VERIFICATION OF ETHERNET PROTOCOLS VIA PARAMETRIC COMPOSITION OF PETRI NET

INCOM'2006: 12th IFAC/IFIP/IFORS/IEEE/IMS Symposium Information Control Problems in Manufacturing May 17-19 2006, Saint-Etienne, France

VERIFICATION OF ETHERNET PROTOCOLS VIA PARAMETRIC COMPOSITION OF PETRI NET Zaitsev D.A., Zaitsev I.D. Dmitry A. Zaitsev National Telecommunication Academy Kuznechnaya, 1, Odessa 65029 Ukraine http://www.geocities.com/zsoftua

Ivan D. Zaitsev Novosibirsk State University Pirogova, 2, Novosibirsk 630090 Russia [email protected]

Abstract: The verification of Ethernet protocols represented by Petri net models is implemented. A model consisting of an arbitrary number of workstations on bus is composed out of model of a single workstation via the fusion of contact places. The invariance of the model with an arbitrary number of workstations is proved on the base of the invariance of separate workstation’s model, which is the functional subnet. Parametric equations and invariants are used. Copyright @ 2006 IFAC Keywords: Ethernet, Petri net, invariant, composition, functional subnet

1.

INTRODUCTION

constitutes an actual area of investigation requiring rapidly growing attention since the process of new protocols appearance becomes more and more dynamic. It was shown (Girault, Valk, 2003) that Petri net model of a correct telecommunication protocol has to be invariant one.

The verification of protocols (Marsan, et al., 1987) is the traditional area of Petri nets (Girault, Valk, 2003) application. Really, the majority of network protocols assume an asynchronous character of systems’ interaction, which makes its description with sequential models such as, for instance, flow block, difficult.

The goal of the present work is to prove the invariance of the local area network Ethernet protocols with the bus topology for an arbitrary number of workstations. In the presence of early known detailed models of Ethernet (Marsan, et al., 1987), formal proof has not been implemented due to the enormous dimension of Petri net. The application of decomposition technique (Zaitsev, 2004a) with properties of functional subnets’ invariants (Zaitsev, 2004b; 2005) usage allowed the solution of the mentioned task. Notice that, the class of functional subnets essentially distinguishes from the subnets studied in (Christensen and Petrucci, 2000; Haddad, at al., 2002; Juan, at al., 1998).

For the investigation of protocols, as a rule, two basic tasks are solved: correctness proof and performance evaluation. The first of the above mentioned tasks is named also as verification of protocol. It is the highest priority indeed, since the presence of defects in source specifications constitutes the most expensive type of errors. Since in the case an incorrect protocol would be implemented into either software or hardware, the expenses concerned with the debugging became enormously huge. During the verification of a protocol the opportunity to repeat unlimitedly the process of systems’ interaction using the elements of limited capacity is established. In negative form these conditions may be represented as the absence of mutual blockings (deadlocks) and also the absence of sequences of actions bringing to overflow of stores. The verification of telecommunication protocols

Moreover, as distinct from (Zaitsev, 2004c; 2004d), the parametric approach is applied. On the base of the regular structure of the model, composed as the repetition of the model of workstation, we prove the invariance for an arbitrary number of workstations. Invariants are obtained in symbolic form of algebraic expression, which allows its easy calculation for

261

concrete values of parameters. This approach may be applied successfully for large-scale systems with regular structure, such as, for instance, computer hardware and software (Cortadella, at al., 2003).

investigated. The start and finish of frame's transmission and the processes of carrier spreading on media in time are considered as well as the collisions’ detection and recognition. Notice that, carrier appearance and cessation spread on the bus during the time interval, so the workstations have got different information about channel’s state in the same chosen moment of time.

2. PROTOCOLS OF ETHERNET In Ethernet network the protocols of multiple media access with carrier sense and collision detection (CSMA/CD) are implemented. Source specifications of protocols are regulated by standards IEEE 802.3.

