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A Verification Concept for SDL Systems and its Application to the Abracadabra Protocol

A Verification Concept for SDL Systems and its Application to the Abracadabra Protocol
A Verification Concept for SDL Systems and its Application to the Abracadabra Protocol

A Verification Concept for SDL Systems and its Application to the Abracadabra

Protocol

Markus Rinderspacher

Institut für Logik, Komplexit?t und Deduktionssysteme

Interner Bericht 14/94

Abstract

SDL is a specification language to specify distributed systems. Especially it is suitable for communication protocols. In some cases however it is not enough to describe just the behaviour of a protocol, but there are formulated some additional properties as requirements of the SDL system. A formalism convenient to describe them is for example first order logic. Our approach is to prove such properties with methods of automated reasoning after transforming the SDL specification into a first order logic specification. The proofs are done with the program verification system Tatzelwurm, especially with its prover.

Practical experience shows that it is convenient to do a proof in two steps. In the first step the behaviour of the system is calculated out of the behaviour of the agents. The proofs of this step is independent of the property to prove. In this report we give a proof methods containing instructions how the arguments are applied during these proofs. It is shown how reachability analysis is done during a formal proof and how fairness arguments are applied.

The report contains two papers, where the first one describes the formal basis of the method and shows the proof obligations occurring verifying a communication protocol. The second paper shows how some tedious tasks can be done more elegant using rewrite rules and recursive equations.

In the appendix we give two examples out of the verification of the Abracadabra Protocol.

Contents

1. Proof Structuring with Fixed Point Theory4 1.1. Introduction4 1.

2. Fixed points of Functions over Lists5 1.

3. Development of a Correctness Proof6 1.

4. Practical experience13

1.5. Conclusion13

2. Systematic Verification of in SDL specified Protocols14 2.1. Introduction14 2.2. The Description of the behaviour of a SDL system15 2.

3. Explicit behaviour21 2.3.1. Solving System Equations21 2.3.1.1.An example21 2.3.2. Liveness27

2.4. Evaluation28

3. References30 Appendix:

I. The specification of the Abracadabra Protocol31 II.The complete presented example33 III. Another example48

1. Proof Structuring with Fixed Point Theory

Abstract

Starting from a purely functional description of a communication protocol, we present a method how correctness proofs including safety- and progress properties can be developed systematically with an automatic theorem prover. We show how a complex proof can be divided into smaller ones due to proof arguments typically occurring in the area of protocol verification. Experience with this method shows that proofs can be developed with an acceptable amount of work.

1.1. Introduction

The objective we pursue with our work is to verify properties of communication protocols. We start with an automata based description of the protocol and a property p given in terms of first-order logic. Due to Broy [Bro87,91] we model the behaviour of the protocol agents with functions over streams and the system behaviour with its least fixed point c. If we use this formalism, we can reduce the verification problem to a proof of the theorem

lf(c) → p(c).

lf is a logical formula formalizing that c is the least fixed point of the system regarded. To find the proof of this theorem we use the program verification system Tatzelwurm [K?U891] with its integrated tableau prover. This prover is designed to prove formulae occurring in program verification. It is enlarged by decision procedures [K?U892] for theories typically occurring in the verification of programs and protocols, like arithmetic, records and lists. Furthermore Tatzelwurm supports interactive theorem proving with a powerful user-interface. Interactive proving is necessary, since automatic proofs of such complex problems seem to be impossible because of the enormous search space, which must be managed.

Nevertheless interactive theorem proving is not the solution of all problems. If we try to find a proof for the above theorem, we will get completely lost, if we have no method how to find this proof.

The presented method is part of our experience with the verification of the Abracadabra protocol [Isc89]. There we took a paper [Bro87], which suggested, how the behaviour of the Abracadabra-protocol can be modelled and how its correctness proof can be found. We did this proof with Tatzelwurm and were successful after we had made some essential changes to the suggestions of Broy.

Respect to our experience we suggest for future verification projects two steps:

(1) calculate the fixed point of the system explicitly and

(2) prove the property.

Calculation of the fixed point is a subgoal occurring in every protocol verification project. It causes a lot of work, so that here a method reduces proof expense.

We organize this paper by giving in Section 2 some results of fixed point theory and especially

the fixed point theorem, the basis of our reasoning.

In Section 3 we will give a formalization of a protocol and show that a correctness proof can be systematically developed applying well-known arguments like reachability and fairness. Section 4 collects some experience obtained proving with Tatzelwurm and finally Section 5 concludes the results and gives hints, how to continue in future.

1.2. Fixed points of Functions over Lists

In this section we want to give some results of fixed point theory, which is the basis for our later argumentation. We regard fixed points in complete partial orders. The following definitions and theorems can be found in [Loe87].

