Damien Aza-Vallina

Bruno Denis

Jean-Marc Faure

Communications reliability analysis in networked embedded systems

This paper presents a novel method to obtain an analytical expression of the reliability of a data transmission between two terminals of a networked system where components may fail in different manners, e.g. silent or babbling, and some failures may propagate to the adjacent components. This method is exemplified on a typical architecture for critical networked systems; comparison to a common approach which does not consider multiple failure modes and propagating failures shows the benefits of this contribution.

It is then possible to factorize this expression:

(π X 0 d d .π X 0 e e ) + (π X F d d .π X 0 e e ) + (π X 0 d d .π X F e e ) (11)

INTRODUCTION

Communications between the components of control systems (controllers, sensors, actuators) have been only implemented in the past by point-to-point connections. These solutions are gradually replaced, even in critical systems, by communication networks, like CAN (Controller Area Network), FlexRay, but also Ethernet-based technologies, such as Ethernet/IP, Ethernet Powerlink, AFDX (Avionics Full DupleX) in aeronautics, for cost reduction reasons.

Parallel to these industrial changes, numerous research works have been performed to facilitate the development and operation of networked embedded systems. The results of these works are methods to assess time performances [START_REF] Marsal | Evaluation of Time Performances of Ethernet-based Automation Systems by Simulation of High-level Petri Nets[END_REF] [START_REF] Georges | A design process of switched Ethernet architectures according to real-time application constraints[END_REF] [START_REF] Bauer | Improving the worst-case delay analysis of an AFDX network using an optimized trajectory approach[END_REF] or reliability [START_REF] Ghasemzadeh | Kterminal network reliability evaluation using Binary Decision Diagram[END_REF]) of these systems. This paper focuses only on this latter feature, which is a fundamental dependability attribute when the network is embedded in a critical system.

Reliability analysis of a networked system is based on a topological model of interconnected components which are terminals, data senders and/or receivers like controllers or smart sensors/actuators, or data transmission devices, such as hubs and switches. A common assumption is that there is only one failure mode for each component [START_REF] Ball | Computing network reliability[END_REF]) [START_REF] Rosenthal | Computing the reliability of complex networks[END_REF]; hence, the behavior of a component can be described by a continuous, often homogeneous, Markov chain that comprises only two states: faultless and faulty. The reliability of a data transmission between two terminals is defined as the probability that there exists at least one path between these two terminals which includes only faultless components; it can be computed from the analysis of the topological model of the network and the knowledge of the behavioral models of the components.

The common two-states behavioral model (binary model) suits only hardware components, however. More complex behavioral models, with several faulty states, must be introduced for the components which contain hardware and software (including software implementation) because they may fail in different manners, e.g. fail-silent or -babbling. A good classification of failure modes of these components can be found in [START_REF] Papadopoulos | Component Failure Behaviour: Patterns and Reuse in Automated System Safety Analysis[END_REF]; according to this reference, six failure modes may occur, which can be ordered in three groups:

• data transmission failures: transmission omission or commission,

• data values failures: the value of the transmitted data is higher or lower than the real value,

• transmission time failures: the transmission happens too early or too late. Only failures of the first group are considered in this study. Moreover, some transmissions commissions, e.g. fail-babbling, may propagate, i.e. their occurrence prevents other components from communicating, even if they are faultless. Consequently, a path is faultless not only if all components which belong to this path are non-faulty but, furthermore, if no propagating failure that impacts one of these components has occurred.

On the basis of these two observations, this paper proposes a novel method to obtain the reliability of a data transmission between two terminals of a networked system (Fig. 1). This method is appropriate to systems which include components whose behavioral models may comprise several faulty states and failure modes may propagate. The models on which this contribution relies are presented in the next section, while section 3 details the proposed method. Application to a typical critical networked system, in section 4, permits to show the interest of this proposal.

PROBLEM MODELING

The model of the network topology which was selected is first presented in this section. Then, a generic model of the behavior of a network component is proposed.

Network topology modeling

A communication network can be modeled as a nondirected graph G = (N , E) where:

• N is a set of nodes, a node representing a network component,

• E is a set of non-directed edges between couples of elements of N . An edge represents a physical link between two nodes.

In this graph, the set of paths which permit to ensure data transmission between two nodes i and j will be noted P ij ; it may contain one or several paths. An element of P ij will be noted P k ij . Two nodes are adjacent if there exists at least one edge between these nodes.

Component behavior modeling

As mentioned in the introduction, the aim of the paper is to propose a reliability assessment method that is appropriate to networks where components may fail in different manners, some failure modes being able to propagate to the adjacent components. Hence, the common two-states (faultless or faulty) behavioral component model is no more suitable.

The behavior of a network component will be then described by a continuous Markov chain M C i ((Cassandras & Lafortune 2006)) :

M C i =< X i , p i , π i (0) > (1)
where:

• X i is the set of states of the Markov chain,

• p i is the transition matrix,

• π i (0) is a vector which contains the probabilities that the different states be active at the initial date.

This chain may include more than two states according to the number of failure modes that are selected. These states can be however gathered in three sub-sets which define a partition on X i :

• X 0 i sub-set of correct operation states,

• X P i sub-set of propagating failure states. A component failure is termed "propagating" when its occurrence entails that every adjacent component becomes unable to ensure any communication, even if it is itself failure-free.

• X F i sub-set of non-propagating failure states. A component failure is termed "non-propagating" when its occurrence does not impact the behavior of adjacent components.

with

X 0 i ∩ X F i = X 0 i ∩ X P i = X F i ∩ X P i = ∅ and X 0 i ∪ X F i ∪ X P i = X i
The probability that the active state at date t is the state α will be noted π α i (t) ; the probability that the active state at date t belongs to one of the three subsets X 0 i , X F i , X P i will be noted respectively π

X 0 i i (t), π X F i i (t) ,π X P i i (t).
In the sequel of this paper, some limitations are introduced in this modeling:

• Fault occurrences are independent events.

• The number of states of correct operation is equal to one (card(X 0 i ) = 1).

• The components are not repairable; there is no transition starting from a failure state to the correct operation state.

• The failures are persistent; there is no transition between two failure states.

The third assumption has been introduced because no reparation is allowed during a mission of embedded system. The fourth assumption comes from industrial design rules that specify that a component becomes faulty as soon as a fault occurs, whatever the duration of this fault.

With these assumptions, the behavior of a component can be described by the model of Figure 2(a), if all failure states are explicitly depicted on Figure 2(b) if only a description by the three states sub-sets which have been previously defined is selected. It matters to note that these models do not contain degraded operation states; a component is either in a correct operation state or in a failure state. In the latter case, the failure may impact the ability of every adjacent component to transmit data (propagating failure) or not (non-propagating failure).

X F X P 0 1 i-1 i n λ 0 1 ( t ) λ0i-1 (t ) λ 0i (t ) λ 0 n ( t ) (a) λ 0F (t) = i-1 k=1 λ 0k (t) λ 0P (t) = n k=i λ 0k (t) X 0 X F X P λ 0 F ( t ) λ 0 P ( t ) (b)
Figure 2: Component behavior modeling (λ XY : failure rate)

TRANSMISSION RELIABILITY EVALUATION

From the models given in the previous section, a method to evaluate the reliability of a transmission between two terminal nodes i and j, probability that there exists at least one path which permits to ensure data transmission between these two nodes, has been developed. The aim of this method is to provide an analytic expression of this reliability. It is also aimed at avoiding components models composition so as to prevent from the classical state space explosion issue which occurs easily when composing discrete state space models. This method comprises three steps which are performed sequentially:

A Research, for each minimum-length path between i to j, of the components states combinations such as the transmission through this path is possible.

B Determination, for the whole set of minimumlength paths between i to j, of the components states combinations such as the transmission is possible.

C Determination of the analytical expression of the transmission reliability.

These three steps will be detailed in the following sub-sections and illustrated on an example issued from [START_REF] Barranco | ReCANcentrate: a replicated star topology for CAN networks[END_REF] (Fig. 3(a)). This communication network is composed of three CAN terminals which are interconnected by two stars (redundant medium); each star is implemented by a hub. The topological model of this network is given Figure 3 A transmission by the path1 P k ij between the nodes i and j is possible if:

• every component which is represented in the topological model by a node which belongs to the path P k ij is in the correct operation state;

• every component which is represented in the topological model by a node which is adjacent to a node which belongs to the path P k ij is in the correct operation state or in a non-propagating failure state;

• all other components are in any state.

