Hiba Zbib
email: hibazbib@hotmail.com

Sandrine Mouysset

Simon Stute

Jean-Marc Girault

Jamal Charara

Sylvie Chalon

Laurent Galineau

Irène Buvat

Clovis Tauber
email: clovis.tauber@univ-tours.fr

Optimized spectral clustering for segmentation of dynamic PET images

The quantification of dynamic PET images requires the definition of regions of interest. The manual delineation is a time consuming and unreproducible process due to the poor resolution of PET images. Approaches were proposed in the literature to classify the kinetic profiles of voxels, however, they are generally either sensitive to initial conditions or favor convex shaped clusters. Recently we have proposed a kinetic spectral clustering (KSC) method for segmentation of dynamic PET images that has the advantage of handling clusters with arbitrary shape in the space in which they are identified. However, its use for clinical applications is still hindered by the manual setting of several parameters. In this paper, we propose an extension of KSC to make it automatic (ASC). A new unsupervised clustering criterion is tailored and a global optimization by a probabilistic metaheuristic algorithm is used to select the scale parameter and the weighting factors involved in the method. We validate our approach with GATE Monte Carlo simulations. Results obtained with ASC compare closely with those obtained with optimal manual parameterization of KSC, and outperform those obtained with two other approaches from the literature.

I. INTRODUCTION

Positron Emission Tomography (PET) is a powerful technique for the in vivo study of physiological and biochemical processes of organs. Such studies require the definition of regions of interest (ROI) to extract the time activity curves (TACs) that are used into compartment models to quantify the target. The definition of regions of interest is generally performed by an expert operator. These manual delineations are a time-consuming, subjective and unreliable process due to noise and poor spatial resolution of PET images. As a result, there is a growing interest in the development of clustering methods that aim at classifying the TAC of voxels in order to separate the PET image into functional regions. Wong et al. [START_REF] Wong | Segmentation of Dynamic PET Images Using Cluster Analysis[END_REF] proposed to segment dynamic PET images using a Kmeans algorithm, Kim et al. [START_REF] Kim | Segmentation of VOI from Multidimensional Dynamic PET Images by Integrating Spatial and Temporal Features[END_REF] integrated the spatial and temporal information in an hybrid method that uses both cluster analysis and region growing. Maroy et al. [START_REF] Maroy | Segmentation of Rodent Whole-Body Dynamic PET Images: An Unsupervised Method Based on Voxel Dynamics[END_REF] estimated the pharmacokinetics far from organ borders and segmented the rodent whole-body PET images by local means analysis followed by hierarchical linkage algorithm to merge regions with similar TAC. Kamasak [START_REF] Kamasak | Direct reconstruction of kinetic parameter images from dynamic PET data[END_REF] presented a parametric iterative coordinate descent algorithm for direct nonlinear estimation of kinetic parameters at each voxel from dynamic PET sinogram data.

Recently we have proposed a kinetic spectral clustering (KSC) approach for the segmentation of dynamic PET images, which has the advantage of handling arbitrary shaped clusters [START_REF] Mouysset | Segmentation of dynamic PET images with kinetic spectral clustering[END_REF]. It achieved increased precision in the differentiation between functional regions. However it requires manual setting of the parameters involved in the method. In this paper we extend the KSC method by making it automatic (ASC) and therefore more suited for clinical applications. We change the last step of the method to make spectral clustering deterministic, and we combine an original unsupervised clustering criterion and a probabilistic metaheuristic to automatically estimate both the scale parameter of the method and the weighting scheme of the frames of the dynamic PET sequence. In order to validate our approach, GATE Monte Carlo simulations of the Zubal head phantom and a sphere phantom were performed. The ASC was successfully compared with the best result of KSC when the parameters are selected manually and with two other methods from the literature.

II. PROPOSED METHOD

A. Notations

Let us denote K the number of functional regions supposedly known a priori by the user, T the number of frames contained in the PET image series, N the number of voxels to be clustered and

X i ∈ R T the measured TAC in voxel i ∈ [1...N ].

B. Kinetic spectral clustering for segmentation of dynamic PET images

The KSC of voxel TAC consists in mapping the data into a high dimensional space to increase the separation between clusters, then clustering the data in a reduced space where they are linearly separable [START_REF] Mouysset | Segmentation of dynamic PET images with kinetic spectral clustering[END_REF].

1) Mapping into a high dimensional space:

We build the similarity matrix between all pairs of TACs by the calculation of the similarity function S defined as follows:

S(X i , X j ) = exp( -d(X i , X j ) 2σ 2 , (1)
where X i and X j are the TACs of voxels i and j, σ is a scale parameter that adjusts the distance of patterns mapped into the feature space, and d(X i , X j ) is a weighted euclidean distance that we define as:

d(X i , X j ) = ( T l=1 w l (X i (l) -X j (l)) 2 ) 1/2 (2)
where w l is the weighting factor of frame l. The noise level in dynamic PET images differs significantly between frames, therefore we include the noise variance in the calculation of the distance between TACs to increase the confidence in frames having less noise. In KSC the weighting factors of each frame were calculated as a noise estimate based on frame duration, time since injection and the number of counts. While this approach allows increased precision in the results, it does not consider the actual activity separation of ROIs. In this paper we propose to consider the weighting factors as parameters of the model and estimate them in a global optimization scheme.

2) Projection into a low dimensional space:

We adopt a Laplacian random walk matrix L rw [START_REF] Meila | A random walks view of spectral segmentation[END_REF]. Spectral clustering using L rw can be interpreted as trying to find a partition of the graph such that the random walk stays long within the same cluster and seldom jumps between clusters. The first K eigenvectors of L rw are calculated, they constitute a new space where the data are projected, which contains the information needed for clustering.

3) Clustering of the projected data:

In KSC the projected data are clustered by Kmeans algorithm [START_REF] Mouysset | Segmentation of dynamic PET images with kinetic spectral clustering[END_REF]. Kmeans depends on the initial condition and several runs from different starting centers can be necessary to obtain the best result. To avoid any dependency on initial condition, we propose to use a Global Kmeans approach (GKM) [START_REF] Likas | The global K-means clustering algorithm[END_REF] to classify the projected data in the last step of spectral clustering. Compared to KSC, the clustering process of ASC is therefore deterministic, which is a necessary condition for the global optimization of its parameters.

