The Semantics of Data Flow Diagrams. P.D. Bruza. Th.P. van der Weide. Dept. of Information Systems, University of Nijmegen

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1 The Seantics of Data Flow Diagras P.D. Bruza Th.P. van der Weide Det. of Inforation Systes, University of Nijegen Toernooiveld, NL-6525 ED Nijegen, The Netherlands July 26, 1993 Abstract In this article we rovide insight as to how seantics can be attached to Data Flow diagras. We rst resent a ethod for transforing a Data Flow Diagra(DFD) to a Petri-Net (PT-net) which secies the synchronization asects of a DFD. Secondly, we sketch how a Data Flow diagra ay be transfored to exressions whose seantics are described in ters of nite autoata. Published as: P.D. Bruza and Th.P. van der Weide. The Seantics of Data Flow Diagras. In N. Prakash, editor, Proceedings of the International Conference on Manageent of Data, Hyderabad, India, Introduction In this docuent we resent a way in which seantics can be attached to Data Flow diagras (see [4] or [9]). DFD's often for an iortant role in the design of inforation systes. Their intention is to odel the rocess asects of an inforation syste (IS). They are often used in the rst hases of inforation analysis to establish a global odel of an inforation syste which can be further rened. In this docuent we use a rigid foralis to describe the (dynaic) behaviour of the concets that are used to secify rocesses and their relations. Usually this dynaic behaviour is considered to be self exlaining. When building inforation systes, however, there is a need to be very recise about these asects. Without recision, the odeller can have no clear idea of the eaning of the secication under develoent. The intention of this aer is to foralise the intuition behind rocess interaction in DFD's with the end that these can 1

2 be used ore recisely by the designer. For the ileentor, a recise secication fors a ore secure latfor on which to base an ileentation. Figure 1: External entity Figure 2: Data Flow Figure 3: Process We will now give a brief introduction to the structure of a DFD. A DFD is a diagra built u fro eleentary building blocks. These are denoted in gures 1 to 4. An exale of how these eleentary coonents can be connected to for a DFD is given in gure 5. A DFD can be viewed as directed grah wherein the nodes are external entities, rocesses or data stores and the edges are data ows. There exist rules for the construction for DFD's. For exale, there can be no data ow fro a data store directly to another data store. We will not deal further with the structure of a DFD. A foral secication of DFD's is given in [3]. The intention of a DFD is to describe an asect of the inforation syste at a articular abstraction level. A useful feature of DFD's is that a rocess can be further rened by another DFD. This DFD is a ore detailed descrition of the rocess at a lower level of abstraction. DFD's are a useful descrition echanis for odelling rocess architecture, but they do not allow sucient inforation to give a clear interretation of what is being described. This roble is already identied in [1]. We resent the roble on the basis of the DFD given in gure 6. This DFD has ultile interretations. For exale, 1. If rocess P receives 1 and 2, then P is activated and roduces both 3 and If P receives 1 alone, it is activated and roduces If P receives 2 alone, it is activated and roduces 4. The above are not the only interretations ossible. It is obvious that the DFD's ust be enriched with further inforation so that they can clearly be interreted. This is the toic of the following section. 2

3 d Figure 4: Data store e q 4 d r 5 7 f Figure 5: Exale of DFD Figure 6: Proble DFD 3

