tiv e a lg o rith m for fin d in g sequences of m ax im al c o n c u rre n t ste p s on a P la c e /T ra n s itio n n e t from th e n e t’s sequence of sin g le ste p s F in a lly , som e conclusions a re given in S ectio n 5

2 N o rm a l fo r m o f t r 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 a te d by A M a rz u rk ie w ic z in [4] a s an a tte m p t

to provide a good m a th e m a tic a l d escrip tio n 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 tr a c e gives a n o p tim al w ay to p erfo rm a p ro ce ss re p re se n te d

55

56 H oa n g Chi T h a n h

by th e trace In th is sectio n we will c o n stru c t a sim p le ite r a tiv e a lg o rith m for finding th e n o rm al form of a trace

2.1 In d e p e n d e n c e r e la tio n a n d tr a c e s

D e f in i t io n 2.1: L et E be a n a lp h a b et

1 A n independence relation over z is a sy m m etric a n d irreflex iv e b in a ry

re la tio n over z

2 A reliance a lp h a b et is a couple c = (E, I), w h ere E is an a lp h a b e t a n d I is an

in d ep en d en ce re la tio n over s

C o n sid erin g a d ja c e n t in d e p e n d e n t sym bols in a s tr in g to be co m m u tin g , one can re la te d ifferen t s trin g s as follow

D e f in i t io n 2.2: L et c = (Z, I) be a relian ce 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 T he C -equivalence relation =c c is d efin ed a s th e le a s t eq u iv alen ce

re la tio n over E* c o n ta in in g = c

T h u s, =c id e n tifie s ‘co m m u tativ ely sim ila r’ s trin g s — each such group of strin g s is called a trace

D e f in i t io n 2.3: L et c = (£, I) be a relian ce 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 =c containing X.

2 A t r a c e over c is a set t of strings over E such th a t t = [*]c for some X e £*

and X is called a rep resen ta tive of t

3 A se t of tra c e s over c is called a trace la n g u a g e o v er c

E x a m p le 2.4: L et c = (E, I) be th e relian ce a lp h a b e t given by th e u n d irec te d

g ra p h in Fig 1

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

2.2 N o r m a l form o f tr a c e s

One can d efine a com position o p eratio n on tra c e la n g u a g e

D e f i n i t i o n 2.5: L e t c — (£, I) be a relian ce a lp h a b e t.

F o r tra c e s t h t 2 over c , the trace com position of tj a n d t2, d en o ted by t,.t2, is

defined by t , t 2 = [Xj.x2]c> w here Xj, x 2 € £* are rep resen tatives of tỵ and t 2

respectively

In g e n e ra l, a tra c e c an be o b ta in e d as a tra c e com position of o th e r trac es b u t trac e d eco m p o sitio n s of th e given tra c e do not h av e to be u n iq u e Hence it is

d e sirab le to h a v e ‘n o rm a l fo rm ’ decom position of trac es

D e f i n i t i o n 2.6: L e t c = (£, I) be a relian ce a lp h a b e t a n d / be a tra c e over c

T he follow ing d eco m p o sitio n t = t ,.t 2 □ 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 [«;], w here Uị G £*, #a(u,)=l for each a e

alph(u,), an d (a ,b ) e I for a ll a, b e alph(Uj) such t h a t a # b ; a n d

3 for a ll 1 s ỉ' s m -1, if Í, = [Uj] and t Uj = [ui+1] th e n , for each a e a lp h ( u 1+i),

there ex ists b e alph (ii(), such th a t (a , b ) Ể I; is called the norm al form of the trace t.