3. MODEL OF ETHERNET In the present work the model of Ethernet with bus topology and an arbitrary number of workstations attached is investigated. It constitutes slightly modified model described in Marsan’s work (Marsan, et al., 1987). In the mentioned work, it was pointed to the constructed model is detailed and precise enough implementation of source specifications. But due to the huge dimension even for the minor number of workstations the investigation of the model is too hard. That is why, it was suggested in (Marsan, et al., 1987) to investigate the simplified model.

Each of workstations listens to the media and in the case of carrier absence may start the transmission of data. If several workstations start the transmission of data simultaneously, then information overlapping occurs, that is called a collision. The workstation has facilities to detect collisions. In the case the collision being detected, the transmission of data is broken and then is resumed in a random interval of time. The specifications of protocols regulate also the format of data (frames) transmitting. In the present work the structure of Ethernet data frame is not used and, moreover, the network with bus topology is

Fig. 1. Model of Ethernet workstation

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The Petri net composition technique application allows the verification of detailed model, including an arbitrary number of workstations. Marshan’s source model is modified in such a way that the model of separate workstation constitutes the functional Petri net (Zaitsev, 2004a; 2005). For these purposes contact places were added. The composition of local area network model was implemented by the workstations’ contact places fusion. Thus, there is the natural decomposition of the entire model into functional subnets.

dotted line. The upper part models the behavior of the workstation, whereas the lower part describes the interaction of the workstation with the channel and the process of data transmission among neighboring workstations. The duplicate notation of Petri net elements is used: semantic names convenient for description and numbers used in the calculations of invariants. Place and transition think models the internal activity of the workstations not concerned with information interchange. In place sense the listening of media is executed. If, as the result of listening, it has been established that the channel is free, then transition idle becomes permitted. While firing it moves a token into place tx (transmit) indicating that the workstation has started transmission of data and also moves a token into place ps (propagation start) modeling the beginning of carrier spread on the network. The presence of token into place tx makes transition tx permitted, modeling data transmission on the network. When the firing of transition tx has been finished, the workstation completes the transmission of data and returns to the initial state think. Moreover, a token is moved into place pe (propagation end) modeling the spread of carrier cessation on the network.

The model is represented in the form of timed Petri net with inhibitor arcs (Girault, Valk, 2003). Notice that, transitions of three types were used: firing instantly represented by narrow bars; firing in deterministic time, represented by filled bars; and firing in random uniformly distributed time, represented by wide unfilled bars. The inhibitor arcs are used for zero marking check. Instead of the arrow such arcs contain the little circle. The contact places are drawn in larger size. 4. MODEL OF WORKSTATION Let us describe in detail the model of the workstation represented in Fig. 1. For convenience of description the model is divided into two parts with the aid of

Z1

1 p17

2 p21

p122

1 p18

p123 p124

p121

Z2

Zk

2 3 p17 p21

k −1 k p17 p21

2 p22

2 3 p18 p22

k −1 k p18 p22

k p18

1 p19

2 p23

2 3 p19 p23

k k −1 p19 p23

k p19

p120

2 p24

2 3 p20 p24

k k −1 p20 p24

k p20

k p17

Fig. 2. Model of Ethernet with bus topology At that, or at any moment of time, while the frame is being transmitted, place chstate (channel state) contains more than two tokens, then instant transition coll (collision) is fired modeling the recognition of collision in the channel. The displacement of token into place jam and firing of transition jam models the actions aimed to cessation of data transmission at collision recognition. The displacement of token into place back initiates the process of repeated transmission of data after collision detection. An arbitrary time of transition back firing models the delay before repeated transmission.

(persist). As soon as the channel becomes free the instant transition pend (pending) becomes permitted and fires. It moves a token into place start and after a determined time delay represented with transition start the transmission of data begins. In the lower part of the net the central element is place chstate representing the state of the channel recognized by the workstation. As the appearance and cessation of carrier constitute the processes, the time of realization for which is taken into consideration in the model, so each workstation may recognize different states of the channel at the same time. If place chstate is empty, then the channel is available; the presence of exactly one token testifies

If, at media listening in place sense, it has been determined that the channel is busy, then the transition busy fires moving a token into place pers

263

that the channel is busy; if this place contains at least two tokens, then there is a collision.