4 2 Enhanceent of DFDs In this section we resent a nuber of extensions to DFDs to increase their inforation content. This is iortant because it is necessary to have a clear interretation of a DFD before we are in osition to secify its seantics. The rst enhanceent we resent is the transforation of DFDs to a variant of Petri-Nets ([8]). We follow [1]. We have chosen Petri-Nets because they allow synchronization asects to be straightforwardly odelled. They also are suorted with an easy to understand grahical notation that can easily be incororated into a DFD. The transforation of a DFD to a Petri-Net can be accolished by transforing every data ow as in gure 7. The lace holder allows the odelling of essage ow along the data ow. For exale, Figure 7: Transforation of a Data Flow the ossible interretations cited with the DFD of gure 6 are deicted in gure 8. The resulting Petri-Net is an ad hoc variant of Petri-Nets. This is because in ure Petri-Nets all inut laces ust have tokens before the transition can re. Fro the above we see that this is not always the only way in which a rocess can re. To solve this we adot the view of [1]. In this solution we attach to each rocess its activation ossibilities. In the exale above we can denote this as follows: : ( 1 ^ 2! 3 ^ 4 ) : ( 1! 3 ) : ( 2! 4 ) For the exale of gure 5 this leads to the PT-net of gure 9, which we also enriched with its activation ossibilities. We ake soe further enhanceents to the Petri-Nets. In any cases we wish to odel that a data ow can suort a nuber of essages at the sae tie. This is called the caacity of the data ow and is denoted as in gure 10. A rocess ay `read' ore that one essage at a tie fro an incoing data ow. We ter this as the the deand of the rocess on that data ow, and is denoted in gure 12. This gure deicts the deand of rocess on dataow as being x essages. Conversely, we introduce the notion of the yield of a rocess onto a articular data ow. This reresents the nuber of essages `written' at the sae tie onto a data ow by a given rocess. Note that if rocess P 1 sends inforation to rocess P 2 via data ow and further the yield of P 1, (yield(p 1 ; )) or deand(p 2 ; ) is less that the caacity of, then this is a deiction of the roducer-consuer roble. The data ow is actually a buer. 4

5 Figure 8: Activation Possibilities e 6 q d : ( 1 2 ) : ( 3 2 ) : ( 6 2 ) q: ( 2 5 ) q: ( 2 4 ) r 7 f r: ( 5 6 ) r: ( 5 7 ) Figure 9: Exale of DFD transfored into PT-net 5

6 c Figure 10: Caacity x Figure 11: Activation Possibilities Figure 12: Deand Soeties it useful to secify an ordering on the buer. For exale, FIFO in which case the buer is a stack, or LIFO, in which case the buer acts like a queue. Finally, we introduce the notion of the load of a data ow. This is the nuber of essages currently on the data ow. The activation of a rocess can now be dened as follows: Denition If k ^ :: ^ l! z ^ :: ^ z q is an activation ossibility of rocess P, where P is not active, and further, for all inut data ows i (k i l) the condition load( i ) deand(p; i ) holds, then rocess P is tered activated and load( i ) := load( i )? deand(p; i ). This denition corresonds to the idea of the triggering of a rocess. The trigger condition is that there is sucient inut stiulus to wake the rocess. The de-activation of a rocess can be conceived of as the rocess having done its work sending its oututs and going to slee. Denition If a rocess P has been activated by the activation ossibility k ^ ::^ l! z ^ :: ^ z q and for all outut data ows z j ( j q) the condition load(z j ) + yield(p; z j ) caacity(z j ) holds, then rocess P de-activates and load(z j ) := load(z j ) + yield(p; z j ). Note: If a rocess P is activated it ust be de-activated before it can be re-activated. 6

7 This coletes the list of enhanceents to a DFD. The result basically is a PT-net but with a dierent ring echanis. Fro this oint on we will refer to this net as a rocess structure (PS). 3 DFDs as Path Exressions In this section we resent a ethod to convert a rocess structure (the PT-net for of a DFD) to an algebraic exression. This is a convenient reresentation of rocess structures (including its seantics). An advantage of these exressions is that they can be aniulated algebraically. This is helful for exale when establishing rocess structure equivalence. This reresentation will also be helful for the transforation of a rocess structure to ileentation in ters of a set of rogras. The syntax of the exressions is based on the language COSY ([6]) which was develoed for secifying systes of concurrent rocesses sharing distributed systes of resources. The intention of this language was an ileentation indeendent vehicle with which concurrent and distributed algoriths for oerating syste functions could be dened. For this reason the language is caable of secifying colex rocess interaction, but the full ower of the language is not used in connection with DFDs because these are only caable of secifying rather sile rocess interaction. The seantics of COSY is given in ters of PT-nets. In [5] it is shown how every COSY construct can be exressed as a PT-net. In this article we investigate the other direction, naely to acquire a COSY-like exression fro the rocess structure as dened in the revious section. We start with a short introduction to the syntax and seantics of the kind of exressions that will be generated. The work is based on [2], [5] and [6]. We distinguish four tyes of rocess synchronization: sequencing, selection, reetition and concurrency. All of these can be exressed by a ath exression as follows: sequencing A sequence of rocesses eans that the rocesses ust execute in the order given. Given that ; q; r are rocesses, then we denote sequentialization with a `;'. For exale, ath exression ; q; r eans that ; q; r are to be executed one after the other in the order given. selection Selection fro a set of rocesses erits only one to activate. Suose the executions of rocesses ; q and r are to be selectively synchronized then this is denoted by the ath exression ; q; r. The above two synchronization schees are fundaental and can be cobined to secify colexer synchronization schees. For exale, ; (q; r); s. This ath exression denotes two ossible scenarios: Either executes before q before s, or before r before s. Next we introduce the concets reetition and concurrency: 7

