A formal semantics of synchronous interworkings

Transcription

1 A formal semantics of snchronous interorkings S. Mau a, M. van Wijk b and T. Winter c a Dept. of Mathematics and Computing Science, Eindhoven Universit of Technolog, P.O. Bo 513, 5600 MB Eindhoven, The Netherlands b Programming Research Group, Universit of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands c Philips Research Laboratories, P.O. Bo 80000, 5600 JA Eindhoven, The Netherlands Interorkings can be considered as a snchronous variant of Message Sequence Charts. In this paper e present the formal semantics of interorkings in an algebraic frameork. We concentrate on the main operators for combining interorkings: the interorking sequencing and the interorking merge. 1. INTRODUCTION Interorkings are used in the analsis phase of the development process at PKI (Philips Kommunikations Industrie) Nürnberg for describing the message interactions beteen blocks. The specification of one functional block contains the most common sequences of message interactions ith all the neighbour blocks it communicates ith. Interorkings are a snchronous variant of message sequence charts mentioned belo. The are used, for eample, for the specification of radio communication sstems. In close cooperation ith PKI Nürnberg a project is being performed in hich an Interorking Tool Set (ITS) is specified and prototped that can process, analse, and combine interorkings. Message sequence charts have been used for a long time b CCITT Stud Groups in their recommendations and ithin industr, according to different conventions and under various names. Interorkings also pla an important role in other areas than the telecommunication industr. Several object-oriented methods have absorbed Message Sequence Charts (ObjectOr [5], Rumbaugh [9]). Compared to other trace languages interorkings have the advantage of a clear graphical laout and structuring. Hoever, interorkings are onl suitable for the description of relativel small parts of the sstem behaviour. Therefore, in order to describe the behaviour of a sstem more completel, composition operations on interorkings similar to the composition operations in for eample LOTOS are required. In the literature several solutions have alread been proposed in this area. In [8] the snta of message sequence charts is enhanced ith the concept of conditions providing the means to combine message sequence charts sequentiall. The sequential composition operation is onl described implicitl in this document. No other composition operations on message sequence charts are defined. In [3] a sstem is described as a hierarch of services increasing in detail. The loest level of services in the hierarch is described in terms of

2 message sequence charts. For the services several composition operations are defined (e.g. sequential-, parallel-composition, choice-operator). Although services can be combined the document does not discuss the related composition of the message sequence charts. In [10], a formalisation of message sequence charts is given in terms of set theor. This formalisation has the advantage that it onl uses basic constructs but has the disadvantage that the parallel composition operator can not be easil epressed in the formalism. A collection of interorkings describes the behaviour of a sstem on a high level of abstraction. Each interorking is a projection of a part of the communication behaviour of a sstem onto a set of entities. In a telecommunication contet here for eample SDL is used as description language, an entit ma be a process or a set of processes combined into one functional block. Interorkings are event oriented instead of entit oriented and onl snchronous communication actions can be modeled. This means that messages can not be delaed as in message sequence charts here communication is asnchronous. Because in practice, this does not seem to be a restriction, e concentrate ourselves in this document on the simpler model of snchronous communication and keep the asnchronous variant for further stud. For the formal definition of interorkings and the composition operators e eplore techniques from process algebra [1]. Process algebra is a commonl accepted technique for the description of communicating sstems. Algebraic reasoning is ver suitable for formal verification activities hich concern in our case the consistenc checking of interorkings. Additionall, via process algebra, it is eas to generate prototpes implementing the various operators, and a link to other algebraic description methods such as LOTOS can easil be made. In this paper e first give an introduction to interorking diagrams and operators. After that e present the formal semantics and some useful properties. We ill not eplain the hole interorking language but e restrict ourselves to the main operators. In [6] snta and semantics of the complete language is given. Acknoledgements Thanks are due to man people for giving fruitful comments on the documents. Special thanks go to Jan Bergstra for his constructive advice concerning both the documents and the project, and to Jos Baeten for his comments on process algebra. WealsoanttomentionLeAugusteijnforhishelpithregardtheElegantcompiler generator tool set hich e used for the development of prototpe tools and finall Michel Reniers and David Riemens for their support ith proving and programming. 2. INTERWORKINGS 2.1. Interorking diagrams The basic graphical language of interorking diagrams is ver simple and resembles the snta for basic sequence charts [7]. Vertical lines are used to represent entities ithin a sstem and horizontal arros are used to represent communications beteen to entities. No global time ais is assumed for one interorking. Along each vertical entit-line the time runs from top to bottom, hoever, no proper time scale is assumed. Events of different entities are ordered onl via messages. Within interorkings, no dela beteen a message output and the corresponding message input is eplicitl modelled. Therefore, an interorking imposes onl a partial ordering on the events being contained. In figure 1,