C. Normalized minimal distance

In the segmentation of dynamic PET images two phenomena frequently occur: the separation and the fusion of clusters. To detect these two phenomena, several unsupervised criteria were proposed: Davies and Bouldin [START_REF] Davies | A cluster separation measure[END_REF] proposed an index based on a ratio between the sum of within cluster scatter and between cluster scatter. Dunn [START_REF] Dunn | A fuzzy relative of the isodata process and its use in the detecting compact well-separated clusters[END_REF] proposed a cluster validity index to identify compact and well separated clusters, and defined the diameter of a cluster as the maximal intra cluster distance between all pairs of points. In this paper, we propose a criterion that consists in a robust extension of the Dunn index. We define the normalized minimal distance (NMD) criterion as: where δ min is a minimal inter-cluster distance that decreases when voxels representing similar time courses are assigned to different clusters. It is defined as follows:

N M D = - δ min ∆ max (3)
δ min = min 1≤o<p≤K ||C o -C p ||, (4)
where C o and C p are the centers of cluster o and p respectively and ||C o -C p || is the euclidean distance between these two centers. ∆ max is a robust maximal intra-cluster distance, that tends to increase when a fusion of clusters occurs. It is defined as:

∆ max = max 1<z<K ( i,j∈z ||Xi-Xj ||A(||Xi-Xj ||≥dz 95 th ) Nz×0.05
),

(5) where dz 95 th is the 95 percentile of intra cluster distances of cluster z, N z is the number of voxels belonging to z and and A is a Boolean function such that A(Y ) = 1 if Y is true and 0 otherwise. The proposed NMD criterion increases when a fusion or a splitting of clusters occurs. Compared to the Dunn index which is based on individual points, NMD is less sensitive to outliers and it better represents the quality of clustering for dynamic PET image segmentation.

D. Global optimization of NMD by simulated annealing

We propose to estimate automatically the scale parameter and the weighting factors w l of each frame by a global optimization scheme using the simulated annealing algorithm [START_REF] Kirkpatrick | Optimization by simulated annealing[END_REF]. At each iteration the parameter values of ASC are randomly perturbed and the NMD cost function is calculated. Downhill steps are always accepted while uphill steps are accepted to step out of a local minimum under a probability acceptance function that depends on the objective functions and the current temperature.

We bound the search of the scale parameter to the interval between the minimal distance and the 90 th percentile of distances between all pairs of TACs. Indeed, in ASC the scale parameter represents a soft threshold of distance between voxels TACs. When σ has a magnitude equivalent to the maximal distance between points, most patterns in the feature space are too close to each other. While when its magnitude is equivalent to the minimal distance, all points are far from every other. We initialize σ by the average distance between every point and its closest neighbor. The bounds of the weighting factors are set as ∀ l, w l ∈ [0 1], while the initial value for all frames is set to w l0 = 1.

III. VALIDATION AND EVALUATION

A. Simulation and experimentation

We performed GATE Monte Carlo simulations of Gemini GXL PET 4D acquisitions. A phantom consisting of 12 spheres of different diameters (simulation 1), and the labeled MR image of the Zubal head phantom (simulations 2 and 3) [START_REF] Zubal | Computerized three-dimentional segmented human anatomy[END_REF] were used as voxelized sources. The regions TAC were generated according to the three-compartment model as proposed by Kamasak et al. [START_REF] Kamasak | Direct reconstruction of kinetic parameter images from dynamic PET data[END_REF] and Maroy et al. [START_REF] Maroy | Segmentation of Rodent Whole-Body Dynamic PET Images: An Unsupervised Method Based on Voxel Dynamics[END_REF]. We used six regions of the Zubal head phantom for the simulations (figure 1). Each sequence consisted in 5 × 30s followed by 15 × 60s dynamic frames. The reconstruction of dynamic PET images was performed with a fully 3D ANW-OSEM iterative method. In order to assess the quality of segmentation in different conditions of noise level, we used several sets of reconstruction parameters summarized in Table I.

B. Evaluation by supervised criteria

1) Adjusted Rand Index (ARI): Adjusted Rand Index [START_REF] Rand | Objective criteria for the evaluation of clustering method[END_REF] evaluates the consistency between ground truth regions and those of the segmented image. This criterion takes into account the number of pairs of objects grouped into same or different clusters.

2) Pratt's Figure of Merit (PFOM):

Pratt's figure of merit measures the precision of edges locations in segmented images, compared to their ground truth location. The PFOM was calculated as follows:

P F OM = 1 max(N I , N D ) ND i=1 1 1 + αd 2 i (6)
where N I and N D are respectively the number of ideal and detected edge voxels, d i denotes the distance from the ithdetected edge voxel to the nearest ideal edge voxel and α is a scaling constant set to 1/9 as in Pratt's work [START_REF] Pratt | Digital Image Processing[END_REF].

These two supervised criteria lie between 0 and 1. When the segmented results agree perfectly with the ground truth, these criteria are equal to 1.

IV. RESULTS

A. Comparison between the supervised criteria (ARI, PFOM) and the unsupervised criteria (NMD, Dunn index)

We compare our proposed NMD criterion with the Dunn index (to show the improvement that NMD brings in the calculation of the modified intra cluster distance ∆ max ) and with the supervised criteria ARI and PFOM. The Dunn index and the absolute value of NMD were plotted with the supervised ARI and PFOM criteria versus the values of sigma tested by the simulated annealing (Fig. 2). The Dunn index was more sensitive to noise and outliers, which was predictable as it calculates the intra and inter cluster distances using individual points. NMD criterion was found more robust and representative of cluster distances. As shown in Fig. 2 the Dunn index had more variance compared to other criteria while NMD criterion varied in agreement with the supervised ARI and PFOM criteria. The ARI, PFOM and NMD criteria were found to be associated. We compared the proposed ASC method with the EM algorithm [START_REF] Ashburner | A cluster analysis for the characterisation of dynamic PET data[END_REF], and the best result that could be obtained with the KM over 100 runs on the 3 simulations. Figure 3 displays the results for simulation 1. The EM and KM algorithms (Fig. 3(c,d)) correctly detected two big spheres, merged the small spheres and separated the background into two parts. ASC correctly identified all regions (Fig. 3(e)). Two opposite tendencies were observed regarding the small spheres: EM and KM methods tended to overestimate their size while ASC underestimated them. ASC was the only method able to distinguish the two small spheres.

B. Comparison of ASC with Kmeans (KM) and Expectation-Maximization (EM) approach

Figure 4 (first row) shows the results for simulation 2, where the level of noise was the lowest of the simulations. The EM algorithm (Fig. 4(d)) merged the occipital lobe with the parietal lobe and several ROI were associated with the background. The KM (Fig. 4(c)) also produced a splitted background while it detected more correctly the regions than EM. However, compared to the ground truth the size of the parietal and occipital lobes were bigger, which was not the case with ASC (Fig. 4(e)) where all regions were correctly and more precisely detected.