Notice that, at arrival of data transmission beginning signal either from the left or from the right (tl4, tr1), a token enters into place chstate modeling that the channel is busy is known by the workstation. The cessation of data transmission by the workstation situated either on the right or on the left in bus leads to the firing of transitions efl, efr correspondingly. In such a way the release of channel is modeled.

Transitions tl1, tl2, tl3, tl4 model the delays of carrier spread (cessation) to the left neighbor workstation. Analogous part transitions tr1, tr2, tr3, tr4 play for the right neighbor workstation. Notice that, as it is assumed to compose the general model through the fusion of places, so the firing times of transitions mentioned should be chosen equal to half of an actual delay.

Therefore, the net constructed has input places X = { p17 , p18 , p23 , p24 } and output places Y = { p19 , p20 , p21, p22 } and according to the definition (Zaitsev, 2004a; 2005) is a functional Petri net.

When the workstation begins the transmission of data and puts a token into place ps, instant transition ps fires and puts a token into each of places chastate, pstl, pstr. The presence of tokens in places pstl, pstr models the propagation of signal to the left and to the right in bus.

5. MODEL OF NETWORK

When the workstation stops the transmission of data and puts a token into place pe, instant transition pe fires; it gets a token out of place chstate and puts a token into each of the places petl, petr. The presence of tokens in places petl, petr models the cessation of signal in bus.

Let us consider the rules of model construction for Ethernet network with bus topology consisting of k workstations. The general scheme of such a model is represented in Fig. 2. The models of workstations Z i are combined by means of contact places' fusion. At that, each workstation interacts exactly with two neighboring workstations. Let's consider, for Z2 . Contact places instance, subnet 2 p21 , p22 , p23 , p24 of subnet Z are merged with contact places p17 , p18 , p19 , p20

The signal of data transmission beginning in bus may arrive also from left neighbor station into place pstr and from right neighbor station into place pstl. The cessation of data transmission in bus is modeled by the arrival of token from left neighbor station into place pefl and from right neighbor station into place pefr.

Table 1. Basis invariants of workstation’s model 1 1

1 2 3 4 5 6

2 1

3 1

4 1

5 1

6 1

7 1

8 1

9

10 1

11

12 1 1

13

14

1

17 1

1

18

19 1 1

20

21

22 1

23

24 1

1

1

1

1 1 1

places p21 , p22 , p23 , p24 of

1

1 1

1

telecommunication protocol has to possess such properties as boundness, safeness, liveness.

of subnet Z 1 correspondingly. Moreover, contact places p17 , p18 , p19 , p20 of subnet Z 2 are merged with contact

16 1

1 1

15 1

subnet

To determine the mentioned properties Petri net invariants are applied (Girault, Valk, 2003). In this case the model of a correct protocol ought to be invariant one. Since known methods of invariants calculation (Kryviy, 1999) have exponential complexity, their application becomes practically impossible in the investigation of large-scale nets.

3

Z correspondingly. The belonging of elements to concrete subnet is represented with the aid of an upper index equaling to the subnet’s number.

Notice that, the constructed model has natural decomposition into functional subnets {Z i } . This fact will be used further for model’s properties analysis.

In (Zaitsev, 2004b) the technique of Petri net invariants calculation on the base of invariants of its functional subnets was presented. For investigation of an arbitrary given Petri net its decomposition into functional subnets is implemented preliminarily. The technique of decomposition was described in (Zaitsev, 2004a; 2005).

