Marie-Pierre Comard

Rachel Calvet
email: calvet@enstimac.fr

John A Dodds

Henri Balard

Inverse gas chromatographic study of the surface properties of talc impregnated with different acidic and basic polymers

Keywords: Inverse gas chromatography, Talc, Polymer, Surfactant

A severe limitation of inverse gas chromatography at infinite dilution (IGC-ID) is that it is mainly sensitive to high-energy sites making it difficult to obtain information about the surface of heterogeneous solids, e.g. the talc. In this paper, it is shown that a progressive impregnation of the talc surface with a suitable polymer can saturate the high-energy sites making the probes access less energetic free sites, specifically the basal surfaces of talc. Here, such impregnations are carried out using a surfactant (cetyltrimethylammonium bromide (CTAB)), different molecular weights polymers (polyethylene glycol (PEG) 2000, 20 000, 35 000, and 100 000) and polymers with different acidic or basic character (polyethylvinylether (PEVE), polyvinyl chloride (PVC), polymethylvinylketone (PMVK), and polystyrene (PS)). One of the aims of this study is to estimate the ratio of lateral and basal surfaces.

Introduction

Inverse gas chromatography (IGC) is a useful method for studying the surface properties of finely divided solids [1][2][START_REF] Papirer | Adsorption on New and Modified Inorganic Sorbents[END_REF]. It consists of injecting clearly identified molecules, called probes, into a column packed with the solid to be tested. The injected probes undergo interactions with the solid surface as they are displaced through the column by a carrier gas stream. The complexity of these interactions depends on the amount of probe injected. Two IGC techniques may be distinguished: infinite dilution IGC, or finite solute concentration IGC. The present study has been carried out at infinite dilution (IGC-ID). In this case, very small amounts of probes are injected in such a way that the interactions between probe molecules themselves can be considered to be negligible and only interactions between an isolated probe and the solid surface are important. By IGC-ID, one can obtain the apparent dispersive component of the free surface energy c S d (injection of linear alkanes), the increment of energy linked to specific interactions (injection of polar probes) and the nanomorphological index I m (injection of bulky probes). The latter gives information on the morphology of the solid at the molecular scale. In the work reported here, we use IGC-ID to study the surface properties of talc impregnated with increasing amounts of a surfactant (cetyltrimethylammonium bromide (CTAB)) or of different polymers (polyethylene glycol (PEG) or polyvinyl derivatives).

The structure of talc directs the choice of the molecules used for impregnation. Talc is a lamellar phyllosilicate with the chemical formula (Mg 3 Si 4 O 10 (OH) 2 consisting of octahedral brucite sheets [Mg 3 (OH) 2 ] n 2n À sandwiched between two tetrahedral silica sheets, [Si 2 O 5 ] n 2n , giving superimposed layers which form the structure of talc. The bonds between two sheets are van der Waals interactions. The crystal presents basal nonfunctionalised hydrophobic surfaces with Si -O -Si groups, as well as lateral surfaces with Si-OH groups and residual magnesium cations. The industrial use of talc depends on the surface properties of its lamellar structure and therefore on the ratio of basal to lateral surfaces. Different techniques can be used to measure these parameters: immersion calorimetry [START_REF] Michot | [END_REF], adsorption of various gases (nitrogen, argon, water vapour, etc.) [5,6] and measurement of contact angles [7]. IGC-ID represents valuable addition to these techniques because, instead of giving a global measurement, it examines the solid surface at the scale of the injected probe. The variety of available probes then make this technique highly attractive, since the probe can be adapted to the nature of the solid provided that the probe can be vaporised in the carrier gas stream. Another important aspect of IGC-ID is that it is mainly sensitive to the presence of sites with high interaction energies. In the case of a heterogeneous solid such as talc, and because of the small amounts of probe injected, only the higher energy sites are visited by the probes. In the case of talc, the probes mainly visit the more energetic sites located on the lateral surfaces, i.e. acidic silanols Si -OH and residual magnesium cations. The aim of the experiment protocol of coupling an impregnation of the solid surface by a polymer with IGC-ID is to progressively saturate these high-energy sites making the probes successively access less energetic sites which remain free. In the case of talc, it seemed appropriate to first impregnate the surface of this solid with a basic polymer, polyethylene glycol (PEG), as it is reasonable to suppose that this polymer will saturate the more energetic sites by preferential interactions with the acidic lateral surfaces. After the lateral surfaces have been covered by polymer, the probes will then access the less energetic sites on the basal surfaces. This method should then allow the determination of the respective amounts of lateral and basal surfaces. Conversely, impregnation with an acidic polymer displaying some affinity with the basic siloxane Si-O -Si groups on the basal surfaces of the talc should protect the high-energy sites on the lateral surfaces and lead to the values obtained on talc without any impregnation.