If N k ij denotes the set of nodes which belong to the path P k ij and P N k ij the set of nodes which are adjacent to a node of the path P k ij , the set of allowed states, states that allow data be transmitted through the path

P k ij , for a component represented by a node l (l ∈ N ), is X P k ij l
, with:

X P k ij l =    X 0 l if l ∈ N k ij X 0 l ∪ X F l if l ∈ P N k ij X l else (2)
For the example, there are two minimum-length paths to transmit data between the nodes a et b, noted P 1 ab and P 2 ab (Fig. 4). Then, the allowed states of the components, defined by (2), are given in Table 1.

The components a and b must be in their correct operation state (X 0 a and X 0 b ), whatever the path. On the opposite, the component c does not belong to any path but is always adjacent to a node belonging to a path (adjacent to d for the P 1 ab and to e for P 2 ab ); it must

P 2 ab a X 0 a X 0 a b X 0 b X 0 b c X 0 c ∪ X F c X 0 c ∪ X F c d X 0 d X 0 d ∪ X F d e X 0 e ∪ X F e X 0 e
Table 1: Set of allowed states for each component and each path not be in a propagating failure state.

X P 1 ab c / ∈ X P c i.e. X P 1 ab c ∈ X c -X P c = X 0 c ∪ X F c .
The active state of the network at a given date is the combination of the active states of the components at this date; this combination comprises n terms, with n = card(N ). The set of the allowed combinations for a transmission through a path is then obtained from the sets of the allowed states of the components in the following way:

C P k ij ij = l∈N X P k ij l (3)
For the two transmission paths between a and b, these sets are:

C P 1 ab ab = X 0 a × X 0 b × (X 0 c ∪ X F c ) × X 0 d × (X 0 e ∪ X F e ) (4)
C P 2 ab ab = X 0 a × X 0 b × (X 0 c ∪ X F c ) × (X 0 d ∪ X F d ) × X 0 e (5)
It can be noted that each set contains only four combinations.

Determination of the components states combinations for all paths

The set of allowed states of a component depends on the studied path; a component may for instance belong to a given path P m ij and be adjacent to a component of path P n ij , where P m ij and P n ij are two different paths by which data can be transmitted from/to i to/from j. The set of the allowed components states combinations for all paths, noted C ij , is then obtained by union of the sets of allowed combinations for each path:

C ij = P k ij ∈P ij C P k ij ij (6)
For the example, this set is obtained from (4) and ( 5). Then it comes:

C ab =X 0 a × X 0 b × (X 0 c ∪ X F c )× (X 0 d × X 0 e ) ∪ (X F d × X 0 e ) ∪ (X 0 d × X F e ) (7)
It must be underlined that this set includes only 6 states combinations among the 243 (3 5 ) of the Markov chain which describes the whole behavior of the network. The steps A and B, which are based on analysis of the network topology and operations on sets, have avoided the construction of this model which is nontrivial-sized even for a simple example.

Analytical expression of the reliability of the transmission

Let c be a components states combination and α c l the state of component l in this combination. The probability that the network be in this combination at date t is noted π c (t). If the probability that the component be in a state α c l at date t is noted π α c l l (t), then π c (t) is computed as follows:

π c (t) = l∈N π α c l l (t) (8)
because faults occurrences are independent events. Therefore, the reliability of the transmission is the sum of the probabilities of the allowed components states combinations:

π ij (t) = c∈C ij π c (t) = c∈C ij l∈N π α c i l (t) (9)
For the example of Figure 3, the reliability of the transmission between the nodes a and b is:

π ab =π X 0 a a .π X 0 b b .π X 0 c c .π X 0 d d .π X 0 e e + π X 0 a a .π X 0 b b .π X 0 c c .π X F d d .π X 0 e e +π X 0 a a .π X 0
This new form of the reliability is structured in a similar fashion to expression (7). This method will be applied in the next section to a network where terminal components communicate by using a redundant bus. This case study will permit to show the interest of the proposed modeling of the components behavior compared to a common modeling where every component owns only one nonpropagating failure mode.

APPLICATION

A bus architecture can be used in a critical system only if medium redundancy is ensured. The method detailed in section 3 will be then illustrated on the generic example of Figure 5 where n terminals are connected by two buses B1 and B2 which will be assumed identical. Moreover, it will be admitted that there is only one failure mode, non-propagating failure, for a bus and two modes, propagating failure and non-propagating failure, for a terminal. Once the behavioral models of components given, the three steps of the method are developed. Last, the results obtained are discussed and compared to those which would have been yielded by a common modeling approach.

Components behavior modeling

With the above assumptions on the number of failure modes, two kinds of behavioral models are to be used.

Component with one failure mode

In this case, the component behavior is described by a Markov chain which comprises two states (Fig. 6(a)): a state to model the correct operation, noted x OK , and a state to model the failure, noted x F . Then it comes:

X l = {x OK l , x F l }, X 0 l = {x OK l }, X F l = {x F l } and X P l = ∅.
It will be assumed that this Markov chain is homogeneous; the failure rate λ l is constant and the probabilities that a given state is the active state at date t are:

π X 0 l l (t) = e -λ l .t π X F l l (t) = 1 -e -λ l .t (12)
4.1.2 Component with more than one failure mode

The following two failure modes are selected for this study:

• transmission omission, what means that a component does not communicate though it should. This failure mode does not propagate to the adjacent components and will be represented by a state x F l .

• transmission commission, what means, on

the opposite, that a component communicates though it should not communicate; as failures are assumed persistent, this unexpected communication is continuously maintained and monopolizes the medium. This failure mode propagates to the adjacent components and will be represented by a state x P l .

The behavior of the component is then described by a Markov chain which comprises three states (Fig. 6(b)) such as:

X l = {x OK l , x F k , x P l }, X 0 l = {x OK l }, X F l = {x F
l } and X P l = {x P l } If this Markov chain is homogeneous, the failure rates λ f l and λ p l are constant and the probabilities that the different states be active at date t are:

         π X 0 l l (t) = e -(λ a l +λ p l ).t π X F l l (t) = λ f l λ p l +λ f l .(1 -e -(λ f l +λ p l ).t ) π X P l l (t) = λ p l λ p l +λ f l .(1 -e -(λ f l +λ p l ).t ) (13)
x

OK l x F l λ l (a) x OK l x F l x P l λ f l λ p l (b)

Transmission reliability evaluation

This analysis will be carried out on the example of the transmission between the terminals 1 and n; it can be easily transposed to any other couple of terminals.

4.2.1 Research of the components states combinations for one path Two paths, noted P 1 1n et P 2 1n , permit data be transmitted between the nodes 1 and n (Fig. 7). For each path and for each component, it is possible to determine the sets of allowed states from (2); the result of this analysis is given in Table 2.

Path Component P 1 1n P 2 1n 1 X 0 1 X 0 1 l ∈ {2...n -1} X 0 l ∪ X F l X 0 l ∪ X F l n X 0 n X 0 n B1 X 0 B1 X 0 B1 ∪ X F B1 B2 X 0 B2 ∪ X F B2 X 0 B2
Table 2: Set of the allowed states for a component according to the selected path

Both nodes 1 and n belong to the path, whatever it could be; hence, they must be in the correct operation state. At the opposite, the nodes 2 to n -1 do not belong to any path but are always adjacent to the bus and therefore must not be in a propagating failure mode; the corresponding unexpected communication would prevent the nodes 1 and n from accessing the bus in this case. Then, the expressions of the sets C 1 1n and C 2 1n are:

C P 1 1n 1n = X 0 1 × n-1 l=2 (X 0 l ∪ X F l ) × X 0 n × X 0 B1 × (X 0 B2 ∪ X F B2 ) (14)
C P 2 1n 1n = X 0 1 × n-1 l=2 (X 0 l ∪ X F l ) × X 0 n × (X 0 B1 ∪ X F B1 ) × X 0 B2 (15)
For each set, there is only one allowed state, the correct operation state, for three components (1, n and the bus which belongs to the studied path) and two allowed states for n -1 components (n -2 terminals and the other bus). The cardinality of these sets is then: card(C

P 1 1n 1n ) = card(C P 2 1n 1n ) = 1 3 .2 n-1 = 2 n-1 .

Determination of the components states combinations for all paths

The set C 1n of the allowed states combinations for the transmission is the union of the two sets C

C 1n = X 0 1 × n-1 l=2 (X 0 l ∪ X F l ) × X 0 n × (X 0 B1 × X 0 B2 ) ∪ (X 0 B1 × X F B2 ) ∪ (X F B1 × X 0 B2 ) (16)
This set contains 3.2 n-2 combinations while 2 2 .3 n combinations are possible with the previously defined behavioral models. For a network with 10 terminals for instance, there are 768 allowed combinations while the Markov chain which models the whole behavior of the network comprises 236,196 states, i.e. around three hundred times more.