Figure 4 (second row) shows the results for simulation 3. For this simulation, the attenuation and the normalization were included, and the number of iterations was higher which resulted in high level of noise. The EM and the KM algorithms (Fig. 4(h,i)) did not detect the thalamus and several ROI were merged with the background. With ASC (Fig. 4(j)), the background was less associated with other regions. The parietal, occipital and frontal lobe were correctly detected, but the thalamus was not detected. In this simulation, the 3 methods misclassified some functional regions and especially the thalamus, parietal and occipital lobes. The pixels of the thalamus were very affected by both the noise and the partial volume effect. In ASC the misclassification of the thalamus forced the splitting of another region which corresponded to the cerebellum.

Table II presents the average and the standard deviation of the quantitative criteria ARI, PFOM, and NMD calculated for 8 2D+t slices from the simulation 1, and 20 slices from the simulations 2 and 3 using KM, EM and ASC. ARI, PFOM, and the absolute value of NMD were the smallest for the EM algorithm, while they were the highest for ASC in the 3 simulations. This improvement can be explained by the ability of ASC to separate arbitrary shaped clusters. It also indicates that the global search procedure used to derive the final partition of the data led to acceptable settings of the parameters.

C. Automatic property of ASC

To illustrate the automatic property of ASC, we determined the maximal ARI and PFOM scores that could be obtained with manual selection of parameters of KSC. Then we calculated for 8 2D+t slices from the simulation 1, and for 20 slices from the simulations 2 and 3, the scores of ARI and PFOM obtained with ASC over the maximal scores that could be obtained manually (Table III). The NMD criterion was efficient to select the parameters that gave good segmentation quality of the simulated dynamic PET images, with a percentage of maximal achievable score comprised between 93% and 98%. V. CONCLUSION An unsupervised criterion was tailored and optimized by simulated annealing to automatically estimate the input parameters of a spectral clustering method. The ASC results were found very close to the best results obtained with manual selection of the parameters. In addition, ASC was favorably compared with two other approaches: KM and EM. It improved the identifiability of functional regions without a priori knowledge on the kinetic models. The ROI definition offered by ASC and the level of automatization achieved might have significant impact for quantification of dynamic PET images.

Fig. 1 .

 1 Fig. 1. Regions of the Zubal head phantom used for image simulation.

Fig. 2 .

 2 Fig. 2. Comparison between the absolute value of NMD, the Dunn index and the supervised ARI and PFOM criteria (Maximum values of all criteria are normalized to 1).

Fig. 3 .

 3 Fig. 3. Axial view of the results obtained for simulation 1: (a) ground truth, (b) one frame from the image series, (c) KM, (d) EM, (e) ASC.

Fig. 4 .

 4 Fig. 4. Sagittal views of the results obtained with the simulations 2 and 3. First row: sagittal view of the results obtained for simulation 2: (a) ground truth,(b) one frame from the image series, (c) KM, (d) EM, (e) ASC. Second row: sagittal view of the results obtained for simulation 3: (f) ground truth, (g) one frame from the image series, (h) KM, (i) EM, (j) ASC.

TABLE I .

 I PARAMETERS USED FOR THE SIMULATION AND THE RECONSTRUCTION OF THE 3 PET SEQUENCES

		Sim.1	Sim.2	Sim.3
	Phantom	Sphere	Zubal	Zubal
	Medium attenuation	yes	no	yes
	Attenuation correction	yes	no	yes
	Normalization	yes	no	yes
	Iteration numbers	2 iteration	5 iterations	10 iterations
	Subset numbers	16 subsets	8 subsets	16 subsets
	Noise level	+ +	+	+ + +

TABLE II .

 II AVERAGE AND STANDARD DEVIATION OF THE QUANTITATIVE CRITERIA.

	Phantoms	Criteria	KM	EM	ASC
	Simulation 1	ARI	0.62 ± 0.32	0.20 ± 0.13	0.79 ± 0.20
	Sphere	PFOM	0.84 ± 0.20	0.49 ± 0.08	0.95 ± 0.04
	(2it,16sb)	NMD	-0.57 ± 0.20	-0.03 ± 0.02	-1.23 ± 0.54
	Simulation 2	ARI	0.73 ±0.13	0.58 ± 0.23	0.86 ± 0.06
	Zubal	PFOM	0.80 ± 0.11	0.76 ± 0.12	0.93 ± 0.05
	(5it,8sb)	NMD	-0.18 ± 0.09	-0.09 ± 0.09	-0.30 ± 0.10
	Simulation 3	ARI	0.54 ±0.23	0.52 ± 0.22	0.75 ± 0.16
	Zubal	PFOM	0.73 ± 0.08	0.72 ± 0.11	0.84 ± 0.09
	(10it,16sb)	NMD	-0.15 ± 0.07	-0.08 ± 0.07	-0.19 ± 0.09

TABLE III .

 III PERCENTAGE OF MAXIMAL SCORES OBTAINABLE FROM MANUAL SETTING

		Sim.1	Sim.2	Sim. 3
	ARI ASC	93 %	98 %	93 %
	PFOM ASC	98 %	97 %	93 %

ACKNOWLEDGMENT

The authors would like to thank the CNRS in Lebanon and the Lebanese university for supporting this work.


Célia Martinie
email: martinie@irit.fr

Philippe Palanque
email: palanque@irit.fr

Martina Ragosta
email: martina.ragosta@dblue.it

Racim Fahssi
email: racim.fahssi@irit.fr

Extending Procedural Task Models by Systematic Explicit Integration of Objects, Knowledge and Information

Keywords: Task modelling, User Centered Design, Model-based approaches

Task analysis can be considered as a fundamental component of user centered design methods as it provides a unique way of analyzing in a systematic way users' roles and activities. A widely used way of storing the information gathered during that phase in a structured and exhaustive way is to build task models which are then amenable to verification of properties or to performance evaluation. In widely used notations such as Hierarchical Task Analysis (HTA) or CTT (Concur Task Tree), information or objects manipulated by the users while performing the tasks does not receive a similar treatment as the sequencing of tasks which is usually carefully and exhaustively described. This paper proposes a systematic account for the various concepts manipulated by the users while performing tasks. Such concepts include different types of knowledge (declarative, situational, procedural and strategic), objects (manipulated by the user) and information. These concepts are systematically represented in a set of extensions of the HAMSTERS notation allowing the analysis of concepts-related properties such as learning curve, complexity, information workload, … We demonstrate the application of the approach on the example of a two players game making explicit the connection between these extended task models and the user interface of the game.