As the aim of this study is to develop a new method for estimating the ratio of lateral to basal surfaces; the evolution of c S d and I m was followed as a function of the surfactant or polymer impregnation ratio. The dispersive component of the surface energy c S d is obtained by analysing the retention times of a series of linear alkanes injected as probes, and gives a measurement of the dispersive London forces. The nanomorphological index I m , obtained from the injection of cyclic or branched alkanes, gives information on the nanomorphology of the solid at the molecular scale. It is a measure of the surface of the solid that can be reached by the bulky probe. I m reflects molecular scale size exclusion effects.

Experimental

Materials

The talc used was supplied by Luzenac Europe and comes from a French quarry. The mean particle size is 10 Am, as measured by Micromeritics 5100 Sedigraph. The argon BET specific surface area is 3 m 2 /g. The different PEGs were obtained from Fluka, the other polymers (polyvinyl chloride (PVC), polystyrene (PS), polymethylvinylketone (PMVK) and polyethylvinylether (PEVE)) from Aldrich and the CTAB from Janssen. The PEGs were purified by solubilisation in chloroform and precipitation in heptane.

Experimental IGC conditions

Two Hewlett Packard 6890 gas chromatographs, each fitted with two flame ionisation detectors (FID), were used with helium carrier gas. The injector and detector are heated to 150 jC. All the columns were conditioned at 120 jC overnight and the analyses made at 100 jC. Net retention times were calculated by subtracting the retention time of a nonretained molecule, methane, from the retention times of the probes.

Preparation of the samples: controlled impregnation of talc

A solution of surfactant or polymer of known concentration is prepared in dichloromethane (for PVC, the solvent was THF). A given volume of this solution is diluted with 5 ml of the same solvent. The mass of talc required to fill the column (around 2 g) is stirred with the polymer solution. The mixture is then placed in a vacuum oven. Using such a procedure, solvent is evaporated and the polymer is deposited on the talc surface. The chromatography column is then filled with the impregnated talc. The percentage of polymer or surfactant per gram of talc can be determined from the volume added, the concentration of the initial solution and the mass of talc used for loading in the column. -For s 1 at 0.02% of PEG 20 M is a first plateau corresponding to a c S d equal to 160 mJ/m 2 . This high value of c S d , also observed without polymer, is typical of lamellar structures [9,10]. It reflects the insertion of linear alkanes into slit-shaped sites located on the lateral surfaces. The ratio s 1 could correspond to the amount of polymer required to fill the lateral surfaces. In fact, the basic character of PEG 20 M should lead to adsorption on the acidic Si -OH groups on the lateral surfaces. The complexation properties towards cations, such as magnesium present on the lateral surfaces, will also first direct the PEG towards lateral surfaces.

Results

Application of the method with PEG 20 000

-For s 2 at 0.084% of PEG 20 M is a first decrease of c S d . This decrease can be attributed to the filling of the wedge-shaped sites between talc crystallites and grain joints. The difference of s 2 À s 1 could correspond to the amount of polymer required to cover the grain joints.