Analytical expression of the reliability of the transmission

Once all allowed combinations determined, the reliability of the considered transmission in function of the probabilities of the states can be defined as follows:

π 1n (t) =π X 0 1 1 (t). n-1 l=2 (π X 0 l l (t) + π X F l l (t)) .π X 0 n n (t) . (π X 0 B1 B1 (t).π X 0 B2 B2 (t)) + (π X 0 B1 B1 (t).π X F B2 B2 (t)) +(π X F B1 B2 (t).π X 0 B2 B2 (t)) (17)
With the behavioral models presented in 4.1 and in particular the expressions ( 12) and ( 13), the analytical expression of the considered reliability is:

π 1n (t) = e -(λ f 1 +λ p 1 ).t . n-1 l=2 λ f l + λ p l .e -(λ f l +λ p l ).t λ f l + λ p l .e -(λ f n +λ p n ).t . (e -λ B1 .t .e -λ B2 .t ) +(e -λ B1 .t .(1 -e -λ B2 .t )) + ((1 -e -λ B1 .t ).e -λ B2 .t ) (18)
It matters to clearly underline that this expression depends on the number of terminals which are connected to the bus. This novel result would not have been obtained by using a binary model for every component.

Numerical application

Assuming the following failure rates:

• λ B1,B2 = 10 -7 h -1 • λ f l = 8.10 -8 h -1 et λ p l = 2.10 -8 h -1
the numerical application of (18), for three values of n: 2, 5 et 10, permits to represent the evolution of the reliability in function of time (Fig. 8). This figure shows clearly that the reliability of the transmission depends on the number of terminals, as pointed out by equation ( 18). Increasing the number of terminals leads to lessen reliability because new propagating failures are thus introduced.

Discussion

Expression (17) must be now compared to those that yield the analyses:

• of an architecture without bus redundancy but based on the same behavioral models of components;

• of an architecture with bus redundancy but using for every component a two-states (correct operation, transmission omission) behavioral model.

Comparison with the reliability of an architecture without bus redundancy For an architecture with only one bus, there is only one path between 1 and n.

From equation (13), it is then possible to define the set of the allowed states for the transmission (here the path):

C 1bus 1n = X 0 1 × n-1 l=2 (X 0 l ∪ X F l ) × X 0 n × X 0 B1 (19)
Then, the analytical expression of the reliability is:

π 1bus 1n (t) = e -(λ f 1 +λ p 1 ).t . n-1 l=2 λ f l + λ p l .e -(λ f l +λ p l ).t λ f l + λ p l .e -(λ f 1 +λ p n ).t .e -λ B1 .t (20)
Hence:

π 1n (t) = π 1bus 1n (t) 1 + e -λ B2 .t ( 1 e -λ B1 .t -1) (21)
On the assumption that λ B1 = λ B2 then π 1n (t) = π 1bus 1n (t).(2 -e -λ B2 .t ). As it might be expected, medium redundancy improves reliability by a factor (2 -e -λ B2 .t ).

Comparison with the reliability of an architecture with bus redundancy but obtained from two-states behavioral models of components To obtain comparable results, the behavioral models of all components are here such as:

• The overall failure rate of the component is unchanged for the terminals λ l = λ p l + λ a l

• The failure is non-propagating for every component.

Then, the set of allowed states is easily obtained from (15) and is:

C 1mode 1n = X 0 1 × n-1 l=2 X l × X 0 n × (X 0 B1 × X 0 B2 ) ∪ (X 0 B1 × X F B2 ) ∪ (X F B1 × X 0 B2 ) (22)
This set contains 3.2 n-2 combinations while 2 n+2 combinations are possible with the hereabove defined behavioral models.

And the analytical expression of the reliability is: Expressions ( 23) and ( 24) show that:

π 1mode 1n (t) = e -λ
• the expression obtained with the common twostates terminal model (expression ( 23)) does not depend on the number of terminals;

• as the multiplier in ( 24) is smaller than or equal to 1, the reliability obtained with the three-states terminal model is always lower than the other one. The reliability obtained with the novel component model which has been proposed (expression ( 17)) is then more pessimistic but more realistic; it is in particular impacted by the number of terminals, the larger this number, the smaller its value at a given date. This knowledge is quite useful when designing networks for critical systems.

CONCLUSION AND PERSPECTIVES

A method to evaluate the reliability of a data transmission between two terminals of a communication network has been proposed in this paper. The novelty of this contribution is that several failure modes of components, e.g. fail-silent and fail-babbling, as well as failures propagation are considered; hence some failures of a component may entail that every adjacent component becomes unable to ensure any communication, even if it is itself failure-free. This method provides an analytical expression of the reliability; this permits the analysis of the evolution of this dependability attribute over time, for maintenance planning purposes for instance. This expression is obtained from the knowledge of the allowed components states combinations which are themselves derived from the analysis of the network topology, and not by composition of Markov chains to avoid (or limit) combinatory explosion.

Several outlooks arise from these results. First, more detailed behavioral models of components are to be developed, by distinguishing the states where the component is active, used for data transmission, from those where it is idle, waiting for a request to be used. Then it will be possible to consider the application of these proposals to design of communication networks for critical systems.

Figure 1 :

 1 Figure 1: Aim of this work

  Figure 3: A five components network (a) and its topological model (b)

Figure 4 :

 4 Figure 4: The two paths to transmit data between a and b

Figure 5 :

 5 Figure 5: Redundant bus (a) and model of its topology (b)

Figure 6 :

 6 Figure 6: Model of the behavior of a component with one (a) and two (b) failure mode(s)

Figure 7 :

 7 Figure 7: The two paths to transmit data between 1 and n

Figure 8 :

 8 Figure 8: Reliability of the transmission between two terminals in function of time and the number of terminals connected to the bus(2,5,10).

For brevity reasons, the term "path" will mean "minimumlength path" in the rest of this paper.


Inès Saddour

The expression of progressivity in Tunisian Arabic: A study of progressive markers in oral retellings of simultaneous situations L'expression de la progressivité en arabe tunisien: étude des marqueurs de progressivité dans des récits de situations simultanées

. Research into the Tunisian Arabic language in general, and the expression of progressivity in particular is very limited and inconclusive. This study seeks to fill in this gap by analyzing the different ways of expressing that a situation is in progress at a certain reference time. It shows that many lexical means are available to Tunisian speakers to express progressivity. Two of those means are undergoing a process of grammaticalization. These are the preverbal marker qa:'id and the post verbal marker fi. The article gives special attention to those two markers especially fi which has completely been ignored in research into progressivity in Tunisian Arabic. It also aims to show the role that lexical aspect plays in the use of each form in the discourse of simultaneous situations.

Résumé

. La littérature sur l'arabe tunisien en général et sur comment l'aspect progressif s'y exprime est très succincte et très peu concluante. Cette étude se propose donc de d'analyser les différents marqueurs de progressivité dans des récits de fiction. Elle montre qu'il existe plusieurs moyens lexicaux possibles en arabe tunisien pour exprimer qu'une situation est en déroulement. Elle se focalise cependant sur les deux marqueurs aspectuels du progressif: le marqueur préverbal qa :'id et le marqueur post verbal fi. L'étude tente aussi d'examiner le rôle que joue l'aspect lexical dans l'expression de la progressivité et dans le choix de ces deux marqueurs dans les récits de situations simultanées.

Introduction

Any investigation of the notion of progressivity or progressivity of a situation, i.e. a process or an event, has to firmly distinguish between the semantic notion, i.e. of a situation in progress at a certain reference time, and its formal manifestation, that is to say, the linguistic devices employed to express it (designated henceforth by PRG markers). In fact, while the semantic progressivity is a universal concept; its expression differs from language to language. It is not necessarily overtly marked by a distinct morphology or form. Only few languages have the PRG category, as reported in the survey of [START_REF] Dahl | Tense and aspect systems[END_REF]. Many others do not code PRG on the verb, but have lexical entities expressing the semantic notion of progressivity, e.g. French with 'en train de' [START_REF] Bertinetto | The progressive in Europe[END_REF][START_REF] Carrol | Language-specific requirements in event construal when grounding events in context: a cross linguistic comparison of advanced adult second language learners[END_REF]).

The present study investigates the expression of progressivity in Tunisian Arabic (TA) in the context of retelling simultaneous situations. It focuses on the constructions that are most frequently found to be used to express progressivity in the oral productions. These constructions can be represented as follows (+ indicate concatenation and the ( ) indicate the optional character.

(qa:'id +) PV (+fi)

We distinguish therefore between four constructions displayed below: We are interested in examining how each one of the four constructions is used in the different retellings of simultaneous situations as we hope to reach an understanding of their systematicity and the context of their use. We also aim at examining if the lexical aspect plays a role in determining the use of each predicate form for expressing progressivity, as it is well established in the literature that the type of lexical content plays a role in the use of PRG devices or not.