Let (S, ≤) be a partial order, then an element m ∈ S is called the least(greatest) element of a set S, if m ≤ s (s ≤ m) for all s ∈ S

Definition 1.2.1.: (Least Upper Bounds)

Let (D, ≤) be a partial order and S a (possibly empty) subset of D. An element u ∈ D is said to be an upper bound of S (in D), if d ≤ u for all d ∈ S; u is said to be the least upper bound (lub) of S (in D), if u is the least element of the set of all upper bounds of S in D.

Note that the least upper bound is uniquely determined, if it exists. In the following a totally ordered subset S of D is called a chain.

Definition 1.2.2.: (Complete Partial Orders)

A partial order (D, ≤) is a complete partial order (cpo), if the following two conditions hold:

1.The set D has a least element.

2.For every chain S in D the least upper bound S exists.

Definition 1.2.3.: (Continuity)

Let (D, ≤) and (E, ≤) be cpo's. A function f: D → E is said to be continuous, if for every chain S in D, f(S) = f(S).

Theorem 1.2.4.: (The Fixed Point Theorem)

Let (D, ≤) be a cpo and f: D → D be a continuous function. Then f has a (uniquely determined) least fixed point mf = {f i(⊥) | i ∈ Nat}, where ⊥ is the least element of the cpo (D, ≤).

Now we want to apply the fixed point theorem to lists and functions over lists. The following theorems are easy to prove. Let G be a set of objects and let L(G) be the possibly infinite lists with elements of G. The application of the fixed point theorem demands cpos and continuous functions. We have to add a least element ω to G and obtain a flat partial order (Gω, ≤). We define a prefix ordering ? on the lists and add an element ε, which is called the empty list. We

consider these lists L(Gω) as a subset of the function set ((Nat → Gω), ≤), which is known to be a cpo. The following theorem establishes that (L(Gω), ?) is a cpo. We simplify our notation in writing L instead of L(Gω), assuming the element sort of the list is a flat partial order.

Theorem 1.2.5.: (L(Gω), ?) is a sub-cpo of ((Nat → Gω), ≤)

Now we study the continuity of functions in more detail. We need some criteria, which allow to check, whether the functions are continuous.

At first we examine the standard list functions cons, car, cdr and an additional function app, which appends a list to a finite list.

Theorem 1.2.6.: The functions cons, car, cdr are continuous in their list argument. The function app is continuous in its second argument.

We write in the following & instead of cons and app. Examining the arguments allows us to determine, which of the constructors is meant. As shortcut we write m k for the list having exactly k elements m. Later we will describe processes by functions over lists. The following theorem allows us to construct continuous functions using continuous functions and some conditions. It will be used to establish the continuity of the functions defined 3.3 and 3.4. Theorem 1.2.7.:

Let I be a finite index set and i, j ∈ I. Let {f i | i ∈ I} ? [L → L] be a finite set of continuous functions and {h i | i ∈ I} ? [G → G]. Let B i ? L, where B i ∩B j = ? and i = L, with:

if l ? l’ and l ∈ B i then l’ ∈ B i.

By the following a continuous function f is defined:

(1)f(ε) = ε

(2)f(cons(g, l)) = cons(h

i (g), f

i

(l)) if cons(g, l) ∈ B i for all i ∈ I 1.3. Development of a Correctness Proof

In this section we want to show, how fixed point theory can be used to develop correctness proofs systematically. The example used here is taken out of the verification of the Abracadabra-protocol [Isc21], which operates over a full-duplex communication medium between two agents. The medium may occasionally lose messages, but will not disorder, corrupt, invent or duplicate messages.

The two stations communicate transferring Protocol Data Units (PDUs). The service, provided by the protocol must be reliable, in the sense that the protocol guarantees the transmission of the data, coming from the user of the service. We assume, that we know in advance which of agents is the sender and the receiver respectively. Also we split up the bidirectional medium into

two unidirectional media mu, having the same behaviour.

Fig 1.3.1. shows the system regarded in this paper, where the names in brackets are the functions describing the behaviour of the agents. The protocol has to transmit a sequence s of data. We have to proof that this sequence is eventually transmitted.

Fig. 3.1. The system regarded

The first problem, to be solved is the calculation of the system behaviour using the behaviour of the cooperating agents. To transmit the data the agents send and receive messages.Lists of messages are called streams. We describe the behaviour of the agents with continuous functions over streams, called stream processing functions. Then we define the behaviour of the system by a recursion equation, which has a solution due to the fixed point theorem.

The sender transmits s to the receiver by sending a stream c(s), which depends beside s on the implementation of the sender and on the input it gets from the medium. Due to [Bro87] c(s) is the solution of the equation c(s) = con(f1(lic(f2(c(s))))). Due to the fixed point theorem the solution is uniquely determined, provided the functions are continuous.