-For s 3 at 0.4% of PEG 20 M is a second plateau corresponding to a c S d equal to 45 mJ/m 2 . This latter value is that obtained by others [11] using Washburn's capillary rise method which, contrary to IGC-ID, is representative of the whole surface and therefore mainly sensitive to the basal surfaces that are predominant in the talc used here. From s 2 to s 3 , only basal surfaces are visible to the linear alkanes. The difference of s 3 À s 2 could then correspond to the amount of polymer required to cover basal surfaces. The basic PEG displays a low affinity for the basal surfaces formed of basic siloxane Si-O -Si bridges.

-After s 3 , a final decrease of c S d down to a value of 28 mJ/ m 2 is observed for 0.8% of PEG 20 M. This value of 28 mJ/m 2 is characteristic of a multi-layer of PEG monomeric units entirely covering the talc surface [12]. This last decrease could correspond to the coverage of the final, less energetic, structural defects. Fig. 2 shows the evolution of the nanomorphological index I m as a function of the PEG 20 M impregnation ratio. I m increases with impregnation ratio of the polymer, thus following an opposite trend to that observed for c S d . For low impregnation ratios, an I m close to 0.1 indicates a high exclusion effect: linear alkanes can be inserted into structural defects from which cyclic alkanes are excluded due to steric constraints. As the polymer is deposited on the solid surface, this phenomenon progressively disappears as structural defects are progressively covered and I m tends towards 1. A value higher than 1 corresponds to the absorption of the probe in a multi-layer polymer.

In the second part of this work, we study the influence of polymer molecular weight on impregnation.

Controlled impregnation with different PEG-influence of polymer molecular weight

The influence of the molecular weight of PEG on both c S d and I m has been studied as a function of impregnation ratio for PEG 2000, 20 000, 35 000, and 100 000. Fig. 3 displays only the variation of the c S d and shows that the polymer molecular weight has no influence on the evolution of the surface properties of talc as a function of impregnation ratio. A translation towards high impregnation ratios is only observed in the case of PEG 2000. This phenomenon is probably due to an overestimation of the polymer mass impregnated due to the evaporation of low molecular weight oligomers during column conditioning. This noninfluence of the molecular weight of the impregnated polymer on solid surface properties has already been observed when grafting of PEG on pyrogenic silicas [13].

Controlled impregnation with polymers or surfactants with different acidic or basic characters

Fig. 4 shows the changes in c S d for talc impregnated with increasing ratios of basic polyethylvinyleather (PEVE), basic polystyrene (PS) and acidic cetyltrimethylammonium bromide (CTAB). The choice of this latter surfactant was guided by a previous study carried out with the same talc [14].

The nature of the impregnation agent completely changes the evolution of c S d as a function of impregnation ratio. Impregnation with PEVE leads to a fast decrease of c S d down to the lowest impregnation ratio. There is no plateau for low impregnation ratios as observed with PEG. At a ratio equal to 0.19%, c S d becomes equal to 45 mJ/m 2 and the decrease then slows down. c S d continues to decrease down to 28 mJ/m 2 after 0.40% of PEVE. Despite the fact that PEVE is a basic polymer like PEG, these two polymers exhibit notable differences in stereochemistry (Fig. 5). PEG has a linear structure whereas PEVE displays some branchings from the linear chain and asymmetric carbons. Because of its structure, PEVE cannot fit against as rough a surface as easily as the linear PEG can.

The curve of c S d as a function of impregnation with PS presents a totally different shape compared with the two other polymers. This polymer is unable to hinder highenergy sites; the value of c S d remains steadily high up to 1-1.5% of PS. The stereochemistry of the macromolecular chain and the low interaction potential with lateral surfaces can explain such behaviour. The polymer probably first covers the basal surfaces which are not visible to IGC-ID, rather than the lateral surfaces which alone contribute to the high value of c S d . After 1 -1.5%, the amount of PS is so high that it covers almost all the surface of the microcrystallites, and c S d falls to reach the energy of a talc surface entirely covered with PS.