The expression of progressivity in language

Progressivity can be expressed through fully grammaticalized means, and the well known example is the English language with the V-ing predicate [START_REF] Bardovi-Harlig | After process, then what? A longitudinal investigation of the progressive prototype in L2 English[END_REF][START_REF] Leclercq | L'influence de la L1 dans l'organisation des discours chez les apprenants avances / quasi-bilingues: Le cas de l'aspect " en déroulement[END_REF].

It is also possibly conveyed through other means such as morphosyntactic means, use of simple tenses, adverbials (Bertinetto et al. 2000, p.520). Among the morphosyntactic; we find verb phrases with a copula as auxiliary: e.g., 'estar + Gerund' in Spanish and 'stare + Gerondif' in Italian; verb phrases with a motion or postural verb as auxiliary e.g., 'walk'; 'go around' in some Germanic languages; verb phrases with a special progressive auxiliary verb, such as 'hold' (hålla) in Swedish; and many other means word order as a syntactic device that expresses progressivity, e.g. Hungarian (Bertinetto et al. 2000, pp.521-523).

When the tools used are the only ones available to express the notion of progressivity, we speak of devices that reached a status of complete grammaticalization (e.g., English, Icelandic, Maltese) (Bertinetto et al. 2000, p.527). Grammaticalized forms are grammatical morphemes which "gradually develop out of lexical constructions in an ever-expanding range of contexts" (Schmiedtová & Flecken 2008, p.5). According to [START_REF] Bybee | The creation of tense and aspect systems in the languages of the world[END_REF], the progressive, "usually" has "a periphrasic expression" in languages. It is "less grammaticized" and therefore shows "less grammaticization of form" compared to the imperfective or perfective aspects (Bybee & Dahl 1989, p.56). When the PRG markers are not fully grammaticalized, there exist more than one lexicalized devices or constructions which compete in linguistic usage; such as the case of French and Albanian [START_REF] Bertinetto | The progressive in Europe[END_REF].

Due to its different realizations in different languages, the question of whether PRG has to be considered an aspect on its own right has also been asked (Bertinetto et al. 2000, p.517). For example, comparing the progressive in English and in French, [START_REF] Vetters | Temps Aspect et Narration[END_REF] considers PRG as an aspect in English due to its obligatory character, as it cannot be replaced by any other form; while in French it belongs to the lexical domain as its use is optional and can be replaced, for instance by the imparfait.

Lexical aspect and the progressive

Predicate type and the inherent temporal properties of the lexical content in an utterance play a significant role in the use and meaning of the progressive form [START_REF] Bardovi-Harlig | After process, then what? A longitudinal investigation of the progressive prototype in L2 English[END_REF]Vendler 1967). Vendler's (1967) description of predicate types shows how the English progressive can occur with certain lexical contents only. In fact, his description of the continuous meaning in English verbs, expressed by the progressive form V-ing in comparison to non-continuous meaning, highlights the notion of phases of an ongoing process and the progressive form highlights one of those phases. He proposes that stative verbs such as love and know do not have progressive forms because of the "internal contradiction between the stativity of the verb and the non-stativity which is an essential feature of the progressive". The progressive is said to be predominantly associated with the verbs belonging to the Vendlerian classes1 of activities and accomplishments [START_REF] Bardovi-Harlig | After process, then what? A longitudinal investigation of the progressive prototype in L2 English[END_REF]. Whether the class of achievement verbs can admit the progressive or not is a subject of disagreement. According to Dowty (1979)2 , achievement verbs force a repeated meaning and they are separated from the verbs that admit the progressive. [START_REF] Binnick | Time and the Verb: A Guide to Tense and Aspect[END_REF] as well as [START_REF] Klein | Time in Language[END_REF] attest the possibility of the occurrence of PRG with achievement verbs. In fact, using a different terminology and a different aspectual analysis framework, Klein (1994, pp.34-35) shows the compatibility of durative adverbials such as for five minutes with an utterance like he opened the window. This adverbial, of course, gives an iterative reading to the content <open the window>.

Many verb classifications have raised problems when applied to different temporal systems (See Binnick 1991 for an overview). The innovation of [START_REF] Klein | Time in Language[END_REF] theoretical framework with relation to lexical aspect is that it accounts for the role of the temporal properties of all the entities included in the predicate, not only those inherent to the verb. As such, his framework accounts for the difference between to write and write a letter, to eat and to eat an apple.

He applies "a single basic criterion" that he believes, is necessary for the analysis of tense and aspect: "the behavior of a lexical content with regard to its linking to some topic time" (Klein 1994, p.80). Lexical contents are understood to be related to the Topic Time (TT); i.e. the time for which a claim is made in three different ways (Klein 1994, p.81):

0-State contents: are lexical contents that are linked to a particular TT, and at the same time, automatically linked to any other TT. They present « no TT-contrast » (Klein 1994, p.101). In

(1) The book is in Russian, the assertion will hold true for the TT the clause is linked to, and also for any other TT.

1-State contents: they involve one TT-contrast. The situation described is not necessarily confined to TT, but does not cover before or after that TT (Klein 1994, p.102). In

(2) John was in Poland the assertion holds true for one contrast between < not be in Poland> vs. <be in Poland>

2-State contents:

They involve two TT-contrasts, e.g.,

(3) John opened the window,

The assertion holds true for two different contrasts < to be closed> vs. <to open>, <to open> vs. <to be open>. 2-state contents include a "source" and a "target state". The TT may be the source or the target state or includes part of both. [START_REF] Klein | Time in Language[END_REF] definition of the progressive accounts for its relation to the lexical aspect, for it is based on the notion of state, and change of state. He states for the English language that « With the progressive form, the TT is properly contained in the first state of the situation (which is the only one for 1-state situations and which has no TT-contrast for 0-state situations) ».

The expression of progressivity in Tunisian Arabic

The expression of progressivity in spoken Arabic languages 3 has not been very enthusiastically addressed. Some research studies have nevertheless informed about the existence of emergent not fully grammaticalized PRG markers in many spoken Arabic languages (e.g., [START_REF] Cuvalay | The expression of durativity in Arabic[END_REF][START_REF] Al Nasser | Grammaticalization of lexical markers in Kuwaiti Arabic[END_REF]. Those PRG markers are found to compete with other lexical markers used for the same aspectual value, namely with a bare prefixed form of the verb (PV). Consider for example, the PRG affixal markers bi-in Egyptian Arabic [START_REF] Mitchell | Colloquial Arabic: The Living Language of Egypt[END_REF], ka-in the "spoken dialect" of Fes, and the preverbal markers qa:'id in the "spoken dialect of Tunis" (Cuvalay 1991, p.143) and gaa'id in Kuwaiti Arabic [START_REF] Al Nasser | Grammaticalization of lexical markers in Kuwaiti Arabic[END_REF].

Descriptions of the Tunisian Arabic temporal system in general and of the progressive in particular are very limited. To our knowledge no investigation of progressivity in spoken Arabic languages is based on a corpus-based analysis. An insightful contribution based on examples taken from different oral and written sources is made by [START_REF] Cuvalay | The expression of durativity in Arabic[END_REF]

2.

The present study This study aims at filling a gap in the literature about progressivity in TA. It proposes an investigation of its expression in context in order to provide a clearer picture of the predicate forms used and what can eventually be considered as PRG markers in TA.

We look at the expression of progressivity in oral retellings of simultaneous situations by 19 Tunisian native speakers aged between 24 and 40 years old. We define simultaneity relation after [START_REF] Acsu-Koç | Temporal relations in narrative: Simultaneity[END_REF] as any type of overlap or inclusion between two situations (processes, events or states). In fact; "Two events, processes or states are simultaneous if they share one value on the time axis. their boundaries need not coincide" (Acsu-Koç & Von Stutterheim 1994, p.397) The retellings of simultaneous situations have proved to be an effective tool for eliciting the use of PRG markers as the research studies of [START_REF] Schmiedtová | At the Same Time: The Expression of Simultaneity in Learner Varieties[END_REF] and of [START_REF] Leclercq | L'influence de la L1 dans l'organisation des discours chez les apprenants avances / quasi-bilingues: Le cas de l'aspect " en déroulement[END_REF]2008) show.

Method

The informants were presented with eight short video scenes in the same order on a computer screen. Each scene was presented as many times as they needed. After making sure they had watched the scene enough times to do the task, the video was stopped, and the informants were asked to retell what happened in each scene. They were all asked the same question in the past tense: "shnuwa sa:r fil mashhad hedha?" (What happened in this scene?). The narratives were recorded using a digital recorder and a microphone.

Concerning the visual stimuli used, they all include two simultaneous situations related to two different entities which we call participants.