3 A B C v u z D Figure 1. Eample interorking diagram the messages u and v are not related and no ordering eists beteen them. This also holds for the messages and z. A collection of interorkings describes the behaviour of a sstem on a high level of abstraction. Each interorking describes a common sequence of communication actions performed b the contained entities. Note that the interorking does not specif communication actions ith entities that are not contained. The behaviour of a single entit is thus onl partiall described b the events being contained in the interorking. The actual behaviour ill be an interleaving ith communication events concerning other entities. Beteen the output of v and the input of, entit A in figure 1 for eample can have interactions ith an entit E hich is not in the diagram. Hoever, it can not have a communication action ith either one of the entities B, C, or D beteen the output of v and the input of Interorking sequencing An interorking diagram can be constructed from atomic communication actions and applications of the interorking sequencing operator (denoted b i ). The interorking sequencing, or simpl the sequencing of to interorking diagrams is the vertical concatenation of the to diagrams (See figure 2). In this case, here the interorking A B A B o i A B Figure 2. Sequencing diagrams have all entities in common, the sequencing corresponds to a real sequentialisation in time. The operands need not necessaril have all entities in common. In the net eample (figure 3) the to interorkings onl have entit B in common. If interorkings contain disjoint entities, the behaviour described b means of their sequencing ill in most cases not be equal to simple sequential composition of the behaviours. Events that do not involve the same entit or are not related via messages ill be unordered in the sequential composition. An eample of the sequencing of to interorkings that contain

4 A B B C o i A B C Figure 3. Sequencing interorkings ith one entit in common unrelated messages is given in figure 4. Note that the right-hand side describes a single interorking in hich the vertical position of the arros does not indicate an ordering. A B C D o i A B C D Figure 4. Sequencing interorkings ith disjoint entities 2.3. Interorking merge Theinterorking merge, or simpl called themerge(denotedb i ), ofto interorkings is their interleaved composition ith the restriction that the interorkings are forced to snchronise on a set of communication actions. This set consists of the communication actions concerning ever pair of entities hich the interorkings have in common. In A B C B C D v i A B C D v Figure 5. Merge figure 5 to interorkings are merged, hich have entities B and C in common. In those cases here the interorking diagrams describe disjoint entities, the merge is equal to real parallel composition. An eample of the merge of to interorking diagrams in hich disjoint entities are described is given in figure 6. Note also that the merge of these to interorking diagrams is equal to the sequencing of these diagrams (See figure 4). An interorking does not specif communication actions ith entities that are not contained. The behaviour of a single entit is onl partiall described b the events contained in an interorking. The merge of to interorkings ma return an interorking in hich the sequence of events for a single entit is the result of interleaving the sequences of events in

5 A B C D i A B C D Figure 6. Merging interorkings ith disjoint entities the original interorkings. As an eample, figure 7 shos the merge of to interorking diagrams that have onl one entit (B) in common. For the interorking resulting from A B B C i A B C + A B C Figure 7. Merge ith interleaving of events the merge in figure 7, nothing can be said about the order of the events for entit B. The input of can equall ell occur before or after the output of. Note that there is no single interorking diagram that describes this merge. In this case e need a collection of to different interorking diagrams Consistenc If to interorkings are merged, the interorkings have to snchronise on all communication actions beteen entities hich the have in common. In case this snchronisation succeeds, e ill call the to interorkings merge-consistent. Hoever, if to interorkings have multiple entities in common but do not snchronise on all the communication actions beteen these entities then the interorkings are not merge-consistent and the merge of the interorkings ill contain so called deadlock or inaction. The to interorking diagrams in figure 8 for eample are not merge-consistent. A B C B C D v i z A B C D v Figure 8. Inconsistent interorkings The tetual and graphical snta for interorking diagrams does not contain a construct to represent deadlock. In figure 8, the deadlock is depicted b means of a double horizontal