INTRODUCTION

To accomplish tasks, users may need to manipulate objects or information about current situation of the system and its environment, and knowledge about which actions to perform and how to perform them. When designing interactive systems, the phase of tasks analysis usually focuses on: identifying goals which should be reached, grouping activities that have to be accomplished, understanding execution order of these activities and identifying objects required to perform the tasks. Existing analysis techniques and notations do not provide full support for explicitly and distinctively describing concepts related to the notions of Object, Knowledge and Information (OKI). In this paper, we focus on task modeling notations, which provide a widely used way of structuring and storing information gathered during the task analysis phase. Although some of them provide support for describing manipulated objects, most of these notations are focused on representation of procedures and methods for reaching a goal rather than the knowledge, information and objects involved. In this paper, we propose a systematic account for the various types of concepts required and manipulated by the users while performing activities including objects (manipulated by the user), information, and different types of knowledge (declarative, situational, procedural and strategic). We present how such information can be captured in task models provided some extensions are made. We exemplify these extensions on HAMSTERS notation. This proposal aims at providing the same level of support for describing concepts involved in users' tasks, as temporal ordering and hierarchical structuring of user tasks provided by most of the notations dedicated to tasks modeling. The underlying philosophy of the approach is similar to computer science view where, in the early days, data structure [9] was considered separately from control structure [9] until the unifying concept of object-oriented programming [9]. The example presented in the paper aims at demonstrating that integrating concepts and procedures within a single tasks model can provide multiple benefits throughout the development process from user needs identification to evaluation and training. The paper is structured as follows. Next section presents a review of literature from cognitive psychology research about mental representation of knowledge and allows us to identify various types of knowledge (such as declarative and procedural as well as situational and strategic). Section entitled "Knowledge, information and objects in task analysis and modeling" presents a review of the state of the art of task modeling notations to assess their capability of handling (in an explicit and systematic way) the types of knowledge identified in the previous section. Section entitled "Integrating knowledge into a procedural task modeling notation" summarizes the concepts in HAMSTERS notation for task modeling and proposes extensions to that notation by means of a concept map notation element (for knowledge, objects and information representation) and the integration of these concepts in the hierarchical description of tasks (with distinctive representations). These extensions are exemplified on a simple case study in section entitled "Illustrative example". Lessons learnt and conclusions are then presented in eponym sections.

KNOWLEDGE AND INFORMATION IN OUR MIND: STRUCTURE, MENTAL REPRESENTATION, AND LIFECYCLE

This section summarises contributions in the area of cognitive psychology dealing with the various types of knowledge processed by humans and how this knowledge is acquired, stored and retrieved. The Oxford Dictionary 1 defines knowledge as "facts, information, and skills acquired by a person through experience or education; the theoretical or practical understanding of a subject." This definition exhibits two critical points. The former, "facts, information, and skills acquired by a person through experience or education" involves a relationship between facts, information and skills that have to be acquired to become knowledge. This acquisition can occur either through experience or through education. The latter, "the theoretical or practical understanding of a subject" suggests different types of understanding, theoretical and/or practical. These two aspects are discussed further in this section that is structured as follows. The first sub-section deals with types of knowledge. The second subsection is dedicated to mental representation of each types of knowledge, and the third one describes knowledge lifecycle. The last one summarises the lessons learnt from previous sub-sections and identifies requirements to explicitly represent objects, knowledge and information in task models.

Types of knowledge

There are two main types of knowledge: declarative and procedural [9]. Declarative knowledge is factual knowledge that is true or false. It describes objects by specifying the properties which characterize them. This type of knowledge does not pay attention to the actions needed to obtain a result, but only to their properties [START_REF] Codd | Relational Database: A Practical Foundation for Productivity[END_REF]. Alternatively, procedural knowledge corresponds to the knowledge exercised in the performance a task. It is directly applied to a task and is acquired while performing activities [9]. Declarative knowledge corresponds to the "I know that" while procedural knowledge corresponds to "I know how". They have different modes of mental representation and involve different memory processes and brain areas [START_REF] Codd | Relational Database: A Practical Foundation for Productivity[END_REF]. Beyond this declarative/procedural distinction, according to [9], there are other two types of knowledge: situational and strategic. Situational knowledge is related to case-based reasoning and contains domain specific information, while strategic knowledge is associated to a plan, a parallel checking and an analysis of possible choices. They have to be considered as refinements of the two main types (declarative and procedural).

The mental representation of declarative and procedural knowledge

After acquisition through senses, each sensed data is elaborated in different ways and using different cognitive processes to become information. Information can be elaborated as mental representations [9] which are hypothetical internal cognitive symbol that represents external reality, or a mental process that makes use of such a symbol [9].

Mental representation of declarative knowledge: structuring, grouping and connecting knowledge and information

According to [START_REF] Codd | Relational Database: A Practical Foundation for Productivity[END_REF], declarative knowledge is represented by mental images and/or symbolic forms such as words. The image is analogue to the object that it represents (e.g. a physical object or information in the real world) and is made up of concrete 1 http://oxforddictionaries.com/us/definition/american_english/kno wledge#m_en_us1261368 last accessed February 2013 attributes and features that can be concurrently captured. Conversely, the word is a symbolic representation and the relationship between this symbolic form and the object is arbitrary.

Knowledge structuring and grouping through concepts

A concept provides a mean for understanding the world and is a mental organization of knowledge. It is the fundamental unit of declarative knowledge, and often a single concept may be captured in a single word. They have different characteristics which ensure a wide flexibility in using them: such as (a) concepts can contain other concepts, (b) concepts can include typical and generic events, (c) concepts can present different abstraction degrees and (d) they can comprehend information about relationships [9].

Connecting through networks A semantic network is a network which represents semantic relations between concepts and can be represented as a web of interconnected elements made up of nodes and labels [9]. Nodes represent concepts and labels denote relationships which may involve category membership, attributes, or other semantic relations. These labels connect concepts in memory for allowing persons to link different nodes through their meaning. Semantic networks are given different names according to their scope and use. For instance, they are called associative networks when focusing on links and concept maps when focusing on concepts. As presented in [START_REF] Codd | Relational Database: A Practical Foundation for Productivity[END_REF] concept maps are also used as a direct method for assessing knowledge and can be used for representing the transition from novice to expert performance [9,[START_REF] Codd | Relational Database: A Practical Foundation for Productivity[END_REF]. As claimed by Sowa [START_REF] Codd | Relational Database: A Practical Foundation for Productivity[END_REF] all these semantic networks share a graphical representation that can be used either to represent knowledge or to support automated systems for reasoning about the represented knowledge.