S1 ---------------------------------- S2 ---------------------------------- Earthsea, Birds S1 ---------------------------------- S2 |------------------------------ Inclusion of S2 in S1 Kabaret, Wakeup, Fire, Soup, Salmon S1 --------------------------------- S2 |-----| |-----| |-----|
The videos used are listed in Table 1. All of S1 situations in all the videos are durative. S2 are different. While in Breakfast it is perfectly parallel to S1, in Earthsea and Birds, S2 slightly starts after S1. In the second type of simultaneity, S2 is composed of short sub-situations which are interlocked in the time in which S1 takes place.

Data Analysis

The video retellings were transcribed using the CHAT format on the CHILDES programme, and coded using CLAN. The retellings were organized into utterances (an utterance being an eventuality containing a verb or not)5 . Every predicate form was coded for the situation they describe in relation to the visual stimuli. The aspectual relation expressed in each predicate was also coded. This report focuses only on the progressive constructions. All verb forms, as well as the lexical content of each predicate, were coded using the 3-class categorization of [START_REF] Klein | Time in Language[END_REF].

The different forms expressing progressivity

The several different forms found to express that a situation is in progress are listed in detail in Table 2 according to the frequency of their occurrence. Out of a total of 1180 utterances, there are 335 progressive predicates; 319 in the present tense and 16 in the past. As Table 2 shows, the forms that are most used to express progressivity are the prefixed verb form (PV) that we will call the unmarked form PV (25%) 6 , The prefixed verb followed by the locative preposition fi (in) making up the construction PV + fi (33%). This latter can be preceded by the preverbal marker qa:'id making up the periphrasis qa:'id + PV + fi (14%). This periphrasis is also used at a rate of 20% without the preposition fi (qa:'id + PV). Past: 5% (n=16)

q'ad (SV) + PV - 2% - - - 15% 4% - 2% q'ad (SV) + AP - 2% - 2% - - - 12% 1%
Bqa(

(SV) + PV - - - 2% - - - - 0.3%
The video retellings are classified in Table 2 according to the use of the progressive in them. The progressive aspect is highly used in most of the video retellings. The forms which express progressivity can be classified into the following 3 types:

1) Simple entities such as the active participle (AP) or the prefixed verb form (PV);

2) Periphrases that are only used to encode progressivity, qa:'id + PV, qa:'id + PV +fi and PV + fi, due to the presence of the aspectual PRG markers qa:'d and/or fi;

3) Combined predicates composed of preverbs or "coverbs"7 . The progressive aspect is expressed in these predicates by those preverbs due to their lexical semantics, e.g bqa (he stayed) and kammal (he finished).

This article focuses on the forms encircled in the table above and gives special attention to the two PRG markers in TA namely; qa:'id and fi.

The markers of progressivity in Tunisian Arabic

The preverbal marker qa:'id:

Qa:'id is the masculine singular active participle (AP) derived from the verb q'ad 8 (q'ad being possibly translatable into 'he sat' or 'he stayed'). The AP qa:'id can be used alone in a predicate (4):

(4) Qa:'id fi(j)jarda sit&AP&PS3M9 in-the-garden He is sitting in the garden In a preverbal position, qa:'id is a PRG marker. It is inflected for number and gender as the example (5) and shows.

(5) A4, Soup

Weh id qa:'id ye-kol One PRG PS3M-eat Somebody is eating.

It generally precedes a PV and possibly another AP. This use is very rare in our data there are only two instances of "qa:'id + AP" which are limited to the verb rqad (he slept). We illustrate this by example (6) (the sound /g/ is an allophone of the sound /q/ produced in some regions in Tunisia).

(6) A11, Wakeup ra:jil ga:'id ra:gid . man PRG sleep&AP&PS3M A man is sleeping

We hypothesize that the PM cannot PV precede the PVs conveying a static position such as rqad (he slept). It is rather the AP that is used instead.

As such we cannot say (7) *qa:'id yo-rqod PRG PS3M-sleep He is sleeping To sum up, qa:'id plays a purely aspectual role in progressive contexts. It is a preverbal PRG marker that is in a process of grammaticalization as it did not completely lose its semantic value and it is still inflected for gender and number, contrary to [START_REF] Cuvalay | The expression of durativity in Arabic[END_REF] remark. Furthermore, 8 There is no infinitive in Tunisian Arabic. The root is generally believed to be the three consonants making up the inflected forms [START_REF] Bulos | The Arabic triliteral verb: a comparative study of grammatical concepts and processes[END_REF][START_REF] Fleisch | Etudes sur le verbe arabe[END_REF][START_REF] Mitchell | Colloquial Arabic: The Living Language of Egypt[END_REF]. The consonantal root serves as a tool to make entries in the dictionaries of Modern Standard Arabic for example. In the case of q'ad (he sat), the consonantal root is composed of the three letters q,' and d. Given the absence of the infinitive the masculine singular suffixed form of verbs generally stand for the entry of any verb in the dictionaries. It is therefore translated in this paper as inflected in the masculine singular in the past tense.

it is not systematically used when a situation is described as ongoing at a certain time. Its lexical source is a verb of posture (to sit). Therefore, the formation of this PRG marker in TA is not unusual; it is also the case of PRG markers of Italian and Spanish languages.

The post verbal marker fi

The PRG marker fi is originally a preposition of location which means (in). Fi obligatorily follows a verb which needs a direct object complement in an utterance that expresses that a situation is in progress (see ( 8)). Fi is exclusively used to express progressivity. Its use in contexts expressing a different aspectual value makes the utterance unacceptable (consider example (9) which describes a bounded finished event with the use of the suffixed verb form kla (he ate)).

In example (8) below, fi is employed together with the Preverbal PRG marker qa:'id. When this latter is removed, the utterance preserves its progressive reading (see ( 10)). But when fi is removed, the utterance becomes unacceptable (Example ( 11)). When fi is the only PRG marker in an utterance, removing it changes the progressive value to another aspectual one. In example ( 12) where we have a narrative sequence, the use of the PV yekol conveys a bounded event. To conclude, fi is obligatorily used when the situation is viewed in progress and when the utterance contains a direct complement. The use of the preverbal PRG marker qa:'id shows more optionality as with certain verbs, removing it from the utterance does not change its progressive reading. Yet in spite of this apparent free variation of the use of the different forms qa:'id + PV / PV and the equivalent predicates that require complements qa:'id + PV + fi / PV + fi, we believe that there should be a possible systematic description of each one. In order to examine this systematicity, we investigate the lexical contents selected with each construction in retelling the simultaneous situations.

Lexical aspect

Based on the three-class categorization of lexical contents by [START_REF] Klein | Time in Language[END_REF], the lexical contents selected with the four most used constructions are analyzed and displayed in Table 3 (the percentage of the lexical content used with each form is calculated and put in brackets). We examine whether there is a relation of dependence between the type of lexical content selected and the use of one construction or the other. The lexical contents used with a progressive aspectual value are of two types: Predicates which present only one TT contrast, 1-State lexical contents(1S); called "activities" in other theoretical frameworks, and predicates that present two contrasts TT contrasts creating a change from a source state to a target state (2S).

Table 3 shows that the progressive aspect is predominantly expressed using 1-State lexical contents, i.e. with activity verbs. In fact, the frequencies and rates of use show that PV forms used alone to express the PRG aspect tend to select almost only 1-State contents. The 2-State contents selected for the same form are very limited in number and represent only 4% of the total PVs used to express the progressive.

We can clearly notice that the presence of the PRG markers qa:'id and fi allows expressing progressivity when the verbs used present two state contents. In order to test the null hypothesis of independence between the type of lexical content used and the use of the aspectual marker qa:'id, we did a standard χ 2 test (see Table 4 below).

Our null hypothesis (H 0 ) postulates that there is no significant relationship between the use of the preverbal PRG marker qa:'id and type of lexical content. We examined the difference between the observed and the expected frequencies of each predicate form (with qa:'id and predicate forms without qa:'id) with 1-S and 2-S lexical contents. Our calculated value of chi square is 2.58%. It indicates that there is less than 5% probability that the relationship we have stated is random. This allows us to reject the null hypothesis of independence at the standard 5% significance level.

We furthermore tested the hypothesis (H 0 ) of independence between the predicate form chosen to express the progressive and the type of lexical content chosen in an utterance (see Table 5). A χ 2 was also conducted with each of the 5 predicate forms listed in the following table: H 0 is yet again strongly rejected given the chi square value obtained.

In conclusion, we postulate that progressivity in Tunisian Arabic has more affinities with lexical contents which present an activity (1-State contents). The PV form alone used with 1-State contents, unless explicitly marked otherwise by an adverbial for example, expresses unambiguously the progressive aspect. Lexical contents which present a source state and a target state however, employed with PV form with no progressive markers take a different aspectual value, for e.g., the imperfective value, and in our data it takes the value of the narrative present (13)).