6 line. It should be noted that deadlock is not related to an contained entit in particular. Deadlock is a propert of the interorking as a hole. 3. FORMAL SEMANTICS In this section e ill first give an introduction to BPA, hich is a basic theor for process description. Then e define the to interorking operators and the class of interorkings. Finall e give some useful properties of interorkings Basic Process Algebra BPA (Basic Process Algebra) is an algebraic theor for the description of process behaviour [1, 2]. We consider to basic notions: atomic actions and processes. An atomic actionisanindivisibleunitofbehaviour,suchastheinsertionofacoininacoffeedispenser or the communication of some piece of information beteen to agents. A process is the description of the (possible) behaviour of a sstem. Atomic actions ill be denoted b a,b,... and processes b,,... Ever atomic action ill be considered as a process. Processes can be constructed from simpler processes ith the use of to operators. The sequential composition of processes and (notation or for short) is the process that first eecutes, and upon completion of starts. The alternative composition of and (notation +) is the process that either eecutes, or eecutes (but not both). We ill not specif ho this choice is made. These operators are defined b the equations from table 1. Table 1 Basic Process Algebra + = + ( + ) + z = + ( + z) + = ( + )z = z + z ()z = (z) The equations state that the alternative composition is commutative, associative and idempotent, and that the sequential composition is associative onl. Note that the onl distributes over a +-operator on the left-hand side. In order to describe processes hich ma not terminate successfull, e etend BP A ith features for unsuccessful termination. We use the term deadlock for unsuccessful termination. The special atomic action δ denotes a deadlocked process. Furthermore e introduce the encapsulation function, hich renames atomic actions from a given set into δ. It is denoted b H, here H is a set of atomic actions. The equations in table 2 define deadlock and encapsulation. The first equation sas that no deadlock ill ever occur as long as there is an alternative that can proceed. The second equation states that after a deadlock has occurred, no other actions can possibl follo. The predicate isdelta determines hether a process equals δ, and the predicate df determines hether a process is deadlock-free or not. The equations from tables 1 and 2 define the theor BPA δ, H,df.

7 Table 2 Deadlock +δ = δ = δ H (a) = a if a H H (a) = δ if a H H (+) = H ()+ H () H () = H () H () isdelta(a) if a δ isdelta(δ) isdelta(a) = isdelta(a) isdelta(+) = isdelta() isdelta() df(δ) df(a) if a δ df(a) = df() if a δ df(+) = df() df() if isdelta() and isdelta() 3.2. Interorking operators Basics The collection of atomic actions can be considered as a parameter of BPA. In the setting of interorkings, e ill onl consider communication actions of a fied format and some primitive actions, so e specialise the set of atomic actions. Let EID and MID be finite sets, containing entit identifiers and message identifiers. Define the folloing collection of communication actions. A = {c(p,q,m) p,q EID,m MID} Action c(p,q,m) means that entit p sends message m to entit q. If e etend this collection of atomic actions ith δ e get A δ. Unless stated differentl, all variables a,b,... range over A δ. Furthermore, e need an auiliar function E, hich determines the entities involved in an action. This function is defined in table 3. Table 3 The Entit function E(δ) = E(c(p,q,m)) = {p,q} E(a) = E(a) E() E(+) = E() E() if a δ Interorking Sequencing The interorking sequencing of to processes (notation i ) is their interleaved composition ith the restriction that actions involving common entities are performed first

8 b the left-hand process and then b the right-hand process. First e define the set of actions generated b the entities of a given process. α E () = {a A E(a) E() } The definition of the interorking sequencing in table 4 resembles the definition of the communication free merge from [1]. We use the auiliar operators left sequencing (L i ) andrightsequencing(r i )hichhavethefolloingintuitivemeaning. Theleftsequencing of to processes means that the left operand is forced to do the first step. The right sequencing of to processes means that the right operand has to do the first step, but it ma onl eecute this step if it is not blocked b some action from the left operand hich involves the same entit. The encapsulation operator ( H ) is used for blocking unanted actions. Table 4 Interorking Sequencing i = L i + R i al i = a R i a = α E ()(a) al i = a( i ) R i a = α E ()(a) ( i ) (+)L i z = L i z + L i z R i ( +z) = R i + R i z Eample 1 Assuming that p, q, r, s and t are all different entities, e have the folloing derivation. c(p,q,m) i (c(p,r,n) i c(s,t,l)) = c(p,q,m) i (c(p,r,n) c(s,t,l)+c(s,t,l) c(p,r,n)) = c(p,q,m) (c(p,r,n) c(s,t,l)+c(s,t,l) c(p,r,n)) +c(s,t,l) c(p,q,m) c(p,r,n) Interorking Merge Theinterorkingmergeoftoprocesses(notation i )istheirinterleavedcomposition, ecept that the processes are forced to snchronise on a set of atomic actions (see table 5). This set consists of the actions concerning ever pair of entities hich the processes have in common. We define the function α CE, hich determines the actions performed b common entities as follos. α CE (,) = {c(p,q,m) A p,q E() E()} First e define the S-interorking merge (notation S i) for a subset S of A. As in the definition of the parallel composition in ACP (the algebra of communicating processes, [1]) e need to auiliar operators: the left interorking merge ( S ) and the snchronisation interorking merge ( S i). The left interorking merge forces that the first step i is taken from the left argument, provided that this action is not an element of the set S. The snchronisation interorking merge can onl eecute an action if both operands can perform this same action and this action is an element of the set S. The function γ S determines hether to actions have to snchronise.