Mental representation of procedural knowledge: structuring, grouping and connecting knowledge and information

Mental representation of procedural knowledge has always been considered hard to identify and of course to prove and most of the results currently available derive from computer simulations. As claimed in [START_REF] Codd | Relational Database: A Practical Foundation for Productivity[END_REF] such knowledge could be represented as scripts which is a structure that describes sequences of events in a particular context. The structure is an interconnected net made of slots influencing each other.

Knowledge structuring and grouping through routines and subroutines

Procedural knowledge can be structured as a set of rules governing a production. As defined in [START_REF] Codd | Relational Database: A Practical Foundation for Productivity[END_REF] production system is the entire set of rules to perform a task or use a skill. These rules are organized into routines that are sets of instructions for accomplishing a task. They can be refined in subroutines that are sets of instructions to accomplish a subtask. These routines are typically organized according to basic control structures in computer science such as sequence, alternatives and iteration [9,9].

Connecting through hierarchical networks Due to the structuring in routines and subroutines it is possible to adopt a hierarchical representation to describe them [START_REF] Codd | Relational Database: A Practical Foundation for Productivity[END_REF]. In such hierarchical networks information known at a higher level in the hierarchy is known at lower levels.

Even though we presented declarative and procedural knowledge in an independent way, both knowledge types corresponds to different aspects of the same concept. For this reason, next section describes research contributions aiming at proposing integrative models for representing declarative and procedural knowledge.

Mental representation of declarative and procedural knowledge adopting integrative models: structuring, grouping and connecting

Adaptive Character of Thought-Rational (ACT-R): structuring, grouping and connecting

This approach integrates a network representation for declarative knowledge and a production system representation for procedural knowledge. On one hand, declarative network includes storing and retrieving of information (called compilation process) and, on the other hand, procedural system implements the processes of automation or proceduralization [9]. According to ACT-R [9], all knowledge begins as declarative information. Procedural knowledge is learned by making inferences from already existing factual knowledge. New production rules are formed by the conjunction or disjunction of existing production rules [9]. This demonstrates the importance of dealing with declarative knowledge in an explicit and exhaustive way.

Parallel Distributed Processing (PDP): structuring, grouping and connecting

Neuropsychological research work [9] on amnesic patients and psychobiological studies on animals and human brain indicate that a large part of human cognition is based on parallel processing in which multiple operations are performed simultaneously. These experiments suggest that "non-declarative knowledge" may encompass a broader range of mental representations than just procedural knowledge [START_REF] Codd | Relational Database: A Practical Foundation for Productivity[END_REF]. All these forms of non-declarative knowledge are implicit and not easily explainable as they are tacit [9]. This highlights the needs for dealing in an explicit way with such information in order to be sure that concepts (and more precisely OKI) are carefully analysed and represented. The Parallel Distributed Processing (PDP) approach suggests that humans are able to carry out a large number of cognitive operations simultaneously by the use of countless numbers of neural processors located in the brain [START_REF] Codd | Relational Database: A Practical Foundation for Productivity[END_REF]. In neural networks the configurations, also called patterns, carry more information than single units generating knowledge representation.

Lessons learned from this related work and requirements for task modelling

This section has described research work in cognitive psychology around the notion of knowledge classification and structuring. This related work has identified two main types of knowledge that have to be taken into account when analysing users' activities. The first one is declarative knowledge that can be structured and grouped using concepts (objects or information from the outside world and knowledge from the user), which can then be connected ending up in a semantic network or concept map. The second main type of knowledge is procedural knowledge which is structured by mean of routines and subroutines, which can be connected together and thus represented in a hierarchical structure. Two sub-types of knowledge should also be taken into account when analysing users' activities: situational and strategic. Both sub-types help refining declarative and procedural knowledge to represent more precisely users' activities. The integrative approaches suggest that declarative and procedural knowledge can be structured, grouped and connected together as they are closely intertwined while users perform tasks. This related work analysis allows us to identify a set of requirements for including in a systematic way both declarative and procedural knowledge not only during tasks analysis and also during tasks modelling:

-REQ_1: Concepts should be used to represent explicitly declarative knowledge.

-REQ_2: Concepts can be refined into objects and into information and these two entities should be made explicit.

-REQ_3: Semantic networks may be used to represent the concepts and their relation-ships.

-REQ_4: As concepts relate to each other, these connections should be made explicit to structure and group declarative knowledge.

-REQ_5: Declarative knowledge can be refined into strategic and situational and these two types of knowledge should be represented explicitly.

-REQ_6: Routines and subroutines should be used to represent explicitly procedural knowledge.

-REQ_7: As routines and subroutines are related to each other, these relationships should be made explicit in order to structure and group procedural knowledge.

-REQ_8: Hierarchical structures (trees) may be used to represent both routines and sub-routines and their relationships.

-REQ_9: Procedural knowledge can be refined into strategic and situational and these two types of knowledge should be represented explicitly.

-REQ_10: All the information above should be integrated in a single model as they are closely intertwined while users perform tasks.

It is important to note that this related work section has focussed on the data manipulated by users when performing tasks either being of knowledge, information or objects nature. Cognitive process manipulating this data (analysis and decision) or detailed interaction with the real world (motor and perception) have not be detailed in this related work as they are already rather extensively covered by notations for task models.

KNOWLEDGE, INFORMATION AND OBJECTS IN TASK ANALYSIS AND MODELLING

This section proposes a detailed analysis of a number of established task modelling techniques (including their notations and support tools) to assess their capability to cover the requirements detailed above. As the tools associated to the notation sometimes cover only part of the notation and sometimes go beyond the notation (with the help of simulation or analysis for instance), last row to "Tool support" for each task modelling notation. If the box shows a tick "√" means that the task modelling notation satisfies the corresponding requirement, while if the box shows a cross "x" means that the task modelling notation does not satisfy the requirement. "Partially with objects" sentence indicates that the task modelling notation partially satisfies the requirements i.e. does not deal with knowledge and information. "Through precondition" or "Through constraint" sentences indicate that the task modelling notation has attempted to integrate all the requirements via precondition or constraint. Due to space constraints, only task modelling techniques are presented with references but the tools supporting editing and sometimes simulating models are indicated (without reference).