In [1] J I A a lb e rsb e rg a n d G R ozenberg p o in te d o u t t h a t every trac e can be

u n iquely decom posed in to a n o rm a l form, i.e a m in im a l n u m b e r of ‘m axim al

in d e p e n d e n t p a rts T hey h a v e b u ilt two a lg o rith m s for fin d in g th e n o rm al form of traces T he f ir s t a lg o rith m is b a se d on in te g e r p o in te rs a n d th e second one is b ased

on d ependence g ra p h s

In th e n e x t su b se c tio n we b u ild a sim ple ite r a tiv e a lg o rith m for finding th e norm al form of a tra c e from its re p re s e n ta tiv e

2.3 A lg o r ith m fo r f i n d i n g the n o rm a l fo rm o f tr a c e s

We give a n in tu itiv e d e sc rip tio n of th e a lg o rith m F ir s t we co n sid er th e in p u t

s trin g a s a se q u e n c e of sin g le p a rts , every p a r t c o n sists of one symbol T he

a lg o rith m re p e a te d ly goes b ack on th e sequence of p a r ts a n d sh ifts a sym bol from a

p a rt onto th e p re v io u s p a r t if th e sym bol is in d e p e n d e n t w ith every sym bol belonging to th e p re v io u s p a rt W hen no sym bol can be sh ifted , th e alg o rith m

te rm in a te s

A l g o r i t h 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) a n d 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 a n d 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 e v ery a € u(j) do

5 begin

58 H oan g Chi T h a n h

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

7 w hile V b € v(i), (a ,b ) € I do begin i := i - 1 ; OK := true end;

8 if OK th e n begin v(j) := v(j) \ {o}; ư(i) := v(i) u {a} end;

9 end;

10 if v(j) - 0 th e n ignore v(j) an d 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 h e strin g s u (l), u(2), u(k)

E x a m p le 2.8: L et c be th e relian ce a lp h a b e t given in E x am p le 2.4 a n d let

U) = a e c b b e d Com puting 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 alg o rith m is u( l ) = ab , u(2) = bee , u(3) = de

Form alizing the above we get the'following result.

T h e o r e m 2.9: L et c = (£, I) be a relian c e a lp h a b e t 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 n o rm al form of [w].

The a lg o rith m is very sim ple an d easy to im p lem en t on co m p u ter I t tra n s fe rs

a se q u e n tia l process re p re se n te d by a s trin g in to a c o n c u rre n t process T he

com plexity of th is a lg o rith m is 0 ( k 2).

F ir s t of all, we recall som e n o ta tio n s concerning P e tr i n e ts

3.1 P la c e /T r a n s itio 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 a n d F c ( P x T )

u (T X P) is a re la tio n , so-called the flow relation of th e n e t N.

A n e t is sim ple if a n d only if its two d ifferen t e le m e n ts h av e no common p re ­

se t and p o st-set A sim p le n e t is used to re p re se n t s ta tis tic a l s tr u c tu r e of a system From a sim ple n e t one can co n stru ct d ifferen t n e t m odels by adding some

co m p o n en ts for re p re s e n tin g dynam ical s tru c tu re of th e system T he

Place/Transition net is such a net and is defined in [6] as follows:

D e fin itio n 3.1: The 6-tuple X = (P, T, F, K, M°, W) is called a

an elem ent of T is called a tra n sitio n

2 K : p N <u {cc} is a function showing a capacity on each place.

3 w : F -> N \ {oc} is a function a ssig n in g a w eight on each a rc of th e flow

relation F.

: p u ^ is a n in itia l m a rkin g , w hich is n o t g r e a te r th a n cap acity

on each place, i.e.: V p e p, M°(p) < K(p).

The in itia l m arking rep resen ts given tokens on each place of a net The tokens

a re no g r e a te r th a n th e c a p a c ity of th e co rresp o n d in g place I f to k e n s on each place

belonging to the pre-set of som e transition are greater than or equal to w eigh t of the

arc connecting th is place to th e tran sitio n , i.e it is enough for “p ay in g ” th en th e

n itial m a rk in g can a c tiv a te th e corresponding tra n s itio n A fter p erfo rm in g th e