To define the behaviour of the agents we assume that a message is one of the in pairs different objects {em, cr, cc, dt, ak, dr, dc}. It is dt a two place, ak a one place function symbol and the rest are constant symbols. dt takes as argument a data and a control-bit and ak a control-bit. em is used to model, that the medium does not deliver one of the other messages. The control-bit is assumed to be one of the boolean objects {tt, ff}. We have a function symbol non, to define a function mapping tt to ff and vice versa. In the following are m, m’arbitrary messages, c, c’ streams, d, d’ data, s, s’ list of data and a, b boolean.

Definition 1.3.1.: (Medium Function)

A medium function f: L → L is a function having the following properties:

1f(ε) = ε

2.car(f(m & c)) = m or car(f(m & c)) = em

3.f(m & c) = f(m & ε) & f(c)

Lemma 1.3.2: A medium function is continuous

Proof: (a sketch) It is a well-known fact, that it is enough to prove that f is monotone and

f(S) ? f(S) for every chain S with an infinite number of elements

1.Monotony: Let c ? c’. If c is infinite, c’ is also infinite and c = c’. Then f(c) = f(c’),

which is sufficient for f(c) ? f(c’).

If c is a finite stream, we can write c’ = c & c’’. Since f(c) ? f(c) & f(c’’), f(c) ? f(c’) holds

2.f(S) ? f(S): Let S = {c i | i ∈ Nat} be a chain. Then f(S) := {f(c i ) | i ∈ Nat} is also a chain. Since S and f(S) have an infinite number of elements, so S and f(S) are infinite streams. Since f(S) is an infinite stream and f(S) ? f(S), f(S) = f(S) holds.

We show a sketch of the behaviour of the system in Fig. 1.3.1. It is shown by the following state transition diagrams and the parallel operator . The state labeled with x means termination and the transitions are labeled with pairs (i; o) of I/O-signals, where - means no input (output).We denote the indeterminism of the medium by τ-transitions. The other indeterminisms are occurring, because the diagram does not describe all the details. It is omitted that the agents‘sender and receiver have an additional boolean control variable, which makes them deterministic.

Fig. 1.3.2. The components of the system

Sender:Receiver:Medium:

em; em

dt; ak

em; cr

ak; dt em; dr The full behaviour of these agents is given by the following functions. For each state we define a function typically having three arguments, the input stream, the data to be transmitted, and the state of the control variable. If arguments are not necessary, we omit them.

Definition 1.3.3: (Sender)

A sender-function con is a function, having the following properties:

(Con)con(c, s) = cr & itc(c, s)

(Itc 1)itc(cc & c, d & s) = dt(d, tt) & itt(c, d & s, tt)

(Itc 2)itc(em & c, s) = cr & itc(c, s)

(Itt 1)itt(ak(a) & c, d & d’ & s, a) = dt(d’, non(a)) & itt(c, d’ & s, non(a))

(Itt 2)itt(ak(a) & c, d & ε, a) = dr & itd(c)

(Itt 3)(em = m ∨ cc = m ∨ m = ak(non(a)))

→ itt(m & c, d & s, a) = dt(d, a) & itt(c, d & s, a)

(Itd1)itd(dc & c) = ε

(Itd2)? dc = m → itd(m & c) = dr & itd(c)

(Id)For the rest of the streams the functions are the identity on their stream.

Definition 1.3.4: (Receiver)

A receiver-function lic is a function, having the following properties:

(Lic1)lic(cr & c) = cc & lib(c)

(Lic2)? cr = m → lic(m & c) = em & lic(c)

(Lib1)lib(dt(d, a) & c) = lit(dt(d, a) & c, tt)

(Lib2)(cr = m ∨ em = m) → lib(m & c) = cc & lib(c)

(Lit1)lit(dt(d, a) & c, a) = ak(a) & lit(c, non(a))

(Lit2)lit(dr & c, a) = lid(dr & c)

(Lit3)m = dt(d, non(a)) ∨ em = m → lit(m & c, a) = ak(non(a)) & lit(c, a)

(Lid1)lid(dr & c) = dc & lid(c)

(Lid2)? dr = m → lid(m & c) = dc & lid(c)

(Id)For the rest of the streams the functions are the identity on their stream.

Note that these functions are continuous respect to 1.2.7. We have described the system behaviour with stream-processing functions of its components and the equation c(s) = con(f1(lic(f2(c(s))))). To prove properties however we have to solve this equation, which is the same as to calculate the fixed point explicitly.

We do this by manipulating the fixed point equation in a systematic way. In our example we define 6 more equations and fixed points and 16 theorems describing the relations between these fixed points.

At first we give these equations and theorems and then we show using an example how we get the fixed point equations and the theorems successively.