8 reetition This allows the secication of ath exressions that reeat. This is denoted as P, if rocess P can be reeated a zero or ore ties, or P + if P can be reeated one or ore ties. If the rocess never stos then P 1 is used. Reetition is used to secify loos in the PS. concurrency A ath exression describes a sequential syste. Concurrency is odelled by concurrent sequential systes denoted P k Q, where P and Q are ath exressions. For exale, ; r k q; r secies that the rocesses and q execute concurrently before rocess r is invoked. 3.1 The Seantics of Path Exressions A eaning can be attached to each ath exression in the following sense: A ath exression can be viewed as a nite autoaton. In the case of arallel ath exressions, this can be considered as nite autoatons that are oerating concurrently. In short, the seantics can be equated with the language dened by the autoaton in question. We denote this as follows given that is a rocess and P; Q are ath exressions: L() = fg L(P; Q) = L(P ) [ L(Q) L(P ; Q) = L(P )L(Q) L(P ) = L(P ) L(P + ) = L(P ) + L(P 1 ) = all innite strings over L(P ) There is a roble with denoting the seantics of arallel ath exressions. We adot the convention of [6], naely we write the denotations of the individual ath exressions on to of each other. For exale, L(; r k q; r) is denoted as: q r The advantage of secifying the seantics in ters of the languages roduced by the autoatons is that a articular string fro the language can be viewed as a history. We can therefore discuss about the behaviour of DFDs in ters of such histories. We therefore ust be able to reduce a given DFD to a ath exression. This is the toic of the following section. 3.2 Translation of Process Structures to Path Exressions In this section we show how the four tyes of synchronization can be identied in rocess structures. 8

9 sequencing This corresonds with rocesses that are connected via a data ow. If rocess sends essages to rocess q via data ow, then this can be reresented by the ath exression ; q. selection This can occur in two ways in a PS: Selection can arise because a rocess ay re in ore than one way. This often occurs as a selection between rocesses that ay follow a given rocess deending on which way this rocess res. The second way in which selection can occur corresonds with conict situations (in the PT-net sense). These situations are generalized by the PS of gure 13. The q 1 q n Figure 13: Conict Situation conict situation is deterined by the condition caacity() < deand(q 1 ; ) + :: + deand(q n ; ) arallelis All rocesses that are not connected directly or indirectly are considered to be in arallel. reetition Reetition corresonds with loos in the PS. Loos disrut the sequencing by introducing arallelis between rocesses that are directly or indirectly related. We deonstrate this with the exale given gure 14. We consider two cases: 1. The activation ossibilities of q are: q : ( 1! 2 ) q : ( 1! 3 ) Then the corresonding ath exression is: 2. Process q has activation ossibility: q : ( 1! 2 ^ 3 ) (; q) + ; r Then the corresonding ath exression is: ; q; (( k r); q) + 9

10 1 3 q 2 r Figure 14: Exale of a loo 4 A Basis for PS Equivalence The ath exressions introduced in the revious section also have the advantage that they can be aniulated algebraically. This can be used in establishing rocess structure equivalence. We now give the roerties on the ath exression algebra. The reader should kee in ind that the riorities of the ath oerators are as follows (in increasing order): 1., 2. ; k 3. + This eans that + is ost binding, followed by (; k), followed by,. We use arentheses to establish riority in the usual way. Then we have the following equivalence rules for ath 10