6. INVARIANCE OF MODEL As was mentioned in (Marsan, at al., 1987; Girault, Valk, 2003), an ideal model of a correct

264

nonnegative integer coefficient zl . The columns of

As set {Z i } for the constructed model of Ethernet with bus topology defines the partition of Petri net into functional subnets, so the method described in (Zaitsev, 2004b; 2005) may be applied for invariants calculation. Remember that, the invariant is a nonnegative integer solution x of equation x⋅A=0,

matrix correspond to places of net Z i . According to (Zaitsev, 2004b), after calculation of functional subnets’ invariants it is necessary to find the common invariants for contact places. As in the composition of two neighbor subnets Z i and Z i +1 i i i i four contact places { p17 , p18 , p19 , p20 } of the left subnet and four contact places i +1 i +1 i +1 i +1 { p21 , p22 , p23 , p24 } of the right subnet take part, so the system of equation for contact places has the following form:

(1)

where A is the incidence matrix of Petri net for invariants of places or the transposed incidence matrix for invariants of transitions. Notice that, in the constructed Ethernet model all the functional subnets have the same structure and are distinct only in the upper index of their elements. At calculation of subnet Z i invariants the inhibitor arcs and loops were not been taken into consideration, as they do not change the marking of net. Moreover, arcs (t3 , p6 ) , (t6 , p6 ) , (t10 , p13 ) , (t10 , p14 ) , (t11 , p12 ) , (t11 , p15 ) were omitted and arc (t11 , p6 ) was added, as the source model is not invariant. These transformations of matrix correspond to the starting of frame’s retransmission only after cessation signal have been put into the bus. Thus, the following incidence matrix is used: ⎛- 1 0 0 0 0 0 1 ⎜ 0 -1 0 0 0 0 0 ⎜ 1 1 -1 -1 0 0 0 ⎜ 0 0 0 1 -1 0 0 ⎜ 00 00 00 00 10 - 01 - 01 ⎜ 0 0 1 0 0 1 0 ⎜ 0 0 0 0 0 0 0 ⎜ 0 0 0 0 0 0 1 ⎜ 0 0 0 0 0 0 0 ⎜ 0 0 0 0 0 0 0 A = ⎜ 00 00 00 00 00 00 00 ⎜ 0 0 0 0 0 0 0 ⎜ 0 0 0 0 0 0 0 ⎜ 0 0 0 0 0 0 0 ⎜ 0 0 0 0 0 0 0 ⎜ 00 00 00 00 00 00 00 ⎜ 0 0 0 0 0 0 0 ⎜ 0 0 0 0 0 0 0 ⎜ 0 0 0 0 0 0 0 ⎜ 0 0 0 0 0 0 0 ⎝ 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 1 0 0 -1 0 1 -1 0 0 0 1 0 -1 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 1 0 0 0 0 1 0 0 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0

⎧ z1i + z5i = z5i +1 , ⎪ i i +1 i +1 ⎪ z3 = z1 + z3 , ⎨ i i i +1 ⎪ z1 + z 2 = z 2 , ⎪ z i = z i +1 + z i +1 , i = 1, k − 1. 1 6 ⎩ 6

0⎞ 0⎟ 0⎟ 0⎟ 0⎟ 0 0⎟ 0⎟ 0⎟ 1⎟ 0⎟ 0 1⎟ 0⎟ 0⎟ 0⎟ 0⎟ 0 0⎟ 0⎟ 0⎟ 0⎟ 0⎟ - 1⎠

The application of tool Tina (Berthomieu, at al., 2004) gives us the matrix of basis solutions for invariants of places represented in Table 1. Row l of the represented matrix is the basis invariant, which at construction of general solution is multiplied by any

Fig. 3. Model of bus with four workstations

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(2)

It should be noted, that system (2) contains an infinite number of equations. For its solution the sequence of concrete systems for values k = 2,3,... was constructed. Then, these systems are solved using Tina (Berthomieu, at al., 2004) tool. The application of inductive reasoning gives us the parametric form of solutions' representation (see Appendix). System (2) has the following 2 ⋅ k + 4 basis parametric solutions: ⎛ ( z 4i ), i = 1, k ; ⎜ ⎜ ( z 2j , j = 1, k ); ⎜ j ⎜ ( z 3 , j = 1, k ); ⎜ ( z j , j = 1, k ); ⎜ 5j ⎜ ( z 6 , j = 1, k ); ⎜ ⎛ (( z j , z j ), j = 1, i − 1), ( z i ), 1 ⎜ ⎜⎜ 3j 6j ⎜ (( z , z ), j = i + 1, k ) 5 2 ⎝⎝

⎞ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎞ ⎟, i = 1, k ⎟ ⎟ ⎟ ⎠ ⎠

(3)

Notice that, the first and sixth solutions of (3) represent k concrete solutions of system; moreover, solutions are described by indication of nonzero (equalling to unit) elements.