We define fixed points by following equations:

)c(s) = con(f1(lic(f2(c(s)))))

(F

)r1(s) = itc(f1(cc & lib(f2(r1(s)))), s)

(F

1

(F

)s ≠ε→ r2(s) = itt(f1(lib(f2(dt(car(s), tt) & r2(s)))), s, tt)

2

)s ≠ε→ r3(s, a, non(a)) = itt(f1(ak(non(a)) & lit(f2(r3(s, a, non(a))), a)), s, non(a))

(F

3

)s ≠ε→ r4(s, a, a) = itt(f1(lit(f2(dt(car(s), a) & r4(s, a, a)), a)), s, a)

(F

4

(F

)r5(a) = itd(f1(lit(f2(dr & r5(a)), a)))

5

)r6 = itd(f1(dc & lid(f2(r6))))

(F

6

The relations between the fixed points are given by the following theorems:

)c(s) = cr & r0(s)

(S

1

)car(f2(cr & r0(s))) ≠ em → r0(s) = r1(s)

(S

2

(S

)car(f2(cr & r0(s))) = em → r0(s) = cr & r0(s)

3

(S

)s ≠ε∧ car(f1(cc & lib(f2(r1(s))))) ≠ em → r1(s) = dt(car(s), tt) & r2(s)

4

(S 5)

car(f 1(cc & lib(f 2(r 1(s))))) = em → r 1(s) = cr & r 1(s)(S 6)

car(f 2(dt(car(s), tt) & r 2(s))) ≠ em → r 2(s, i, k) = r 3(s, ff, tt)(S 7)

s ≠ ε ∧ car(f 2(dt(car(s), tt) & r 2(s))) = em → r 2(s) = dt(car(s), tt) & r 2(s)(S 8)

s ≠ ε ∧ cdr(s) ≠ ε ∧ car(f 1(ak(non(a)) & lit(f 2(r 3(s, a, non(a))), a))) ≠ em → r 3(s, a, non(a)) = dt(car(cdr(s)), a) & r 4(cdr(s), a, a)(S 9)s ≠ ε ∧ car(f 1(ak(non(a)) & lit(f 2(r 3(s, a, non(a))), a))) = em

→ r 3(s, a, non(a)) = dt(car(s), non(a)) & r 3(s, a, non(a))

(S 10)s ≠ ε ∧ car(f 2(dt(car(s), a) & r 4(s, a, a))) ≠ em → r 4(s, a, a) = r 3(s, non(a), a)

(S 11)s ≠ ε ∧ car(f 2(dt(car(s), a) & r 4(s, a, a))) = em

→ r 4(s, a, a) = dt(car(s), a) & r 4(s, a, a)

(S 12)s ≠ ε ∧ cdr(s) = ε ∧ car(f 1(ak(non(a)) & lit(f 2(r 3(s, a, non(a))), a))) ≠ em

→ r 3(s, a, non(a)) = dr & r 5(a)

(S 13)car(f 2(dr & r 5(a))) ≠ em → r 5(a) = r 6

(S 14)car(f 2(dr & r 5(a))) = em → r 5(a) = dr & r 5(a)

(S 15)car(f 1(dc & lid(f 2(r 6)))) ≠ em → r 6 = ε

(S 16)car(f 1(dc & lid(f 2(r 6)))) = em → r 6 = dr & r 6

As an example we show how we can deduce the fixed point equation F 2 and the theorems S 4 and S 5 from F 1.

(F 1) r 1(s) = itc(f 1(cc & lib(f 2(r 1(s)))), s)

Due to the indeterminism of the medium we have to treat two cases.

1. Suppose f 1(cc & lib(f 2(r 1(s)))) = cc & f 1(lib(f 2(r 1(s)))).

If we assume, that s ≠ ε, we get with the definition (Itc 1 ) of 1.3.3.

r 1(s) = dt(car(s), tt) & itt(f 1(lib(f 2(r 1(s)))), s, tt), which leads to

cdr(r 1(s)) = itt(f 1(lib(f 2(dt(car(s), tt) & cdr(r 1(s))))), s, tt),

which is a fixed point equation, which has a least fixed point, we name it r 2(s). We get:

(F 2) s ≠ ε → r 2(s) = itt(f 1(lib(f 2(dt(car(s), tt) & r 2(s))), s, tt).

Since the least fixed point is uniquely determined, it is r 1(s) = dt(car(s), tt) & r 2(s).

We get (S 4) by collecting the assumptions we made.