INTEGRATING KNOWLEDGE INTO A PROCEDURAL TASK MODELLING NOTATION

The comparison of existing task modelling notations has demonstrated the limited support for explicitly representing knowledge and for integrating it into the usual procedural-centred description of tasks. As a consequence, task models produced using these notations only capture partial information on the tasks which may in the end result in an increase the design effort (increased number of prototyping-evaluation cycles for example for correcting mistakes). In order to overcome this issue, our proposal aims at providing support for representing in a complete and systematic way information and knowledge users need when accomplishing tasks while using interactive systems.

This section presents how a task modelling notation can be extended in order to provide support for complete and systematic description of required knowledge when using an interactive system. In this paper we used the HAMSTERS notation as a basis but this approach is also compatible with other proceduralcentred notations such as CTT for instance. The approach is similar to previous work we have recently conducted in order to address scalability issues in task models by adding structuring mechanisms [9]. In this work HAMSTERS was also used to convey the concepts but the extensions are applicable to any other procedural-centred notation.

HAMSTERS task modelling notation and tool

HAMSTERS 2 is a tool-supported graphical task modelling notation aiming at representing human activities in a hierarchical and ordered way. Goals can be decomposed into sub-goals, which can in turn be decomposed into activities, and output of this decomposition is a graphical tree of nodes. Nodes can be tasks or temporal operators. Tasks can be of several types (as illustrated in Figure 1) and contain information such as a name, information details, critical level. Only the high-level task type are presented 2 http://www.irit.fr/recherches/ICS/softwares/hamsters/index.html here (due to space constraints) but they are further refined (for instance the cognitive tasks can be refined in Analysis and Decision tasks [9]).

For the procedural description concepts are very close to the ones of CTT such as the task types and the operators. Temporal operators are used to represent temporal relationships between sub-goals and between activities (as detailed in can also be tagged by temporal properties to indicate wh not they are iterative, optional or both [9].

Figure 1. High-level task types in HAMSTERS

As explained above; composition and structuration mechanisms have been introduced in order to provide support for description of large amounts of activities [9]. One main element of these mechanisms is subroutine. A subroutine is a group of activities that a user performs several times possibly in different contexts and which might exhibit different types of information flows. A subroutine can be represented as a task model and a task model can use a subroutine to refer to a set of activities. This element of notation enables the distribution of large amount of tasks across different task models and factorization of the number of tasks. This corresponds precisely to the way procedural knowledge is mentally represented as explained in the dedicated section. HAMSTERS also provides support for representing how particular objects (data, information….) are related to particular tasks. Figure 2 illustrates the three relationships (input, output or both) between objects and tasks that can be expressed with HAMSTERS notation. Object (data, information…) can be needed as an input to accomplish a particular task (as illustrated in Figure 2a by the incoming arrow). Particular tasks may generate an object or modify it (as illustrated in Figure 2b) here (due to space constraints) but they are further refined (for instance the cognitive tasks can be refined in Analysis and For the procedural description concepts are very close to the ones uch as the task types and the operators. Temporal operators are used to represent temporal relationships between goals and between activities (as detailed in Table 2). Tasks can also be tagged by temporal properties to indicate whether or level task types in HAMSTERS As explained above; composition and structuration mechanisms have been introduced in order to provide support for description ]. One main element of these mechanisms is subroutine. A subroutine is a group of activities that a user performs several times possibly in different contexts of information flows. A subroutine can be represented as a task model and a task model can use a subroutine to refer to a set of activities. This element of notation enables the distribution of large amount of tasks across zation of the number of tasks. This corresponds precisely to the way procedural knowledge is mentally represented as explained in the dedicated section. HAMSTERS also provides support for representing how particular objects (data, information….) are related to particular rates the three relationships (input, output or both) between objects and tasks that can be expressed with HAMSTERS notation. Object (data, information…) can be needed as an input to accomplish a particular task (as illustrated arrow). Particular tasks may generate an object or modify it (as illustrated in Figure 2b)

Table 2. Temporal ordering operators in HAMSTERS

(a) (b) Figure 2.

Relationships between tasks and objects in HAMSTERS

(such a precondition is graphically visible on the task at the top of the hierarchy of the task model in Figure 10 object flow between tasks can also be captured by means of input and output ports (as shown in Figure 3).

Figure 3. Input/output ports and their relationship with objects in HAMSTE

Amongst the requirements identified in paragraph entitled "Lessons learned from this related work and requirements for task modelling", HAMSTERS fulfils requirements 6, 7 and 8 thanks to its hierarchical decomposition and subroutine mechanisms, which can be used to describe procedural knowledge. HAMSTERS partially fulfils requirements 1 and 2 thanks to the explicit representation of objects that can be integrated in the task model by means of pre for instance (partly covering requirement 10).

Extensions to integrate knowledge representations

This section presents the extensions to HAMSTERS notation in order to allow the representation of all the elements of declarative knowledge and their integration within the procedural knowledge description of HAMSTERS. The main changes correspond to the explicit handling of procedural knowledge various elements within a concept map and the explicit representation of these concepts within the task tree.

Extensions for the representation of declarativ HAMSTERS capabilities to describe relationships between concepts and tasks have been extended. Knowledge of declarative type and its refinement in strategic and situational dimensions can be represented using the corresponding boxes illustrated in Figure 4. Relationships between the represented knowledge and the tasks can be represented using input/output relationships represented with arcs as for the objects.

(c) Figure 2. Relationships between tasks and objects in HAMSTERS

(such a precondition is graphically visible on the task at the top of Figure 10) for the tasks and object flow between tasks can also be captured by means of input Figure 3).

Figure 3. Input/output ports and their relationship with objects in HAMSTERS

Amongst the requirements identified in paragraph entitled "Lessons learned from this related work and requirements for ", HAMSTERS fulfils requirements 6, 7 and 8 thanks to its hierarchical decomposition and subroutine n be used to describe procedural knowledge. HAMSTERS partially fulfils requirements 1 and 2 thanks to the explicit representation of objects that can be integrated in the task model by means of pre-conditions on tasks ent 10).

Extensions to integrate knowledge representations

This section presents the extensions to HAMSTERS notation in order to allow the representation of all the elements of declarative knowledge and their integration within the procedural knowledge cription of HAMSTERS. The main changes correspond to the explicit handling of procedural knowledge various elements within a concept map and the explicit representation of these Extensions for the representation of declarative knowledge HAMSTERS capabilities to describe relationships between concepts and tasks have been extended. Knowledge of declarative type and its refinement in strategic and situational dimensions can be represented using the corresponding boxes Relationships between the represented knowledge and the tasks can be represented using input/output relationships represented with arcs as for the objects. Concepts which are of information type can also be represented in a non-ambiguous way using the Information box illustrated Figure 5. As for an object, it is possible to represent the relationships between a task and information (input, output and input-output), as described in previous paragraph.