1 2 2 3 3 4 4 ( p13 , p120 , p124 , p13 , p20 , p13 , p20 , p13 , p20 ); 1 1 1 1 1 ( p11, p13 , p13 , p14 , p15 , p16 , p17 , p81 , p10 , p12 , p15 , p17 , p19 , p122 , p124 , 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 p12 , p14 , p16 , p17 , p19 , p12 , p14 , p16 , p17 , p19 , p12 , p14 , p16 , p17 , p19 ); 1 1 1 1 ( p11 , p13 , p15 , p18 , p120 , p122 , p124 , p12 , p22 , p32 , p42 , p52 , p62 , p72 , p82 ,

Let us substitute the basis invariants, corresponding to free variables zl of Table 1, into parametric solutions (3) of system for contact places (2). We obtain the following parametric solutions of the source system (1): ⎞ ⎛ ( p 6i , p8i , p9i ), i = 1, k ; ⎟ ⎜ j j j j =1 ⎟ ⎜ (( p12 , p16 , p19 , p 23 ), j = 1, k ); ⎟ ⎜ j j j j =1 ⎟ ⎜ (( p11 , p15 , p18 , p 22 ), j = 1, k ); ⎟ ⎜ (( p j , p j , p j =1 ), j = 1, k ); ⎟ ⎜ 14j 17j 21j =1 ⎟ ⎜ (( p13 , p 20 , p 24 ), j = 1, k ); ⎟ ⎜ ⎛ (( p j , p j , p j , p j , p j =1 , p j =1 ), ⎞ 24 ⎟ ⎟ ⎜ ⎜ 11 15 18 20 22 ⎟ ⎟ ⎜ ⎜ j =i 1, ii − 1),i i i i i i ⎟ ⎜ ⎜ ( p1 , p 2 , p 3 , p 4 , p5 , p 6 , p 7 , p8 , ⎟ , i 1 , k = i i i i i i =1 i =1 ⎟ ⎟ ⎜ ⎜ p10 , p12 , p15 , p17 , p19 , p 22 , p 24 ), ⎟ ⎟ ⎜ ⎜ (( p j , p j , p j , p j , p j , 14 16 17 19 ⎟ ⎟ ⎜ ⎜ j =12 1 j = 1 j = 1 ⎟ ⎟ ⎜ ⎜ p , p , p ), j = i + 1, k ) 22 24 ⎠ ⎠ ⎝ ⎝ 21

2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 p10 , p12 , p15 , p17 , p19 , p12 , p14 , p16 , p17 , p19 , p12 , p14 , p16 , p17 , p19 ); 1 1 1 1 2 2 2 2 2 ( p11 , p13 , p15 , p18 , p120 , p122 , p124 , p11 , p13 , p15 , p18 , p20 , p13 , p23 , p33 , 3 3 3 3 3 4 4 4 4 4 p43 , p53 , p63 , p73 , p83 , p10 , p12 , p15 , p17 , p19 , p12 , p14 , p16 , p17 , p19 );

1 1 1 1 2 2 2 2 2 3 3 3 ( p11 , p13 , p15 , p18 , p120 , p122 , p124 , p11 , p13 , p15 , p18 , p20 , p11 , p13 , p15 , 3 3 4 4 4 4 4 p18 , p20 , p14 , p24 , p34 , p44 , p54 , p64 , p74 , p84 , p10 , p12 , p15 , p17 , p19 ).