Place bei- 2 n g to the p r e s e t of th is “ ion are

an d tokens o 7 1 1 K r COnneCting ^ cor^ P o n d i n g place to th is tra n s itio n

ea ce beIon^ ng t0 th e p ° s t -se t of th is tra n s itio n a re in creased by

w eight of th e a rc co n n ectin g th is tra n sitio n to th e co rresp o n d in g place I t m u st be

ensured that new tokens on each place are not greater than the O p a city of that place

perform edna n d t h in itia l T r k in g „a ctiv a tes som e tra n s itio n , th e tr a n s itio n is tra n s itio n ĩ n d n we a new m a rk in g ’ th e new m a rk in g can a c tiv a te a n o th e ĩ

ac iv ities h a p p e n e d on a P/T n e t will be m a th e m a tic a lly fo rm 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) a n d

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 a n d th e p o st-se t

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 p erfo rm an ce of

th e tra n s itio n t, we g et th e follow ing new m arking:

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

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

SOrae 0 th er, tra n s iti0 n a n d th e n we s e t a " ° th “

»ím Ĩ This, Z t °11 a, m a rk in g s reach ab le from th e m a rk in g M i s d e n o te d by

narkings and ĩ : Ĩ : F ’ r T b " • P /T "e t' i * * M' ’ M/ M" be a “ «>>•“ * of

, be a seq u en ce of tra n s itio n s of th e n e t I , su ch th a t:

60 H oang Chi T h an h

M‘ '[ í* > M ', ỉ = 1, 2, , n.

T h e seq u en ce M°[ t 1 > M ‘[ t2 > M2 Mn l [ r > M n i l l u s t r a t e s a se q u e n tia l

process of th e n e t a n d th e w ord a = t 1 t2 tn e T * is called a ctivities sequence on

th e n et

The set of all a ctivities sequences on the net £ is called the l a n g u a g e g e n e r a t e d

by the n et £ a n d d en o ted by L(z):

L(E) = { t ể t 2 t n 13 M 1, M2, , M n l, Mn e *[M °] :

M°[ t 1 > M ‘[ t2 > M2 Mn l [ tn > Mn}

B u t th e lan g u a g e g e n e ra te d by a n e t is se q u en tial I t show s us only se q u e n tia l

p erfo rm an ces of th e P/T n e t’s processes

So w hen se v era l tra n s itio n s of th e n e t can be perform ed co n currently

3.2 Sequence o f m a x i m a l co n cu rren t ste p s

L et u c T be a s u b s e t of tra n s itio n s of th e n e t z an d u * 0

D e f i n i t i o n 3.2: T h e su b se t u is called a step on th e n e t I iff th e re is a

m ark in g M 6 |«[M 0) sa tisfy in g th e follow ing in eq u alities:

1 V p e *u, M(p) > £ w (p, t ) an d

teU

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

teU

In such a case, the tran sitions in the step u can be performed concurrently

an d a fte r th e ir p e rfo rm an c e we get th e follow ing m arking:

M ( p ) - ỵ w ( p , t )

ưu , i f p 6 u \ u

M(p) + X w (t’P) , i f p e u * \ * u

M(p) - ỵ W(p,t) + X w (t>p) , i f p G * u n u *

M ’(p) =

W e also denote th a t: M[ u > M ’ a n d th e m ark in g M is called U -activ atin g Such as above, we can find sequences of ste p s on th e n e t As big a re th e steps

a s h ig h concurrency is

E x a m p le 3.3: C o n sid er a P/T n e ts p re s e n te d by th e lab elled d irected b ip a rtite

g ra p h in F ig u re 2

K M°

On this net we recognize the following sequential process:

[ t 4 > M4

0

— - - 4

The in itial m arking M° can activate the subset {t„ t4} and we have:

M° [ {tj, t4} > M r = (1, 2, 6, 0, 1)

So th e su b s e t {th t 4} is a c o n c u rre n t step on th is P/T net.

L et M°ỊUi > M 'fU a > M2 Mk l [Uk > M k be a sequence of ste p s on th e n e t s

The sequence illu stra tes a concurrent process on the net What steps are “biggest”? The following definition answ ers this question.