( Yo-q'od hd e-ha PS3M-sit near-her He sits close to her In (13) b. Taqra fi(j)jari:da (she is reading the newspaper) expresses the progressive aspect while c. and d. contain punctual 2-S verbs which denote bounded events in a sequence.

Discussion and conclusions

By analyzing retellings of simultaneous situations, we examine the linguistic encoding of progressivity in Tunisian Arabic. Two PRG markers are found: qa:'id (originally the masculine singular active participle of the verb q'ad, 'he sat' / 'he stayed') and fi (originally means 'in'). Both of them are locative entities, which are developing into grammatical markers. The process of grammaticalization of lexical markers is a natural process that happens as a result of the weakening of their lexical content by a process of generalization [START_REF] Bybee | The creation of tense and aspect systems in the languages of the world[END_REF].

Qa:'id is a postural verb acting as an auxiliary and is very similar to many Germanic languages allowing postural verbs with the finite form of the main verb, such as Danish, Icelandic, Norwegian and Swedish (Bertinetto et al. 2000, p.523). Given the semantics of qa:'id; the periphrasis qa:'id + PV (+ fi) recalls the PRG devices in languages where verb phrases include a copula as an auxiliary; such as Spanish or Italian (Bertinetto et al. 2000, p.521).

In many other Arabic spoken languages, progressive markers are found to be developing and undergoing a process of grammaticalization; for example, ga:'id is found to have developed into a PRG marker in Kuwaiti Arabic (Al Nasser 1991). Its grammaticalization is accompanied by a phonological reduction "already in process" as it occurs "interchangeably with its shorter form ga'" (Al Nasser 1991, p.669). Other spoken Arabic languages have developed different affixal progressive markers; e.g. ka-in Moroccan Arabic, bi-in Egypt and 'am-in Syria [START_REF] Cuvalay | The expression of durativity in Arabic[END_REF].

The preverbal PRG marker qa:'id in TA has received some attention in some research studies, e.g., [START_REF] Cuvalay | The expression of durativity in Arabic[END_REF]. She has examined the status of qa:'id in the TA that is spoken in the city of Tunis. However, she postulates based on the examples she investigates that as a preverbal marker, qa:'id is no longer inflected for number or gender, and that a more reduced affixal form of it; qa-is used as a preverbal marker in the Jewish Arabic spoken in Tunis (Cuvalay 1991, p.146). In our data, the locative participle qa:'id is found to be inflected for person, number and gender, and no instance of qa-is found.

While the status of qa:'id as a PRG marker is recognized by some studies, nothing can be found about the PRG marker fi. Even though its post verbal position seems uncommon compared to PRG markers in other languages; its lexical source is not that rare. In fact, the Irish copula ag preceding a verbal noun originates from the locative preposition (at) [START_REF] Bybee | The creation of tense and aspect systems in the languages of the world[END_REF]. Moreover, the affix bi-in Syrian spoken Arabic is originally a preposition of location meaning (in) [START_REF] Cuvalay | The expression of durativity in Arabic[END_REF].

The post verbal PRG marker fi seems to penetrate the Time of the situation in such a way that it is possible to focus on one of its internal phases to focus it as a situation in progress. This is possible as its original semantics conveys being inside some temporal interval or space. In the same way, fi can make a real world situation which is perceived as punctual, e.g. <understand somebody> be perceived in progress and focus a possible internal phase. We illustrate this by the following example (14) (the translation sounds unacceptable in English where the type of verb does not really accommodate the progressive marking:

(14) ni-fhim fi:-k PS1-understand PRG-you I (am understanding) you Concerning the role of lexical aspect, the expression of progressivity in TA has close affinities with 1-state dynamic contents. The bare PV used for the progressive aspect is unambiguous as to its aspectual value with this category of lexical contents. The use of PRG markers appears to disambiguate the progressive reading with the prefixed form of 2-S verbs while they are still possibly used with 1-S lexical aspects.

We hope that we could shed some light into the expression of progressivity and the PRG markers in TA. We believe, however, that more analyses are needed ion order to have a better understanding of the use of the non fully grammaticalized PRG marker qa:'id in context, such as the investigation of the use of adverbials.

  It may help to schematize the use of the progressive aspect by the following examples to illustrate its relation to the lexical aspect (the square brackets [.] represent the topic time, ---represents the source state, +++ the target state, and === the only state of situations) Paul was opening the window -----[------]----+++++++++ Paul was running ====[===]=====

  PRG meal the-morningThe man is having breakfast (11) *E(l)-ra:jil qa:'id ye-kil ft u:r e(l)-s beh . The man eats breakfast then he goes to work.

  who compares PRG markers in four Arabic languages spoken in four cities: Fes, Tunis, Damascus and Cairo. For the spoken Arabic of the city of Tunis, she summarises the progressive expression as follows (our added explanations are put between[.]):«The preverbal progressive marker qa:'id in the dialect of Tunis is not obligatory, in the sense that its use is required only in the absence of time adverbials or a specific context to indicate unambiguously that reference is made to an ongoing action(Singer 1984:301). With verbs belonging to the special verbs [of motion and attitude], progressive aspect is designated by the AP [Active participle 4 ]. For all the values that are not expressed by the SF [suffixed verb form], the AP, or the PF [the prefixed verb form] with the future marker, the PF is used in its 'bare' form, i.e. without additional PM [Preverbal Marker].»(Cuvalay 1991, p.148)

Table 1 . Visual stimuli

 1

	Type	Video	Illustration
	Perfect simultaneity	Breakfast
	between S1 and S2

Table 2 . The different forms expressing progressivity Breakfast Earthsea Birds Soup Salmon Kabaret Fire Wakeup Total

 2

	%		17%	16%	14% 16% 13%	12%	7%	5%	335
	Present: 95% (n=319)
	PV + fi		35%	22%	15% 38% 40%	56%	24% 29%	33%
	PV		28%	35%	46% 23% 14%	8%	24% -	25%
	qa:'id + PV		11%	20%	38% 19% 19%	5%	44% 6%	20%
	qa:'id + PV + fi 26%	15%	-	17% 28%	10%	4%	-	14%
	AP		-	-	-	-	-	-	-	47%	2%
	qa:'id + AP		-	2%	-	-	-	-	-	6%	1%
	'amma:l + PV	-	-	-	-	-	-	-	-	1%
	ykammal (PV)	+ PV -	2%	2%	-	-	-	-	-	1%

Table 3 . Lexical contents

 3

		0S		1S			2S	Total (100%)
	PV + fi	0	0%	59	54%	50	46%	109
	PV	0	0%	78	96%	3	4%	81
	qa:'id + PV	0	0%	51	75%	17	25%	68
	qa:'id + PV + fi 0	0%	18	38%	29	62%	47

Table 4 . Role of lexical content in the use of qa:'id

 4

					Expected
		Observed frequencies	frequencies
		1S	2S	Row total	1S	2S
	Progressive predicates
	without qa:'id	137	53	190	128.1107 61.88925
	Progressive predicates
	with qa:'id	70	47	117	78.88925 38.11075

Table 5 . Role of lexical content in the choice of constructions

 5

		Observed frequencies		Expected frequencies
		1S	2S	Row total	1S	2S
	PV + fi	59	50	109	74.52396166	34.47604
	PV	78	3	81	55.38019169	25.61981
	qa:'id + PV	51	17	68	46.49201278	21.50799
	qa:'id + PV + fi 18	29	47	32.1341853	14.86581
					χ 2 = 0.000000000%
	Column total	214	99	313

Vendler (1967) classifies English verbs into four main classes: a. Activity verbs, such as 'run', walk; b. State verbs, such as 'desire', 'love', 'have' c. Accomplishment verbs, such as, 'paint a picture', 'make a chair' d. Achievement verbs, such as 'win the race', 'recognize somebody'

Cited in (Bardovi-Harlig 2008).

We investigate here the language that is spoken in Tunisia. We reject the "dialect" categorization. Accounts about the similarities that exist in the Arabic spoken varieties exist (e.g.[START_REF] Farghali | A case for an inter-Arabic grammar[END_REF]), but are not conclusive. Tunisian Arabic is in fact the spoken language in every day communication in Tunisia. It co-exists with Modern standard Arabic which is taught at school and used only in formal contexts. The discussion of whether Tunisian Arabic is to be considered as a language or a "dialect" is beyond the scope of this paper.

An AP is derived from a verb. It literally means 'doer' of an activity. As its name in the traditional grammars of Standard Arabic denotes; "ism al fa:'il", it refers to 'the name of the actant'. For example, the AP derived from the verb ktib (he wrote) is ka:tib (writer).