These invariants may be generated easily with the aid of the parametric basis (4). (4)

7. CONCLUSION Thus, in the present work the invariance of the Petri net model of local area network Ethernet with bus topology was proved. The model of Ethernet constitutes the composition of functional subnets modeling workstations. The invariance of the general model was proved on the base of invariance of functional subnets. The application of composition technique at invariants investigation allows the analysis of protocols for an arbitrary number of workstations in the network. To handle invariants for an arbitrary number of workstations the parametric form of representation was used.

In the description of solutions conditional elements are indicated, which are present only in the first subnet, because only it has contact places p21 , p22 , p23 , p24 ; in the fusion of places for the others workstations their minor numbers in the left neighbor subnet p17 , p18 , p19 , p20 correspondingly are used.

The parametric composition described may be applied successfully for objects with a regular structure, for instance, in hardware and software design.

Since, for instance, at choice the sum of all the basis solutions, the natural invariant of the entire net is obtained, the model of Ethernet with the bus topology and an arbitrary number k of workstations is a p-invariant Petri net.

REFERENCES

To verify the obtained parametric solution the models for various numbers of workstations on the bus were generated and their invariants were calculated with the aid of tool Tina (Berthomieu, at al., 2004). The results coincide. For example, for the model with four workstations (Fig. 3) the following invariants were obtained:

Berthomieu B., Ribet O.-P., Vernadat F. The tool TINA - construction of abstract state space for Petri nets and Time Petri nets. International Journal of Production Research, 42(4), 2004. (www.laas.fr/tina). Christensen S., Petrucci L. Modular analysis of Petri nets. The Computer Journal, 43(3), 2000, 224242. Cortadella J., Kishinevsky M., Kondratyev A., Lavagno L., Yakovlev A. Logic synthesis of asynchronous controllers and interfaces. Springer-Verlag, 2002. Girault C., Valk R.: Petri nets for systems engineering. Springer-Verlag, 2003. Haddad S., Klai K., Ilie J.-M. An incremental verification technique using decomposition of Petri nets. Proceedings of the second IEEE International Conference on Systems, Man and

( p16 , p81 , p19 );

( p62 , p82 , p92 ); ( p63 , p83 , p93 ); ; ( p64 , p84 , p94 ); 1 1 1 2 2 2 3 3 3 4 4 4 ( p12 , p16 , p19 , p123 , p12 , p16 , p19 , p12 , p16 , p19 , p12 , p16 , p19 ); 1 1 1 2 2 2 3 3 3 4 4 4 ( p11 , p15 , p18 , p122 , p11 , p15 , p18 , p11 , p15 , p18 , p11 , p15 , p18 ); 1 1 2 2 3 3 4 4 ( p14 , p17 , p121, p14 , p17 , p14 , p17 , p14 , p17 );

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Cybernetics, October 6-9, 2002, Hammamet, Tunisia, vol. 2. Juan E.Y.T., Tsai J.J.P., Murata T. Compositional verification of concurrent systems using Petri net based condensation rules, ACM Trans. on Programming Languages and Systems, 20(5), 1998, 917-979. Kryviy S.L. Methods of solution and criteria of compatibility of systems of linear Diophantine equations over the set of natural numbers. Cybernetics and Systems Analysis, 4, 1999, 1236. Marsan A.M., Chiola G., Fumagalli A. An Accurate Performance Model of CSMA/CD Bus LAN. Advances in Petri Nets, LNCS, 266, 1987, 146161. Zaitsev D.A. Decomposition of Petri Nets. Cybernetics and Systems Analysis, 43(5), 2004, 131-140. Zaitsev D.A. Decomposition-based calculation of Petri net invariants. Proceedings of Token based computing Workshop of the 25-th International conference on application and theory of Petri nets, Bologna, Italy, June 21-25, 2004, 79-83. Zaitsev D.A. “Verification of Protocol ECMA with Decomposition of Petri Net Model”. Proceedings of The International Conference on Cybernetics and Information Technologies, Systems and Applications, Orlando, Florida, USA, July 21-25, 2004. Zaitsev D.A. Verification of protocol TCP via decomposition of Petri net model into functional subnets. Proceedings of the Poster session of 12th Annual Meeting of the IEEE / ACM International Symposium on Modeling, Analysis, and Simulation of Computer and Telecommunication Systems, October 5-7, 2004, Volendam, Netherlands, 73-75. Zaitsev D.A. Functional Petri Nets. Universite ParisDauphine, Cahier du Lamsade n° 224 Avril 2005. (www.lamsade.dauphine.fr/cahiers.html).