The acidic character of CTAB, due to the quaternary ammonium function, should direct its preferential adsorption onto basal surfaces, thus providing a behaviour analogous to that of PS. In fact, the insertion of the alkyl chains into the slit-shaped sites on the lateral surfaces must be responsible for the fast decrease of c S d .

Conclusion

A new method for studying the surface of a heterogeneous solid by IGC-ID has been developed. After having saturated the high-energy sites on the solid under study by impregnation with a suitable polymer, the injected probes can visit the low-energy sites that are not visible in conventional IGC-ID. This method is first used to study the surface heterogeneity of a lamellar solid: talc. In a second stage, we obtain information on the interactions between the solid surface and different polymers or a surfactant. However, the interpretation of the results is not as simple as what might be expected from the study of the evolution of the surface properties of talc impregnated with PEG. Obviously, the nature of the polymer and its stereochemistry, or the amphiphilic character of CTAB, has a large influence on the conformation of the adsorbed macromolecules at the talc surface. Three different types of behaviour are observed.

-With PEG, four different trends were observed that can be attributed respectively to: coverage of lateral surfaces, structural defects, basal surfaces and finally, to the formation of a thick layer of PEG.

-With PEVE and CTAB, only three zones can be observed: first, a rapid decrease, followed by a slower decrease, and then, a plateau corresponding to the formation of a thick layer of macromolecules. -With PS, the apparent surface energy of talc remains steady up to very high impregnation ratios. This behaviour can be attributed to a total lack of affinity for lateral surfaces due to both the stereochemistry and the chemistry of the polymer, in particular, the absence of hetero-elements.

Some complementary studies with different pairs of polymer/solids will be necessary to fully interpret these different behaviours.

Finally, it is important to point out that the technique described requires a polymer/solid pair that is thermally stable at the analysis temperature. This condition was not satisfied with polyvinyl chloride (PVC) and polyvinylmethylketone (PVMK) which gave erratic results that can be interpreted as being caused by the decomposition of the macromolecules catalysed by the talc surface.

Fig. 1

 1 Fig.1shows the changes in the dispersive component of the surface energy c S d as a function of the PEG 20 M impregnation ratio. This curve has four critical ratios[START_REF] Comard | [END_REF]:

Fig. 2 .

 2 Fig. 2. Variation of the nanomorphological index I m as a function of PEG 20 M impregnation ratio (between 0 -3% on the first graph and 0 -0.1% on the second graph).

Fig. 1 .

 1 Fig. 1. Variation of the dispersive component of the surface energy c S d as a function of PEG 20 M impregnation ratio (between 0 -3% on the first graph and 0 -0.1% on the second graph).

Fig. 4 .

 4 Fig. 4. Variation of the dispersive component of the surface energy c S d (between 0 -9% on the first graph and 0 -0.8% on the second graph) of talc impregnated with PEVE (4), PS (x) and CTAB ( Â ) (the dotted curve corresponds to the c S d variation observed with PEG 20 M as the reference).

Fig. 3 .

 3 Fig. 3. Variation of the dispersive component of the surface energy c S d of talc impregnated with PEG 2000, 20 000, 35 000 and 100 000 (between 0 -6% on the first graph and 0 -0.3% on the second graph).


Audrey Fertier

Aurélie Montarnal

Anne-Marie Barthe-Delanoë

Sébastien Truptil

Frédérick Bénaben

Reducing information load to enhance collaborative awareness thanks to a pre-selection of information

Keywords: Collaboration, Crisis management, Common operational picture, Information overload, Model driven engineering

There is a need for collaboration support systems, suited to crisis management, able to sustain collaborations in ever more unstable environments. The organizations involved in a crisis response need support in limiting information overload by accessing information suited to their current needs. The collaboration support system proposed in this paper uses a Common Operational Picture (COP), supported by a Geographical Information System (GIS), that consists of information selected according to (i) the on-going collaboration phase, and (ii) the level of commitment within the collaboration of the current user. Additionally, to validate the proposed classifications, the paper demonstrates how the pre-selection can be applied to support crisis collaborations, operating under high stress and high information load.