2. We assume f 1(cc & lib(f 2(r 1(s)))) = em & f 1(lib(f 2(r 1(s)))).

With the definition (Itc 2 ) of 1.3.3. we get

r 1(s) = cr & itc(f 1(lib(f 2(r 1(s)))), s), which leads to

cdr(r 1(s)) = itc(f 1(lib(f 2(cr & cdr(r 1(s))))), s)

Applying the definition of the medium and (Lib 2) of 1.3.4. we get

cdr(r 1(s)) = itc(cc & f 1(lib(f 2(cdr(r 1(s))))), s)

Since the least fixed point is uniquely determined, it′s r 1(s) = cr & r 1(s).

We get (S 5) by collecting the assumptions we made.

We received the theorems S 1, ..., S 16 by manipulating of the fixed point equations. We illustrate them in the following diagram graphically. We got the insight that manipulation of fixed point equations in the described way is a reachability analysis during a proof. The fixed points can be regarded as states and the relations between the fixed points can be viewed as reachability graph. In this case the graph describes the behaviour on the senders point of view. The sender is not able to recognize, whether the medium described by f 2 transmits the signal or not. This is reflected by the τ-transition in the diagram.

Fig. 1.3.3. Relations between the fixed points

em; dt We have examined the system behaviour without fairness until now. Fairness should guarantee that the system eventually leaves a state after entering it. We systematically derive a formulation of the system behaviour including fairness. To speak in terms of Fig. 1.3.3. we show that the small loops eventually terminate, provided that the medium is fair .

Definition 1.3.5: (Fair Medium Function)

A fair medium function f is a medium function with the following additional property

? k. f(m k+1 & c) = f(em k & m & c)

With this property we are able to prove following theorems, which enlarges the first formalization of the system to a formalization containing liveness.

(S a ) ? k. c(s) = cr k+1 & r 1(s)

(S b ) s ≠ ε → ? k. r 1(s) = cr k & dt(car(s), tt) & r 2(s)

(S c ) s ≠ ε → ? k. dt(car(s), tt) & r 2(s) = dt(car(s), tt)k+1 & r 3(s, ff, tt)

(S d ) s ≠ ε ∧ cdr(s) ≠ ε

→ ? k. r 3(s, a, non(a)) = dt(car(s), non(a))k & dt(car(cdr(s)), a) & r 4(cdr(s), a, a)

(S e ) s ≠ ε → ? k. dt(car(s), a) & r 4(s, a, a) = dt(car(s), a)k+1 & r 3(s, non(a), a)

(S

f

) s ≠ε∧ cdr(s) = ε→? k. r3(s, a, non(a)) = dt(car(s), non(a))k & dr & r5(a)

(S

g

) ? k. dr & r5(a) = dr k+1 & r6

(S

h

) ? k. r6 = dr k

It is now possible to prove properties of the formalized system. We prove properties, which are formalized by requirements on fixed points with induction over the list s. Induction over lists

demands that the lists are finite. Note that the loop consisting of state r

3 and r

4

is terminating, if

the list s is finite.

We proof now by induction the property, that the sender transmits all the data of s. Let k be a list of integers. To express a property at first we define functions messc and messt, which extract the transmitted data out of a list of messages. messc and messt have the following properties:

(1) messc(ε) = messt(ε, a) = ε

(2) messt(dt(d, a) & c, a) = d & messt(d, non(a))

(3) messt(dt(d, non(a)) & c, a) = messt(dr & c, a) = messt(c, a)

(4) messc(cr & cr & c) = messc(cr & c)

(5) messc(cr & dt(d, a) & c) = messt(dt(d, a) & c, tt)

(6) messc(cr & dr & c) = messt(dr & c, tt)

(7) exp(ε, k, a) = ε

(8) exp(d & s, i & k, a) = d i & exp(s, k, non(a))

Corollary 1.3.6.: It is messt(exp(s, k, a), a) = messt(exp(s, k, non(a)), non(a)).

Theorem 1.3.7.: If s ≠ε then messc(c(s)) = s for all finite lists of data s.

Proof: (Induction over the length of s)

Assume for an arbitrary s messc(c(s)) = s holds.

There are k, k’, k, k’’ such that c(cons(d, s)) = cr k& dt(d, tt)k’ & exp(s, k, ff) & dr k’’

Since c(s) has the form c(s) = cr k& exp(s, k, tt) & dr k’’,

it’s messc(c(s)) = messt(exp(s, k, tt) & dr k’’, tt) = s and

messc(c(d & s)) = messt(dt(d, tt)k’ & exp(s, k, tt) & dr k’’, tt)

= d & messt(exp(s, k, ff) & dr k’’, ff) = d & s.

1.4. Practical experience

We proved the theorem lf(c) → p(c) given in section 1 with Tatzelwurm in three steps:

(1a)lf(c) → S

1∧ ... ∧ S

16

, using the fact that c is the least fixed point

(1b)S

1∧ ... ∧ S

16

→ S

a

∧ ... ∧ S

h

, using the fairness property of the medium

(2)S

a ∧ ... ∧ S

h

→ p(c)

To do these proofs with Tatzelwurm we needed about 25 h to prove S

1, ..., S

16

, 37 h to prove

S a , ..., S

h

and 12 h to prove p. The proof of the five lemmata took 4 h.