D e f i n i t i o n 3.4: T he seq u en ce of ste p s M ^Ư ! > M1[U2 > M2 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 kl

-V t G u i+1, M''1 is not (UịUỊí}) -activ atin g

If tran sition s of each step can be performed concurrently, then the total time

for p erfo rm an ce of th e process d e crea se s considerably T herefore, we alw ays expect

to find sequences of m axim al concurrent steps and at that tim e, the performance of processes becomes optim al In order to do so, we propose second “sh ift-left” algorithm in the next section.

32 H o a n g Chi T han h

ị A lg o rith 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 s te p s

Let s = (P T F, K, M°, W) be a P/T n et Each se q u e n tia l process: M °[i; > MUi2

> M2 > Mk of th e n e t m ay be considered as a seq u en ce of sin g le step s: M°[{*;} > M '[{í2} > M2 ,Mk l [{£*} > Mk a n d as a n in p u t of o u r a lg o rith m

The algorithm repeatedly goes back on the sequence of step s and shifts a

t r a n s i t i o n from a s te p in to t h e p r e v io u s s te p if w h e n a d d i n g t h e t r a n s i t i o n to t h i s

p r e v i o u s s t e p , t h e o b t a i n s d s t e p b e c o m G S a c t i v a t e d b y t h e c o r r e s p o n d i n g m a r k i n g When no transition can be shifted, the algorithm term inates.

A lg o r ith m 4.1 (“S h ift-le ft” a lg o rith m 2):

I n p u t: A s e q u e n tia l process M°[£;> M'[£2> M2 M k *[£*> M k on a P/T n et

Output' A seq u en ce of m ax im al c o n c u rre n t ste p s on th e n et.

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 tr a n s itio n t e U i+1,

if MM is (U ,u{i})-activatin g then begin we replace:

M ‘(p) =

tẻ ư ịẩ ịtì

tẻ U ịẩ ltl

tè U ịẩ ltl tè U ịẩ lt)

M l I (p)

, if p 6 *(U, u W ) \ (U,

, if p € (U, <j{t}Y \ *(Ui u{i})

, if p e *(U, u{í}) n (U,

, otherwise

an d u, := UịU{í} , Ui+1 := ui+1\{i} , i.e we “sh ift-le ft” th e tr a n s itio n t from the

ỉtep u +1 into the previous step Uj After shifting, if u i+ 1=0then 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 step s maximal concurrent but also reducing the number of steps.

Example 4.2: We apply this algorithm to the P/T net drawn in Fig 2 with tltu inp ut M°[íi > M l [t2 > M 2[t3 > M 3[t4 > M4.

- After 1st going back:

The m arking M° is {tj, t 2}- activating, so we sh ift left t 2 and get the follow ing

sequence:

- A fter 2nd 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:

The obtained sequence M° [{tlft 2} > Ml [{t3, t 4) > M2 is indeed a sequence of

m axim al concurrent steps of the net

Theorem 4.3: When Algorithm 4.1 term inates, its output is the sequence of

m axim al concurrent step s of the P/T net.

The com plexity of th is algorithm is 0 ( 1 T I I p I k 2). So the complexity is square

in the num ber of steps This algorithm is sim ple and easy to im plem ent on computers.

In th e 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 efficien t algorithm to transform sequ en tial processes of a P/T net into concurrent ones T hese algorithm s are not only useful for concurrency control on system s but also su itab le for calculating concurrent behaviours of a system with

(34 H o a n g Chi Thanh

jv n a m ic a l s tr u c tu r e s T he a lg o rith m s m ay be ap p lied for p ro cessin g tra n sa c tio n s on

d a ta b a s e , for fin d in g th e n o rm al form of a sem i-trace

A c k n o w le d g e m e n t- T h is p a p e r w as w ritte n d u rin g my sta y a t De M ontfort

U n iv ersity , L e ic e ste r, UK I would like to th a n k P ro fesso r H ongji Y ang, Dr D ang Van H u n g a n d th e IIST /U N U for my v a lu a b le tim e a t L eicester

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