9 Table 5 Interorking Merge i = α CE(,) i S i = S i + S i + S i a S i = S(a) a S i = S(a) ( S i) (+) S i z = S i z + S i z a if a = b a S γ S (a,b) = δ otherise a S ib = γ S (a,b) a S ib = γ S (a,b) S () a S ib = γ S (a,b) S () a S ib = γ S (a,b) ( S i) (+) S iz = S iz + S iz S i( +z) = S i + S iz Eample 2 Both processes in the folloing merge have to snchronise on c(q, r, l). c(p,q,k) c(q,r,l) i c(q,r,l) c(r,s,m) = c(p,q,k) c(q,r,l) c(r,s,m) 3.3. Interorkings The theor BPA δ, H ith the to operators for interorking merge and interorking,df sequencing ill be called BPA i. An interorking is a process hich can be constructed onl from atomic actions and applications of the interorking sequencing and the interorking merge operator. The set of interorkings is denoted b IW. The set of deadlock free interorkings is denoted b IW df. A process hich is onl constructed of atomic actions and the interorking sequencing operator is called a locall deterministic interorking. The set of locall deterministic interorkings is called IW ld. Likeise, IWdf ld is used for deadlock free locall deterministic interorkings. It is not alas the case that the merge of to deadlock free interorkings is again deadlock free as as shon in figure 8. We sa that to interorkings are consistent if their interorking merge is deadlock free. This notion of consistenc plas an important role hen developing a specification via stepise refinement. The tool set can be used to automaticall check consistenc of interorkings. Eample 3 The folloing eample shos to interorkings hich don t agree on the communications ith respect to entities q and r. c(p,q,k) c(q,r,l) i c(s,t,m) c(r,q,n) = c(p,q,k) c(s,t,m) δ +c(s,t,m) c(p,q,k) δ 3.4. Properties In this section e present some useful properties about BPA i. In general, and z are finite process epressions in BPA i and S is a set of actions, The use of the first proposition is that most theoretical results for BPA can be transferred to BPA i. Proposition 1 (Elimination) (1) Ever process in BPA i is equal to a process hich has one of the folloing forms: δ, a, a, +

10 (2) BPA i is a conservative etension of BPA. Proof Term rerite analsis as in [1]. The operators S iand i are commutative, hile the arguments of the i operator ma onl be interchanged if the have no entities in common. The first rule, for eample, can be used to derive that in figure 6 and are not ordered. Proposition 2 E() E() = i = i (1) S i = S i i = i Proof For (1) use E() E() = L i = R i (2) and (3) are clear b smmetr of the definitions. The operators S iand i are associative, hile associativit for the i operator onl holds for pairise consistent interorkings. The proof of proposition 5 consists of a comple simultaneous induction. Proposition 3 ( i ) i z = i ( i z) Proof Simultaneous induction on the total number of smbols in, and z of the folloing four propositions. (2) (3) (L i )L i z = L i ( i z) (R i )L i z = R i (L i z) ( i )R i z = R i (R i z) ( i ) i z = i ( i z) Proposition 4 ( S i) S iz = S i( S iz) Proof Simultaneous induction on the total number of smbols in, and z of the folloing four propositions. ( S i ) S i z = S i ( S iz) ( S i) S iz = S i( S iz) ( S i) S i z = S i( S i z) ( S i) S iz = S i( S iz) Proposition 5 Let, and z be interorkings, let be of the form, δ or δ, be of the form, δ or δ and z be of the form z, z δ or δ, such that, and z are pairise consistent interorkings, then e have ( i ) i z = i ( i z)