Figure 5. Representation of a concept of information type

Extensions for the representation of procedural knowledge HAMSTERS capabilities to describe relationshi and temporal ordering operators have been extended. Representative distinctions between strategic and situational procedure can be made using two new types of arcs illustrated in Figure 6.

An ordered set of actions related to a strategy t will be highlighted with the blue "St" tagged arcs ( ordered set of actions the user can execute in a given situation will be highlighted with the green "Si" tagged arcs

ILLUSTRATIVE EXAMPLE

To illustrate how several types of knowledge can be represented in complete and systematic way during task modelling and due to space constraints, we have chosen a sim Though this framework is used to analyse activities handled by users of interactive critical systems such as pilots and ground segments engineers, a case study would have required more space in order to describe the application domain concepts is a two player traditional game. Each player takes his/her turn to choose a remaining number (a token) ranging from 1 to 9. The first player whose sum of selected number is exactly 15 wins the game. In this example, we analyse the tasks perfo players using the interactive application depicted c) Concepts which are of information type can also be represented ambiguous way using the Information box illustrated in As for an object, it is possible to represent the a task and information (input, output and output), as described in previous paragraph.

Figure 5. Representation of a concept of information type

Extensions for the representation of procedural knowledge HAMSTERS capabilities to describe relationships between tasks and temporal ordering operators have been extended. Representative distinctions between strategic and situational procedure can be made using two new types of arcs illustrated in An ordered set of actions related to a strategy the user can apply will be highlighted with the blue "St" tagged arcs (Figure 6a). An ordered set of actions the user can execute in a given situation will be highlighted with the green "Si" tagged arcs (Figure 6b).

knowledge refined into strategic a) and situational b)

To illustrate how several types of knowledge can be represented modelling activities, and due to space constraints, we have chosen a simple example.

activities handled by users of interactive critical systems such as pilots and ground segments engineers, a case study would have required more space in order to describe the application domain concepts. Game of 15 is a two player traditional game. Each player takes his/her turn to choose a remaining number (a token) ranging from 1 to 9. The first player whose sum of selected number is exactly 15 wins the the tasks performed by the players using the interactive application depicted in Figure 7. The upper part of the window display who is next player, below are displayed the remaining tokens while the last row tokens taken by each player. More detailed information about behaviour of this application can be found in [ Figure 8 presents the concept map associated to the Game of 15. It represents required declarative knowl to play the game. The concepts which are necessary or useful to perform actions during the game are grouped and linked through a semantic network. It is made up of 4 main nodes Objects, Roles, Rules and Strategies. Structure from left to right shows the refinement of concepts (from abstract to concrete) as well as the instantiation (for instance from the concept of player to the instantiation of player1 and player2). Lastly, relationship between concepts is also made explicit (for instance the use of the link between the concept "list of available tokens" and the concept Rule). These concepts are integrated in task models depicted in Figure 9 and Figure 10 (presented top of next page) making explicit the integration of procedural and declarative knowledge in HAMSTERS.

Figures 9 and 10 present HAMSTERS models of tasks for playing Game of 15 using the interactive application presented Figure 7.

Figure 9 presents the main HAMSTERS model with top task named "Play Game of 15". The hierarchical and temporally tructure from left to right shows the refinement of concepts (from abstract to concrete) as well as the instantiation (for instance from the concept of player to the instantiation of player1 astly, relationship between concepts is also made explicit (for instance the use of the link between the concept "list of available tokens" and the concept Rule). These concepts are in Figure 9 and Figure 10 p of next page) making explicit the integration of procedural and declarative knowledge in HAMSTERS.

present HAMSTERS models of tasks for playing Game of 15 using the interactive application presented in in HAMSTERS model with top-level task named "Play Game of 15". The hierarchical and temporally ordered sub-trees detail in a procedural way the various tasks that have to be known and performed by the players.

First they have to agree on who will start the game. Then they play the game in an iterative way (round-shaped arc at the top right of the "Play the game" abstract task) until they commonly agree to stop (if one player won or if there are no token left or if they just want to stop). The "Play the game" sub-task is a sequence of the following activities: turn taking (for one player), selecting a token (for current player, this task is a subroutine described in detail in Figure 10 and explained in next paragraph), processing combination of tokens (for each player), and giving turn to the other player.

In Figure 9, procedural knowledge required to play Game of 15 is described via hierarchical decomposition and temporally ordering of nodes. Declarative knowledge (refined in objects and rules in the concept map of Figure 8) is represented by mean of violet boxes connected to particular tasks with input arcs are they described required knowledge to accomplish a task. Required information is depicted by mean of orange boxes connected to particular tasks (with input arcs when information is mandatory to accomplish a task and with output arc when information is modified by a particular task).

Figure 10 describes in a hierarchical way the activities one player has to perform for selecting a token. In sequence, the interactive application interface is first displaying remaining tokens. The player then reads the set of available numbers, has to remember existing strategies and to adopt a particular strategy to select a number (sub-trees detail possible choices of strategies and tasks which have to be executed according to the selected strategy).

Once the player has selected a token, he/she inputs the number and the interface displays the token that has been selected. In this task model too, procedural knowledge required to accomplish "Select of token" subroutine is described via hierarchical decomposition and temporally ordering of nodes. Declarative knowledge (extracted from the concept map in Figure 8) is represented by mean of violet boxes connected to particular tasks with input arcs are they described required knowledge to accomplish a task. Strategic knowledge is represented both in a procedural way (using blue arcs in sub-trees dedicated to strategic knowledge below Adopt a Strategy task) and in a declarative way (blue boxes labeled StK (for Strategic Knowledge) connected to tasks related to particular strategies, also coming from the concept strategies in the concept map in Figure 8. Required information is depicted by mean of orange boxes connected to particular tasks. Example of situational knowledge description is not depicted in this example due to space constraints but would be represented in the same way as strategic knowledge (with corresponding color and arc label as described in Figure 6b)..

LESSONS LEARNT AND TOOL SUPPORT

This example demonstrates that integration of both declarative and procedural knowledge in a task model provides additional useful information for making explicit the concepts (objects, information and knowledge) required for reaching a goal as well as the concepts produced while performing the tasks. -Requirements by identifying what the users' needs in terms of knowledge, information and objects -Design by making explicit the information that has to be provided to the user and the type of input that the users will need to perform -Implementation by providing an informal but complete description of the data that the system will have to embed in order to allow performance of the user tasks -Evaluation by providing an explicit representation of the information that will be manipulated and thus identifying candidate values for user testing.