Introduction

When gathered inside one room, the partners of a collaboration can directly access large amounts of information [START_REF] Jongsawat | A Study Towards Improving Web-Based Collaboration Through Availability of Group Awareness Information[END_REF], enabling them to enhance their collaborative awareness. They can identify common goals, critical partners, or share accurate information.

Because the collaborations tend to extend their geographical reach, they cannot gather as easily as before. To help them, [START_REF] Schmidt | Coordination mechanisms: Towards a conceptual foundation of CSCW systems design[END_REF] recommends the use of a common artifact to support cooperative activities that can be both individually conducted and interdependent. The main goal of the artifact is to reduce the complexity of collaborations, including the complexity of their information system due to:

-The amount of information in our daily lives is continually increasing and is multiplied by existing information systems [START_REF] Karlsson | Challenges of Designing Augmented Reality for Military use[END_REF], while our brains can only process a limited amount of complex information;

-Each partner must be able to access a part of the collaborative awareness adapted to their business and their level of responsibility; -Information shared within a collaboration comes from heterogeneous sources, and each has an expiration date before which it must be used to not be lost.

These three issues are particularly true during a crisis situation where the collaboration aims to respond to every risk and consequences due to the disaster [START_REF] Shen | Emergency management information systems: Could decision makers be supported in choosing display formats?[END_REF][START_REF] Lee | Group value and intention to use -A study of multi-agency disaster management information systems for public safety[END_REF][START_REF] Klein | Common Ground and Coordination in Joint Activity[END_REF]: the crisis cells have to face high information load and high time pressure, within complex communication channels, while the collaboration can easily breakdown due to heterogeneous experiences, heterogeneous information accesses and heterogeneous comprehensions.

To support the partners in managing the infomration available within the collaboration, we proposed a collaboration support system able to select information according to (i) the on-going collaboration stage and (ii) the level of commitment of the current user, in order to give each user access to a suited Common Operational Picture (COP), supported by a Geographical Information System (GIS).

A COP is, as defined by [START_REF] Luokkala | Developing a concept of a context-aware common operational picture[END_REF], an operational picture shared by several partners during a particular operation. Its goal is to enables a shared Situation Awareness (SA) within the collaboration. In this case, the term SA can be defined as a model of the environment surrounding the collaboration [START_REF] Endsley | Design and evaluation for situation awareness enhancement[END_REF]. This COP can be displayed through the use of a GIS. According to [START_REF] Luokkala | Developing a concept of a context-aware common operational picture[END_REF], such an information system is a powerful tool to support SA, in particularly during crisis situation where almost all relevant information are spatial.

Our goal is to strengthen collaborative awareness in order to enhance the agility of the collaboration (defined in [START_REF] Barthe-Delanoë | Eventdriven agility of interoperability during the Run-time of collaborative processes[END_REF]) in the face of new threats or opportunities. The collaboration support system described in this paper includes:

-A meta-model, as defined by [START_REF] Bézivin | On the unification power of models[END_REF], to enable a unified approach of interoperability, and its models modelling the collaborative situation. -A GIS that takes the shape of a COP to communicate information from the system to the user; -An automatic classification by collaboration stages to filter information according to the current phase of the collaboration; -An automatic classification by partner roles to filter the information according to the place of the user in the collaboration.

Section 1 presents the two classifications introduced in this paper: the collaboration classification and the partners classification that are used for the pre-selection stage, to obtain a COP suited to the current stage of the collaboration and to the current user of the system. Section 2 proposes to validate these two classifications by using them in case of a very specific type of collaboration: a crisis collaboration.