In TA it is possible to have utterances with no explicit verbs. They are generally called verb-less clauses.

In TA, as well as in all spoken Arabic languages, there are only two possible inflected verb forms the suffixed form and the prefixed form[START_REF] Benmamoun | The role of the imperfective template in Arabic morphology[END_REF][START_REF] Cuvalay | The expression of durativity in Arabic[END_REF][START_REF] Cuvalay-Haak | The Verb in Literary and Colloquial Arabic[END_REF][START_REF] Farghali | A case for an inter-Arabic grammar[END_REF][START_REF] Fleisch | Etudes sur le verbe arabe[END_REF][START_REF] Grand' Henry | Syntaxe du verbe en arabe parlé Maghrébin[END_REF][START_REF] Mitchell | Colloquial Arabic: The Living Language of Egypt[END_REF]. These forms are often equated with an aspectual opposition: completed / incomplete (accompli / inaccompli); pefective, imperfective.

A term used byWilmet (1998, pp.318-21) to designate verbs and auxiliaries in French that are combined to an infinitive in a predicate such as "commencer à" or "venir de".

The '&'symbol in glosses is used to indicate that the affixes merge with the root of the word, '-' is used when they can be separated.


Magdalena Wencka

Simon Jazbec

Zvonko Jaglicic

Stanislav Vrtnik

M Feuerbacher

Marc Heggen

Stefan Roitsch

Janez Dolinsek
email: jani.dolinsek@ijs.si

J Stefan

§ S Jazbec

Z Jagličić

J Dolinšek

Electrical resistivity of the mu-Al4Mn giant-unit-cell complex metallic alloys

Keywords: electrical transport, quasicrystals user supplied): Complex metallic alloys, Electrical resistivity, Al-Mn system Complex metallic alloys, Electrical resistivity, Al-Mn system

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Introduction

Among a vast number of known intermetallic phases, the term "complex metallic alloys" (CMAs) denotes a class of phases with giant unit cells containing some hundred up to several thousand atoms [1]. Examples of CMAs are the cubic NaCd 2 with 1152 atoms per unit cell (u.c.) [2,3], the Bergman phase Mg 32 (Al,Zn) 49 (162 atoms/u.c.) [4], the orthorhombic ξ'-Al 74 Pd 22 Mn 4 (258 atoms/u.c.) and the related Ψ phase (about 1500 atoms/u.c.) [5][6][7][START_REF] Feuerbacher | Quasicrystals, Structure and Physical Properties[END_REF], the cubic β-Al 3 Mg 2 (1168 atoms/u.c.) [START_REF] Samson | [END_REF][START_REF] Samson | Developments in the Structural Chemistry of Alloy Phases[END_REF], the hexagonal λ-Al 4 Mn (586 atoms/u.c.) [START_REF] Kreiner | [END_REF], the Al 39 Fe 2 Pd 21 (248 atoms/u.c.) [12] and the heavy-fermion compound YbCu 4.5 , comprising as many as 7448 atoms in the unit cell [13]. These giant unit cells contrast with elementary metals and simple intermetallics whose unit cells in general comprise from single up to a few tens atoms only. The giant unit cells with lattice parameters of several nanometers provide translational periodicity of the CMA crystalline lattice on the scale of many interatomic distances, whereas on the atomic scale, the atoms are arranged in clusters with polytetrahedral order, where icosahedrally-coordinated environments play a prominent role. The structures of CMAs thus show duality; on the scale of several nanometers, CMAs are periodic crystals, whereas on the atomic scale, they resemble quasicrystals (QCs) [START_REF] See | Quasicrystals[END_REF]. The high structural complexity of CMAs together with the two competing physical length scales-one defined by the unit-cell parameters and the other by the cluster substructure-may have a significant impact on the physical properties of these materials, such as the electronic structure and lattice dynamics. In this paper, we present a study of the electrical resistivity and the magnetic susceptibility of the hexagonal µ-Al 4 Mn phase with 563 [START_REF] Shoemaker | [END_REF]. Magnetic properties of the µ-Al 4 Mn were already studied before [16].

Structural considerations and sample description

While the QC phase in the Al-Mn system forms by rapid cooling at a wide composition range, single-phase icosahedral material is produced only at compositions close to Al 4 Mn [17]. At higher Mn concentrations and lower cooling rates, a decagonal phase is formed. There exist two crystalline phases at the approximate composition Al 4 Mn, µ and λ , both with giant hexagonal unit cells [18]. The µ-Al 4 Mn structure belongs to the P6 3 /mmc space group with a = 1.998(1) nm, c = 2.4673(4) nm and 563 atoms in the unit cell (453 Al and 110 Mn, leading to the formula Al 4.12 Mn) [START_REF] Shoemaker | [END_REF]. The unit cell contains 10 Mn and 32 Al crystallographic sites. Two of the Al sites are partially occupied. The structure consists of interpenetrated icosahedral chains aligned along the [100] crystal direction with pentagonal bi-prisms composed of pentagonal antiprisms (layer thickness 0.235 nm) and pentagonal prisms (layer thickness 0.290 nm) [19].

Along the [021] direction, a pseudo-5-fold symmetry is present. Along the [001] hexagonal direction, the µ-Al 4 Mn structure can be viewed as a layered compound [18].

The layer at z = c/4 (Fig. 1a) is located in a mirror plane and therefore planar; the one centered at z = c/2 (Fig. 1d) is almost planar, although not required to be so by symmetry. The layers centered at z = 0.325c (Fig. 1b) and z = 0.425c (Fig. 1c) are considerably puckered. The layers at z = 0.575c, 0.675c and 3c/4 (not shown) are generated from the layers centered at z = 0.425c, 0. 3 with its long edge along the c hexagonal axis was cut from a large, single-grain crystal. Recently, an X-ray photoelectron diffraction study on the 6fold surface of the same µ-Al 4 Mn material was reported [20], where further characterization of the material can be found.

Magnetic susceptibility

Magnetic measurements were conducted by a Quantum Design MPMS XL-5 SQUID magnetometer equipped with a 50 kOe magnet. In the measurements, the magnetic field was applied parallel to the hexagonal c axis of the sample. The susceptibility data presented in the units emu/mol are calculated per mol of Al 0.807 Mn 0.193 "molecules".

The zero-field-cooled (zfc) and field-cooled (fc) susceptibilities (p = 5.9), the rest being nonmagnetic, we shall nevertheless estimate the fraction of Mn moments for the second scenario. In that case we obtain the magnetic Mn fraction as

( ) = = 2 p p f eff
3 %, so that the moments would be quite diluted.

The ac susceptibility was measured in an ac magnetic field of amplitude H 0 = 6.5

Oe at the frequencies ν = 1, 10, 100 and 1000 Hz. The real part of the susceptibility χ′is displayed in Fig. 3. As expected for a nonergodic spin system, the position of the cusp in χ′is frequency-dependent, shifting to lower temperatures at lower frequencies.

At the lowest frequency of 1 Hz, the cusp in χ′occurs at the temperature T = 2.7 K.

Considering the cusp in χ′ to occur at the freezing temperature, the ( )

Electrical resistivity

The electrical resistivity was measured between 300 and 2 K using the standard fourterminal technique. The current was applied along the hexagonal c axis of the sample. Apart from the low-temperature resistivity minimum, which will be described in the following as a magnetic effect, we proceed with the resistivity analysis in a semiquantitative way by employing the theory of quantum transport of slow charge carriers [START_REF] Trambly De Laissardière | [END_REF], applicable to metallic alloys and compounds in which the electron mean free path

τ v l =
(where v is the electron velocity and τ the electron scattering time) is small compared to the extension of the conduction-electron wave packet, in which case the electron propagation is non-Boltzmann. This theory has recently been successfully applied to the Al 4 (Cr,Fe) decagonal approximant [24,25], where the temperaturedependent resistivity exhibited a maximum similar to that shown in Fig. 4.

According to the

( ) ( ) ( ) τ τ ε τ ε σ σ σ 2 2 2 2 L g e v g e F F NB B + = + = , ( 2
)
where B σ is the Boltzmann contribution and NB σ is the non-Boltzmann contribution.

The scattering rate 1 τ will generally be a sum of a temperature-independent rate 2) can be rewritten as

( ) α α σ CT B CT A SC + + + = 1 1 , ( 3
)
where the subscript " SC " denotes "slow carriers". SC The low-temperature resistivity minimum can be attributed to a magnetic effect.

The magnetic measurements presented in Figs. 2 and3 clearly demonstrate slowingdown of the Mn spin fluctuations upon cooling and their freeze-out on the experimental time scale below the freezing temperature T f = 2.7 K. Static spins provide strong magnetic scattering mechanism for the conduction electrons, so that the electrical resistivity increases quite dramatically on approaching the spin glass phase.