11 exressions. P k Q = Q k P (Coutativity) P; Q = Q; P (P ; Q); R = P ; (Q; R) (Associativity) (P; Q); R = P; (Q; R) (P k Q) k R = P k (Q k R) P ; (Q; R) = P ; Q ; P ; R (Distributivity) P k (Q,R) = P k Q ; P k R P k (Q;R) = (P k Q); R We now give an exale of how the above algebra can be alied. Figure 14 with the following activation ossibility for rocess q: Consider the PS of q : ( 1! 2 ^ 3 ) We can dene a set of ath exression equations based on the DFD. We then solve these equations which results in a ath exression for the DFD. The setting u of the equations can be done in a systeatic way. For exale, the DFD begins with the invocation of rocess followed by the rest. We odel this in the equation P = ; Q where P is the ath exression which describes the DFD and Q is the ath exression which odels the art of the DFD following rocess. Such an equation can be considered as a graar rule for the language L(P ). We can now further dene Q. This begins with rocess q and due to the activation ossibility of q arallelis occurs between P and the rest of the DFD which we odel as R. Q is thus dened as: Q = q; (P k R) Note that if rocess q had the activation ossibilities q : ( 1! 2 ) and q : ( 1! 3 ), then selection would have occurred. The equation for Q would in this case be: Q = q; (P; R) We now dene equation R. This is sily r. On the basis of the equation P; Q and R, we can substitute and get an equation for P which we solve in the following way: P = ; q; (P k r) = ; q; (;q;(p k r) k r) = ; q; (( k r); q; (P k r)) Distributivity = ; q; (( k r); q; (;q;(p k r) k r)) = ; q; (( k r); q; ( k r); q; (P k r)) = ; q; (( k r); q) 1 11

12 Note how the rocess structure described by P never halts. We now wish to introduce the following denition: Denition Two rocess structures are dened equivalent if they can be reduced to ath exressions with the sae seantics. Consider the PS of Figure 9. This has the followind set of ath equations: This results in the following ath exression: P = e; Q Q = ; R R = q; (S; T ) S = d; P T = r; (P; U) U = f P = e; (; q; (d; ; q) ; r) + ; f If rocess q in this gure were to have only the following activation ossibility q : ( 2! 4 ^ 5 ) then the rocess structure has the following ath exression: P = ; q; ( k r); (( k f); q; (d k r)) 1 Thus the two rocess two rocess structures are not equivalent, although there underlying DFD is the sae. This deonstrates that a DFD by itself is not owerful enough to recisely odel the behaviour of an inforation syste. 5 Conclusions and Further Research In this aer we have given soe guidelines as to how seantics can be attached to DFD's. The iortance of such research is clear. Current inforation syste design too often involves foraliss without a clearly dened eaning. This leads to isunderstandings both in the use of the foralis and the interretation of the results. The transforation of the DFD to a PT-net variant lus the associated ath exressions with a clear eaning and algebraic roerties rovide a useful fraework for the investigation of DFD's. In this way we can coe to a better understanding of the. 12

13 In this aer we have only sketched how a DFD ay be transfored to a ath exression. In general this is quite a dicult task. Further research is needed to rovide a general algorith for this transforation. Also the algebra is not colete. Research ust be done to deterine what extra transforations are necessary. Finally, as DFD's are not that dierent fro the A-grahs and I-grahs of ISAC ([7]). We believe that the seantics of these grahs can also be established in an analagous way to that resented in this docuent for DFD's. References [1] J.A. Bergstra and G.P.A.J. Delen. Van dataowdiagraen via etrinetten naar systeeatrixnotatie. Technical reort, Matheatical Centre, Asterda, The Netherlands, [2] R.H. Cabell and A.N. Haberann. The Secication of Process Synchronization by Path Exressions, volue 16 of Lecture Notes in Couter Science. Sringer Verlag, [3] E.D. Falkenberg, R. van der Pols, and Th. P. van der Weide. Understanding rocess structure diagras. Inforation Systes, 16(4):417{428, Set [4] C. Gane and T. Sarson. Structured Syste Analysis: Tools and techniques. IST Databooks. MacDonald Douglas Cororation, St. Louis, [5] P.E. Lauer and R.H. Cabell. Foral seantics of a class of high-level riitives for coordinating concurrent rocesses. Acta Inforatica, 5:247{332, [6] P.E. Lauer, P.R. Torrigiani, and M.W Shields. Cosy - a syste secication language based on aths and rocesses. Acta Inforatica, 12:109{158, [7] M. Lundeberg, G. Goldkuhl, and A. Nilsson. Inforation Systes Develoent - A Systeatic Aroach. Prentice Hall, [8] Peterson. Petri Net Theory. Prentice Hall, [9] R. van der Pols. The SSAD handbook. Technical reort, Digital Equient BV, Utrecht,