APPENDIX SOLUTIONS OF PARAMETRIC SYSTEM I. Sequence of systems: ⎧ z11 + z 15 = z 52 , ⎪ 1 2 2 ⎪ z = z1 + z 3 , k = 2 : ⎨ 13 1 2 ⎪ z1 + z 2 = z 2 , ⎪z1 = z 2 + z 2 . 1 6 ⎩ 6

⎧ z11 + z15 = z52 , ⎪ 1 2 2 ⎪ z3 = z1 + z 3 , 1 1 2 ⎪z + z = z , 2 2 ⎪ 11 2 2 ⎪ z 6 = z1 + z 6 , k = 3: ⎨ 2 2 3 ⎪ z1 + z 5 = z 5 , ⎪z 2 = z3 + z3, 1 3 ⎪ 32 2 3 z z z + = ⎪ 1 2 2, ⎪ z 2 = z 3 + z 3. 1 6 ⎩ 6

... II. Sequence of solutions: k 2

Basis solutions ( z 14 ) , ( z 42 ) , ( z 12 , z 22 ) , ( z 13 , z 32 ) , ( z 15 , z 52 ) , ( z 16 , z 62 ) , ( z 13 , z 16 , z12 ) , ( z11 , z 22 , z 52 ) .

3

( z 14 ) , ( z 42 ) , ( z 43 ) , ( z 12 , z 22 , z 23 ) , ( z 13 , z 32 , z 33 ) , ( z 15 , z 52 , z 53 ) , ( z 16 , z 62 , z 63 ) , ( z 13 , z 16 , z12 , z 23 , z 53 ) , ( z11 , z 22 , z 52 , z 23 , z 53 ) ,

( z 13 , z 16 , z 32 , z 62 , z13 ) .

4

( z 14 ) , ( z 42 ) , ( z 43 ) , ( z 44 ) ,

( z 12 , z 22 , z 23 , z 24 ) , ( z 13 , z 32 , z 33 , z 34 ) , ( z 15 , z 52 , z 53 , z 54 ) , ( z 16 , z 62 , z 63 , z 64 ) , ( z 13 , z 16 , z12 , z 23 , z 53 , z 24 , z 54 ) ,

( z 13 , z 16 , z 32 , z 62 , z13 , z 24 , z 54 ) , ( z 13 , z 16 , z 32 , z 62 , z 33 , z 63 , z14 ) , ( z11 , z 22 , z 52 , z 23 , z 53 , z 24 , z 54 ) .

5

( z 14 ) , ( z 42 ) , ( z 43 ) , ( z 44 ) , ( z 45 ) , ( z 12 , z 22 , z 23 , z 24 , z 25 ) , ( z 13 , z 32 , z 33 , z 34 , z 35 ) ,

( z 15 , z 52 , z 53 , z 54 , z 55 ) , ( z 16 , z 62 , z 63 , z 64 , z 65 ) , ( z 13 , z 16 , z12 , z 23 , z 53 , z 24 , z 54 , z 25 , z 55 ) , ( z 13 , z 16 , z 32 , z 62 , z13 , z 24 , z 54 , z 25 , z 55 ) ,

( z 13 , z 16 , z 32 , z 62 , z 33 , z 63 , z14 , z 25 , z 55 ) , ( z 13 , z 16 , z 32 , z 62 , z 33 , z 63 , z 34 , z 64 , z15 ) , ( z11 , z 22 , z 52 , z 23 , z 53 , z 24 , z 54 , z 25 , z 55 ) .

267

268