In alloys containing magnetic transition-metal impurities, the s-d exchange interaction between the s moments of the conduction electrons and the d moments of the transition-metal impurities contributes a term to the resistivity [START_REF] Mizutani | Electron Theory of Metals[END_REF] (

    + + = D T k Jg S S J g c B F F spin log 4 1 1 2 2 2 0 ε ε π ρ ρ . ( ) [ ] ( ) ( )  
) 4
Here c is the magnetic impurity concentration, S the impurity spin, J the s-d exchange constant, 0 ρ the residual resistivity at 0 → T and ( )

F g ε is the DOS at F ε in a flat
band of width 2D.

The total resistivity can be expressed as a sum of the slow-charge-carrier contribution

SC SC σ ρ 1 =
and the magnetic contribution spin ρ and can be conveniently written in the form

T d SC log + = ρ ρ . ( 5
)
For a negative exchange constant, J < 0 (and hence d < 0), a minimum in the resistivity occurs at low temperatures. Eq. ( 5) was used to fit the resistivity of the µ-Al We should like to stress that, besides the model of slow-charge-carrier resistivity of Eq. ( 3), there exist other models that yield a maximum in the electrical resistivity. An example is the Baym-Meisel-Cote theory of amorphous metals [START_REF] Mizutani | [END_REF] that considers the inelastic electron-phonon interaction to be of prime importance for the electron transport at temperatures below the Debye temperature D θ .

Conclusions

The µ-Al 4 Mn complex intermetallic phase with 563 atoms in the giant unit cell shows complicated temperature dependence of the electrical resistivity that exhibits a broad maximum at about 175 K and a minimum at 13 K. Away from the low-temperature minimum, the temperature dependence of the resistivity was reproduced theoretically by employing the theory of quantum transport of slow charge carriers, which predicts a crossover from the metallic (Boltzmann-type) PTC electrical resistivity at low temperatures to the insulator-like (non-Boltzmann) NTC resistivity at elevated temperatures that yields a resistivity maximum. The low-temperature resistivity minimum was reproduced by considering it to be a magnetic effect due to an increased scattering of the conduction electrons by the Mn spins on approaching the spin glass phase. The spin freeze-out below the spin freezing temperature T f = 2.7 K is evident from the zfc-fc magnetic susceptibility splitting below T f in low magnetic field and in the frequency-dependent cusp in the ac susceptibility.

  325c and c/4, respectively, by the operation of the twofold axis at z = c/2 in the [110] direction. Since the layer at z = has a pseudo-twofold axis in that direction, the layer at c/2 is a pseudo-mirror with respect to its adjacent layers. A triangular distribution of atoms is evident on the c/4 and c/2 layers. In the µ-Al 4 Mn structure, fragments of the Mackay icosahedral cluster occur. The structure is well ordered, showing a very low amount of disorder. Our µ-Al 4 Mn sample with the composition Al 80.7 Mn 19.3 was single-phase material grown from an Al-rich melt by the Czochralski technique. A rectangular bar of dimensions 2× 2× 8 mm

  a low magnetic field H = 8 Oe. The low-temperature susceptibilities between 1.9 and 6 K are shown in Fig. 2a. Below about 3 K, zfc χ and fc χ start to distinguish, demonstrating ergodicity breaking on the experimental time scale, typical of a magnetically frustrated spin glass. The spin-freezing temperature T f = 2.7 K was determined as the temperature where the zfc-fc splitting occurred (this coincides well with the temperature where zfc χ exhibits a cusp).

χ

  K in the magnetic field H = 1 kOe is shown in Fig.2b. The analysis was performed with the Curie-Weiss law, is the temperature-independent part of the susceptibility, C CW the Curie-Weiss constant and θ the Curie-Weiss temperature. The constant C CW gives information on the magnitude of the Mn moments, whereas the type and strength of the coupling between the spins can be estimated from the magnitude and sign of θ . In the inset of Fig.2b, the ( )T χ data are displayed in a ()versustemperature plot. The fit of the high-temperature data T > 40 K (solid line) yielded the parameter values 0 χ = 1.06× 10 -4 emu/mol, C CW = 2.95× 10 -2 emu K/mol and θ = -29 K. The negative Curie-Weiss temperature suggests an antiferromagnetic (AFM)-type antiparallel coupling between the spins. The value of the Curie constant C CW was used to calculate the mean effective magnetic moment B eff eff p µ µ =per Mn ion, where eff p is the mean effective Bohr magneton number that can be calculated using the formula[START_REF] Mabbs | Magnetism and Transition Metal Complexes[END_REF] C CW given in units per mol of Mn atoms). We obtained eff p = 1.09 per Mn atom. Comparison of this value to the Bohr magneton number of a bare Mn 2+ ion (p = 5.9) shows that the Mn moments in the µ-Al 4 Mn phase are significantly reduced, indicating partial screening by the conduction-electron cloud in a conducting environment. While the reduced mean effective Bohr magneton number cannot distinguish between the situations where a large fraction of Mn atoms carry small magnetic moments or a small fraction of Mn atoms carry full magnetic moments

  displayed as an inset in Fig.3, where a logarithmic dependence (base 10) of f T on the frequency is evident. The frequency shift of f T is often quantitatively evaluated by the empirical criterion e., by calculating the relative change of f T per decade ν . For the µ-Al 4 Mn, this ratio amounts 0.013, which is in the range found for metallic spin glasses[START_REF] Mydosh | Spin glasses: an experimental introduction[END_REF] like the AuFe and PdMn. The change of f T with the frequency is thus small, amounting about 1 % per decade ν . The ac susceptibility results also support the spin-glass nature of the µ-Al 4 Mn phase.

  in Fig.4. The temperature dependence of the resistivity is quite complex, exhibiting a broad maximum at about 175 K and a minimum at 13 K.The room-temperature resistivity amounts =

2 L

 2 Einstein relation, the conductivity σ depends on the electronic density of states (DOS) ( ) ε g and the spectral diffusivity ( ) ε D within the thermal interval of a few T k B around the Fermi level F ε . In the case of slowly varying metallic DOS around F ε it is permissible to replace ( ) ε g by ( ) F g ε . For the diffusion constant of slow charge carriers it was shown [23] that it can be written as is the non-Boltzmann contribution to the square of spreading of the quantum state at energy ε due to diffusion, averaged on a time scale τ . ( ) τ L is bounded by the unit cell length and saturates to a constant value already for short averaging time.The dc conductivity of the system can be written as[START_REF] Trambly De Laissardière | [END_REF]

1 0 -τ

 10 due to scattering by quenched defects and a temperature-dependent rate1

2 L 2 L

 22 scattering by phonons and the magnetic scattering by localized paramagnetic spins. In the simplest case, p τ can be phenomenologically written as a power-law of temperature, within a limited temperature interval. Assuming that ( ) τ can be replaced by its limiting value, a constant , Eq. (2) yields a minimum in the conductivity σ as a function of τ or temperature (or equivalently, there is a maximum in the resistivity) at the condition v L = τ . At temperatures below the resistivity maximum, τ is long enough that the system is in the Boltzmann regime ( B σ σ ≈ ) with a power-law positive-temperature-coefficient (PTC) metallic resistivity. At high temperatures above the resistivity maximum, τ is short enough that the system is in the non-Boltzmann regime and the resistivity exhibits a negative temperature coefficient (

σ

  contains four fit parameters A , B, C and α (where the last two always appear in a product α CT ).

4 - 1 ,

 41 Mn and the resulting theoretical curve is shown in Fig. 4. Good agreement with the experiment was found in the entire investigated temperature range from 300 to 2 K, reproducing both the maximum and the minimum of the resistivity. The fit parameters are A = B = 8.43× 10 -4 (µΩcm) -1 , C = 4.83× 10 -3 , α = 0.87 and d = -5.05 µΩcm, where the units of C and d are chosen such that the temperature in the expressions α CT and T d log appears dimensionless.

Fig. 1 :

 1 Figure captions:

Fig. 2 :

 2 Fig. 2: (Color online) (a) The zfc and fc susceptibilities

Fig. 3 :Fig. 4 :

 34 Fig. 3: (Color online) Real part χ′ of the ac susceptibility measured in an ac magnetic field of amplitude H 0 = 6.5 Oe at frequencies ν = 1, 10, 100 and 1000 Hz. The inset

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Acknowledgement: This work was done within the 6 th Framework EU Network of Excellence "Complex Metallic Alloys" (Contract No. NMP3-CT-2005-500140). J.D. acknowledges support from the Centre of Excellence EN→FIST, Dunajska 156, SI-1000 Ljubljana, Slovenia.