What is not included in these times, is the time we spent on planning the proof. It took a very long time (5 months) until we found a way applying the proof arguments of the proofs (1a, b) systematically. The manner of applying these arguments is general so that we hope, that we can reduce the time spending on protocol verification closely to the above times.

1.5. Conclusion

It has been shown that fixed point theory is not only convenient to describe the behaviour of protocols, but it is also possible to find correctness proofs systematically. We saw how well-known arguments as reachability and fairness are applied during the correctness proof.

The systematic application of proof arguments promises soon to reduce verification effort drastically, so that we hope to tackle real world protocols successfully.

Our next subgoal is to implement the described method with our proof programming language.

2. Systematic Verification of in SDL specified Protocols

Abstract

SDL is a language convenient to describe communication protocols. We have to verify them formally, if they are safety critical systems. Formal verification however is a very complex task, which needs to be supported by methods and tools. We present a method how correctness proofs including safety- and liveness properties can be developed systematically. We start from a purely functional description of a communication protocol, which is mechanically derived out of the SDL description. We separate its proof into several parts and show which proof methods are convenient to tackle them. Experience with this method shows that proofs can be developed with an acceptable amount of work.

2.1. Introduction

A SDL system description contains a number of process descriptions, which are in the following called agents. An agent is an extended finite automata, which is built up from states and transitions. A state of an agent consists of one of a finite number of agentstates and of a datastate, a function σ, mapping the agents variables to their values.

There are two possibilities to prove properties of SDL systems. The first possibility is to extend modelcheckers with datastates. A contribution to this task can be found in [Di92]. As alternative we use a theorem prover, which has the advantage that we don’t have to extend it in order to prove system properties expressible in first order logic. The praxis however shows that it is necessary to use a special purpose prover to do verification successfully. A prover which is designed for proofs occurring in program verification is the prover of the Tatzelwurm system [K?u89a,b]. It is a tableau prover [Fit90] extended by reduction procedures treating arithmetic, records and lists, datastructures typically occurring in programs. These extensions are also very helpful for protocol verification problems. However case studies showed that we can make use of the automata structures of the agents in order to define a concept which optimizes the verification process. The introduction of this concept is the objective of this paper.

We suggest to do correctness proofs in two steps. In the first step we calculate the behaviour of the system out of the behaviour of the agents. In the second step we prove the property.

In this paper we show how the first step of the proof can be done systematically. There are reflected typical tasks like reachability analysis and fairness while verifying with the presented method.

The concept bases on the formalism suggested by M. Broy [Bro87, 91], who describes systems of communicating processes with stream processing functions. However the work of Broy lacks an approach to its automatization, which is the contribution of this paper.

Now we want to summarize some characteristic properties and illustrations of the concept.

At first we have to transform the SDL system into a formal system, including a proof theory.

*We transform a SDL process description into a system of conditional rewrite rules.

*We transform the SDL system description into a equation system eq.

*The system behaviour is the solution of eq.

*We formalize the desired property as requirement on the solution of eq.

To calculate the behaviour of the SDL system we have to solve the equation system eq.

*We apply rewrite rules as long as there are some applicable.

*If there are no more rewrite rules applicable, we transform the equation system eq with its solution c to a equation system eq’ with its solution c’ and establish a relation between c and c’. Then we proceed solving eq’.

*The process terminates, if all the derived systems of equations are solved.

The following correspondences illustrate the verification process.

*A transition is simulated in the formal system by the application of a rewrite rule.

* A system state corresponds to a equation system and its solution.

*A relation between the solutions of the equation system corresponds to a system transition.

*All solutions of all systems of equations together with its relations can be seen as reachability graph.

*Additional assumptions about the behaviour of the system, like Fairness lead to a drop of solutions.

The following paper stresses the calculation of solutions of the system of system equations.

The solution is a n-tupel {c

1, ..., c

n

}.

We formulate the system properties by defining predicates which characterize correct solutions.

For example define P to be the predicate, formalizing that c

i and c

j

contain the same sequence

of data. The proof of the validity of P(c

i , c

j

) is a pure predicate logic proof, which can be

done with Tatzelwurm.

2.2. The Description of the behaviour of a SDL system

The following section describes, how a SDL system description is mapped into a formal system. The agents of a SDL system are communicating with each other by sending and receiving messages. The signals, received and sent by the agent are carrying parameters.