-Training operators by making explicit what knowledge they have to master in order to use the system. This systematic account for concepts used in tasks descriptions provides means for analysts to verify the compatibility and completeness of the concept map with respect to the task model. For instance one could check that all the concepts in the concept map appear in the task model and are graphically visible on the user interface. According to the case study this is the case for all the concepts around the notion of tokens such as list of available tokens, tokens selected by each player….

As mentioned above, this work has been done in the area of critical systems [9] where analysis about knowledge and information processing is of prime importance. For instance workload analysis has to be performed and decisions around tasks migration and automation can be fruitfully informed by a more complete account of declarative knowledge involved in tasks performance. All the extensions presented to the original HAMSTERS notation have been implemented in HAMSTERS tool which allows both editing and simulating task models. In previous work [9] we have also demonstrated the possibility to connect task models with behavioural descriptions of user interfaces making coexecution and validation of models interactive. Extensions to the CIRCUS (Computer-aided-design of Interactive, Resilient, Critical & Usable Systems) tool suite (which integrates HAMSTERS and PetShop [9]) are currently under development for supporting the knowledge representation in a similar way.

CONCLUSIONS

This paper has presented how some declarative knowledge and procedural knowledge can be integrated in a notation for modelling tasks. We have grounded this contribution on a detailed study of related work both on cognitive psychology (where how human beings process and represent knowledge has been the focus of attention for many years) and on task modelling. Previous work in this field has mainly targeted at describing procedural knowledge leaving declarative knowledge as a second class citizen with the noticeable exception of [START_REF] Codd | Relational Database: A Practical Foundation for Productivity[END_REF] and but the latter only consider procedural knowledge informally. We have shown on a simple game example that these two representations can be easily integrated providing additional benefits in terms of information flow in the tasks and in terms of information required for performing the tasks and information produced while performing the tasks. This work is currently used in the area of safety critical application (both for satellite ground segments and aircraft cockpits) with the double objective of supporting the early phases of the development process (requirements and design) as well as later ones (evaluation and training).

  DescriptionT1 and T2 are executed at the same time Execution of T2 interrupts the execution of T1 Execution of T2 interrupts the execution of T1, T1 T1 is executed then T2 OR T2 is executed then T1

Figure 4 .

 4 Figure 4. Representation of declarative knowledge a) which can be further refined into strategic b) and situational c)

Figure 6 .

 6 Figure 6. Representation of procedural knowledge refined into strategic a) and situational b)

Figure 4 .

 4 Figure 4. Representation of declarative knowledge a) which r refined into strategic b) and situational c)

Figure 7 .

 7 Figure 7. Example of a user interface for playing Game of 15

Figure 8 .

 8 Figure 8. Concept map of game of 15

Figure 7 .Figure 8 .

 78 Figure 7. Example of a user interface for playing Game of 15The upper part of the window display who is next player, below are displayed the remaining tokens while the last row displays the tokens taken by each player. More detailed information about of this application can be found in[9].presents the concept map associated to the Game of 15. It represents required declarative knowledge necessary to be able to play the game. The concepts which are necessary or useful to perform actions during the game are grouped and linked through a semantic network. It is made up of 4 main nodes Objects,

Figure 9 . 15 Figure 10 .

 91510 Figure 9. HAMSTERS model of game of 15

Table 1 . Comparison of task modelling notations according to suggested requirements

 1

		Requirements /notations [reference]	HTA [9] GOMS [9]	MAD [10]	TKS [9] TAKD [9] GTA [10] UAN [9]	Diane+ [10]	VTML [9] CTT [9] CWA [10]	AMBOSS [9]	DSA [10] HAMSTERS
		REQ_1. Concepts used to represent DK	x	x	x	√	x	X	x	x	x	x	√	x	√	x
		REQ_2. Refinement of concepts into objects and into information	x	x	Partly with objects (explicit)	x Partly with objects	X	x	x	x	Partly with objects	x	x	Partly with objects	Partly with objects
	Declarative Knowledge (DK)	REQ_3. Concepts connections should be made explicit REQ_4. Semantic networks to represent the concepts and their relationships REQ_5. Refinement of DK into strategic knowledge and into situational knowledge	x x x	x x x	x x x	√ x x	x x x	X X X	x x x	x x x	x x x	x x x	√ x x	x x x	√ √ x	x x x
	Procedural Knowledge (PK)	REQ_6 and REQ_7. Routines, subroutines and their interrelationships to represent PK. REQ_8. Hierarchical structures to represent routines, sub-routines and their interrelationships REQ_9. Refinement of PK into strategic knowledge and into situational knowledge	x x	x x	x x	x x	x x	X X	x x	x x	x x	x x	x x x	x √ x	x √ x	√ √ x
	Integration	REQ_10. All the information above should be integrated in a single model	x	Through pre-condition	Through pre-condition	Through pre-condition	Through pre-condition	Through pre-condition	x	Through pre-condition	Through pre-condition	Through pre-condition	Through constraint	Through pre-condition	Through pre-condition	Through pre-condition
	Tool Support		Task-Architect	CAT-HCI and others×	K-MADe ADEPT LUTAKD	GTA and EUTERPE	Quantu m	TAMOT and Isolde	SpecTRM tool Suite	CTTe		AMBOSS SAMANTA HAMSTERS

Some of the most recent contributions are

Caffiau et al. [9]

, Villaren et al.

[START_REF] Codd | Relational Database: A Practical Foundation for Productivity[END_REF]

and

Martinie et al. [9]

demonstrating the still vivid interest from the scientific community for task modelling summarizes for each main type of knowledge (declarative and procedural, column 1), the requirements which have been expressed above (column 2). ("√"=satisfied; "x"= not satisfied)

Table 2 . Temporal ordering operators in HAMSTERS

 2

	Operator type	Symbol	Description
	Enable	T1>>T2 T2 is executed after T1
	Concurrent	T1|||T2 T1 and T2 are executed at the same time
	Choice	T1[]T2 T1 is executed OR T2 is executed
	Disable	T1[>T2 Execution of T2 interrupts the execution of T1
	Suspend-resume	T1|>T2	Execution of T2 interrupts the execution of execution is resumed after T2
	Order Independent	T1|=|T2 T1 is executed then T2 OR T2 is executed then T1

ACKNOWLEDGEMENTS

This work was partly sponsored by CNES (French National Space Studies Center) R&T Tortuga and MARACCASS as well as EUROCONTROL research network HALA! on Higher Automation Levels in Aviation.