11 Eample 4 The fact that this proposition requires that the interorkings have to be consistent is shon b the folloing calculations. c(p,q,m) i (c(p,q,n) i c(p,q,n)) = c(p,q,m) i c(p,q,n) = δ (c(p,q,m) i c(p,q,n)) i c(p,q,n) = δ i c(p,q,n) = c(p,q,n) δ Proposition 6 i δ = δ δ i = δ i δ = δ (1) (2) (3) Proof (1) Simultaneous induction ith L i δ = δ (2) follos from (1) and proposition 2. (3) Simultaneous induction of i δ = δ and iδ = δ. The folloing proposition implies that a deadlock in an interorking is global and ill be reached onl if all admissible actions are eecuted. It can be proven ith simultaneous induction. Proposition 7 Ever interorking is deadlock free, equal to δ or of the form p δ here p is a deadlock free interorking. For the folloing properties e need the notion of a trace of a process. A trace is an element of A + A* δ {δ}. So a trace is a non-empt sequence of atomic actions, ith the restriction that onl the last action ma be a δ. Concatenation of traces is denoted b placing the traces net to each other. We define the set of traces of a process as follos. tr(a) = {a} tr(a) = {at t tr()} if a δ tr(+) = tr() tr() if δ δ A process is deterministic if it is of the folloing form. a i i + j i I j Ja here e require that all a k are different for k I J and all i are deterministic. It follos directl from the definitions of the interorking operators that ever interorking is deterministic. This implies that ever interorking is completel determined b its trace set. See [4] for a proof that this proposition holds for all deterministic processes. Proposition 8 For interorkings and tr() = tr() =

12 B e denote the process here all sequential operators are replaced b interorking sequencing operators. ã = a a = a i + = +ỹ The class IW is closed under this operation. Moreover e have = for ever interorking. This operation can also be defined on traces. ã = a ã = a i if a δ We conclude ith a proposition hich states that ever interorking is completel determined b a set of locall deterministic interorkings. The application of this proposition is in the fact that ever interorking, in particular interorkings as in figure 7 can be represented b a set of interorking diagrams. So the user of the tool set does not have to kno about the and the +-operator. Input and output consist of epressions containing onl sequencing and merge operators. Proposition 9 For ever interorking tr() = t tr() tr( t) 4. CONCLUSIONS The problem of the so called horizontal and vertical composition of interaction diagrams as used for Interorkings or Message Sequence Charts has been tackled successfull b introducing the sequencing and the merge operator. Apart from the clear semantics, the algebraic approach shos some more advantages. The main point is that it enables the formal algebraic verification of useful properties such as consistenc and refinement(hich is the implementation relation beteen to interorking specifications as described in[6]). The second point is that, hen interpreted as a term reriting sstem, the equations provide a straightforard prototping strateg for tool builders. This eas prototping has helped in evaluating the use of interorkings in the aforementioned Interorking project. In [6] e give a semantics of the full Interorking language, hich also includes macros, timers and refinement. Research is going on to provide other features from the field of MSCs ith algebraic semantics. These are asnchronous communication, message overtaking and conditions. Furthermore, the possibilit is being studied to rela the requirement that in a merge the components have to snchronise on all common actions. REFERENCES 1. J.C.M. Baeten & W.P. Weijland, Process algebra, Cambridge Tracts in Theoretical Computer Science 18, Cambridge Universit Press, ISBN , 1990.

13 2. J.A. Bergstra & J.W. Klop, Process algebra for snchronous communication, Inf. & Control 60, pp , V. Encontre, e.a., Combining Services, Message Sequence Charts and SDL: Formalism, Method and Tools, SDL 91, Evolving Methods, Elsevier Science Publishers, ISBN , J. Engelfriet, Determinac (observation equivalence = trace equivalence), TCS 36(1), pp.21-25, I. Jacobson, a.o., Object-Oriented Softare Engineering, A Use Case Driven Approach, Addison Wesle, ISBN , S. Mau, M. van Wijk & T. Winter, Snta and semantics of snchronous interorkings, Philips IST technical report RWB-508-RE-92436, E. Rudolph, Snta and Semantics of Basic Sequence Charts, Contribution to the Stud Group X, WP X/3, Q8, CCITT meeting, Geneva E. Rudolph, P. Graubmann, J. Graboski, Toards an SDL-Design-Methodolog Using Sequence Charts Segments, SDL 91, Evolving Methods, page , Elsevier Science Publishers, ISBN , J. Rumbaugh, a.o., Object-Oriented Modeling and Design, Prentice Hall International, ISBN , P.A.J. Tilanus, A formalisation of Message Sequence Charts, SDL 91, Evolving Methods, page , Elsevier Science Publishers, ISBN , 1991.