We call a (possibly infinite) sequence of messages a stream. We define some operations on streams. The function car takes a stream and returns its first element. The function cdr takes a stream and returns it without its first element. The function & takes either two streams and returns their concatenation or it takes a message and a stream and returns a stream, where the message is added to the input stream. We assume that ε is the empty stream.

We describe the behaviour of the agents by stream processing functions, mapping input streams to output streams.

Fig 2.2.1. shows a typical architecture of a communication protocol. We have two protocol

agents, two medium agents and two user agents. The streams are called c 1, ... , c 8. In 2.2.1.the behaviour of the protocol agents is described by the functions g , h , a user agent is described by a function u , and the medium agents are described by the medium function med .

Fig. 2.2.1. A typical protocol architecture

We define the behaviour of the system to be the solution of the following equation system:c 1 = u (c 2); c 2 = g (c 1, c 4); c 3 = h (c 1, c 4); c 4 = med (c 6);

c 5 = me

d (c 3); c 6 = h (c 7, c 5); c 7 = u (c 8); c 8 = g (c 7, c 5);

If we want to express properties, we do this by defining a predicate which is assumed to evaluate to true , if the property is fulfilled and false if not. For example, we define a predicate P with val (P (c , c ’)) = true iff “c and c ’ contain the same data”, where val is assumed to be an evaluation function of first order formulae.

To prove for example that val (P (c 1, c 8)) = true , we proceed in three steps:

1.We transform the SDL specification into a formal system, including a system of equations and rewrite rules, which are implicit definitions of the functions u , g , h , med .

2.We solve the equation system, using the derived rewrite rules.

3.We prove the required property, using the calculated solution.

In the remaining we describe how we derive a set of rewrite rules out of a SDL process description. If we analyze SDL, we realize that we have four elementary actions.

(1) Agent reads a signal.

(2)Agent evaluates a condition

(3)Agent writes a signal

(4)Agent assigns a value to a variable

In a diploma thesis [Mü93] has been shown that the SDL language constructs can be transformed into a sequence of elementary actions. Assume that the descriptions of the agents only consist of elementary actions. We regard the names of the SDL states as program labels and introduce additional labels g i between elementary actions. The labels are mapped into 2-ary functionsymbols in the formal system. The functions (stream processing functions) take a datastate σ and an input stream i as their arguments and return an output stream. We define that σ[s ] means that σ is evaluated at the place s and that σ[s ← t][s ] = t. The relation

“::=” describes a relation between the functions. A evaluation of a condition is mapped to a conditional rewrite rule, where ? separates the condition from the rewrite rule. The following example shows a SDL state definition and its mapping to a system of rewrite rules.

[b ← tt], i)5

s =

ε

g (σ, i) ::= dt(car(s), tt) & g (σ, i)

5

cr g (σ, i) ::= cr & itc(σ, i) 1

itc(σ, cc & i) ::= g (σ, i)2

σ[s] = ε ? g (σ, i) ::= g (σ, i)23σ[s] ≠ ε ? g (σ, i) ::= g (σ, i)24

dt(car(s), tt)

yes no itc(σ1

dr g (σ, i) ::= dr & itd(σ, i)3Fig. 2.2.2. Transformation of Agent Behaviour

In the next step we simplify the representation omitting the intermediate states, because it is inconvenient to derive such a lot of rewrite rules from the SDL text. We get for every transition path one rewrite rule, if we combine all rules situated on it. The following Fig. 2.2.3. shows the result of the transformation of the SDL fragment to rewrite rules in our formal system.Proceeding in this way we can transform SDL agent descriptions into a system of rewrite rules.The functions describing the behaviour of the system are defined implicitly by the rules, i.e.they are the models of this formal system.

← tt], i)

itc(

Fig. 2.2.3. An agent description and its mapping

to a system of rewrite rules

For further presentation we study a simplified version of the Abracadabra Protocol [Isc21].Fig. 2.2.4. shows the architecture of this version. The protocol description is simplified in two aspects. First we use a simplified user interface. We assume that the user hands over a sequence s of data to the protocol. Second we assume, that we know which of the processes wants to transmit s. We describe the behaviour of the sender with the function con and the behaviour of the receiver with the function lic .

Fig. 2.2.4. A simplified version of the Abracadabra Protocol

We give now an informal description of the behaviour of the Abracadabra Protocol. The two stations communicate transferring Protocol Data Units (PDUs). Communication proceeds in the three sequential phases: Connection , Data Transfer and Disconnection . We assume that the medium is unreliable. An attempt to transmit a message m by a medium may succeed or fail.We model the failure of a transmission attempt by receiving a special message em.

a. Connection Phase

The sender sends a cr , a Connection Request PDU. If it receives cc, a Connection Confirmation PDU, it proceeds to the data transfer phase, if there are some data to transmit. If there are no data to transmit it sends dr, a Disconnect Request PDU and enters the disconnection phase. If it receives em , a cr is sent .

b. Data Phase

The sender sends dt , a Data Transfer PDU. If it receives ak , an acknowledgement PDU it may send a further dt . Each dt and ak bears a control bit, which is assumed to be tt or ff ..Successive dt’s carry alternating values in sequence starting at tt after connection. The correct acknowledgement to a dt contains the control bit of the next dt expected. The reception of an em or the wrong control bit causes a further attempt to transmit dt .

c. Disconnection Phase

The sender sends dr , a Disconnection Request PDU. If it receives a dc , a disconnection confirmation PDU, it considers the disconnection to be complete. If it receives em , it tries another time to transmit dr .

Fig. 2.2.5. shows the behaviour of the agents of the protocol using a finite state machine. Note that this is only an approximation of the behaviour of the protocol, because the datastates are omitted.

Fig. 2.2.5. The components of the Abracadabra Protocol

Sender:Receiver:Medium:

em; em

em; cr cc, ak, em; dr em, dr; dc The rewrite rule representation, given in For. 2.2.1. describes the full behaviour of the protocol, including the datastates. The rewrite rules are constructed in the following way. They are conditioned, where C is assumed to be a quantor free formula, which is evaluated to true or false with respect to the actual datastate σ. f , g are function symbols, m , m ’ are messages and i is a stream.

C (σ) ? f (σ, m & i ) ::= m ’ & g (σ, i )

In some cases the right hand side is just ε. These rules correspond to actions, terminating a process.

pqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq+qüw Definition 2.2.1.: (The behaviour of the sender)w w Connection Phase w w(1)con(i) ::= cr & itc(σ, i)w w(2)itc(σ, em & i) ::= cr & itc(σ, i)w w(3)σ[s] ≠ε ? itc(σ, cc & i) ::= dt(car(σ[s], tt)) & itt(σ[b← tt], i)w w(4)σ[s] = ε ? itc(σ, cc & i) ::= dr & itd(i)w w Data Phase w w(5)cdr(σ[s]) ≠ε ∧ σ[b] = v w w? itt(σ, ak(v) & i) ::= dt(car(cdr(σ[s])), non(v)) w w& itt(σ[(s, b) ← (cdr(σ[s]), non(v))], i)w w(6)cdr(σ[s]) = ε ∧ σ[b] = v? itt(σ, ak(v) & i) ::= dr & itd(i)w w(7)itt(σ, em & i) ::= dt(car(σ[s], σ[b]) & itt(σ, i) w w(8)itt(σ, cc & i) ::= dt(car(σ[s], σ[b]) & itt(σ, i)w w(9)σ[b] = v? itt(σ, ak(non(v)) & i) ::= dt(car(σ[s], σ[b]) & itt(σ, i)w w Disconnection Phase w w(10)itd(dc & i) ::= εw w(11)itd(cc & i) ::= dr & itd(i)w w(12)itd(em & i) ::= dr & itd(i)w w(13)itd(ak(v) & i) ::= dr & itd(i)w w w w Definition 2.2.2.: (The behaviour of the receiver)w w Waiting for a connect request w w(1)lic(cr & i) ::= cc & lib(σ, i)w w(2)lic(em & i) ::= em & lic(i)w w Waiting until the connection is established w w(3)lib(σ, dt(v, tt) & i) ::= ak(tt) & lit(σ[b← ff], i)w w(4)lib(σ, dt(v, ff) & i) ::= cc & lib(σ, i)w w(5)lib(σ, cr & i) ::= cc & lib(σ, i)w w(6)lib(σ, em & i) ::= cc & lib(σ, i)w w(7)lib(σ, dr & i) ::= dc & lid(i)w w Data Transfer Phase w w(8)σ[b] = v’ ? lit(σ, dt(v, v’) & i) ::= ak(v′) & lit(σ[b← non(v’)], i)w w(9)σ[b] ≠v’ ? lit(σ, dt(v, v’) & i) ::= ak(v′) & lit(σ, i)w w(10)lit(σ, em & i) ::= ak(non(σ[b])) & lit(σ, i)w w(11)lit(σ, dr & i) ::= dc & lid(i)w w Waiting for a disconnect confirmation w w(12)lid(em & i) ::= dc & lid(i)w w(13)lid(dr & i) ::= dc & lid(i)w w w w Definition 2.2.3.: (The behaviour of the medium)w w The behaviour of the medium does not depend on a internal datastate, therefore w w we omit this argument. Let m be an arbitrary signal: w w(1)med(m & i) ::= m & med(i)w w(2)med(m & i) ::= em & med(i)w oqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq+q i For. 2.2.1. The Rewrite Rule Representation of the Abracadabra Protocol

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