The Figure 1 illustrates how the collaboration support system, proposed in this paper, operates to adapt its COP to is current user and to the current stage of the collaboration. The design of the system involves the definition of a meta-model, several partners' roles and several collaboration's stages, that have to be common to every collaboration types:

-The meta-model (defined in [START_REF] Bézivin | On the unification power of models[END_REF]) is used to homogenize and organize available information in models. Such a meta-model, dedicated to collaborations, is described in [START_REF] Benaben | A conceptual framework and a suite of tools to support crisis management[END_REF]; -The role is used to select information, according to the need of the partner, using the collaboration support system. For example, if a partner manufactures a product for the collaboration benefit, then he needs market forecasts, sub-contractors contact details or warehouses locations. Conversely, he doesn't need customer directories or marketing information. -The stage is used to select information over time. For example, at the beginning of a new collaboration, the partners need to share their expectations. Conversely, when the collaborative process is running, the partners need to access a list of tasks. -The logs are stored to enable future improvements of the collaboration support system.

The partners classification to ensure confidentiality

The work of [START_REF] Touzi | A model-driven approach for collaborative service-oriented architecture design[END_REF], followed by [START_REF] Bénaben | Conception de Système d'Information de Médiation pour la prise en charge de l'Interopérabilité dans les Collaborations d'Organisations[END_REF], enabled us to identify three partner roles, inspired from the maturity levels of collaborations and described in Table 1. By default, the system does not share information of higher responsibility levels, with lower responsibility levels:

A partner P1, with a role R1 can reach information shared by a partner P2, with a role R2

If only R1 ≥ R2 (1)
Furthermore, during an "update()" operation (cf. Figure 1), a partner can set the default responsibility level of information that he adds to the system. A federated partner can, for example, decide to make its newly added information visible to one, several, or all, open partners.

Table 1: The partners classification by partner roles, defined in this paper

Role of a partner

Definition of the paper Communicating partner A partner that exchanges and shares information with the collaboration Open partner

A communicating partner that shares business services and system functionalities with the collaboration

Federated partner

An open partner that takes part in the collaborative process, and shares the collaboration's goals

The collaboration classification to filter the displayed information

Two previous research works [START_REF] Benaben | A conceptual framework and a suite of tools to support crisis management[END_REF][START_REF] Warner | Cognitive Model of Team Collaboration: Macro-Cognitive Focus[END_REF], have enabled us to identify five main collaboration stages, that are described below in Table 2. Each collaboration stage comes with its own information needs. The table 3 shows how the concepts of one collaboration meta-model (from [START_REF] Benaben | A conceptual framework and a suite of tools to support crisis management[END_REF]) can be classified. For example, the partners of a collaboration need to learn about each other at the beginning of the collaboration during the perception stage. Conversely, the goals of the collaboration are set during the convergence stage, when everyone SA is good enough to support this decision.

The classifications are used to identify the "default information" to be first displayed on the COP, for one given user:

If a partner need additional information, the system does not refer to the collaboration classification, but only to the partners classification that manage responsibility levels When each partner gathers information to improve their situation awareness of the collaboration Comprehension (Cp)

When each partner learns how to adapt its information or its outputs to the other partners Understanding (U d)

When new information is inferred from the information shared between several partners Convergence (Cv)

When common goals are identified, solutions are proposed, a solution is chosen and a collaborative process is designed Monitoring (M g)

When the partners adapt their solution, while the collaborative process runs

Characteristic

Feature due to the nature, the composition or the environment of the collaboration that could generate opportunities or threats

• ----

Capacity

One partner's capability, that can be used in the collaborative process The pair <collaboration stage, user role> enables the generation of a view of the model, suited to the current collaborative situation, in order to feed the COP displayed by the GIS. The Figure 2 shows how the information are selected according to the need of the user. This follows the recommendations of Mica Endsley [? ] about goal-directed task analyses. Fig. 2: The process enabling the collaboration support system to generate views suited to both the responsibility level of its user and the current collaboration stage

• • • • • Objective A goal of,

The case of crisis collaboration

In the aftermath of a disaster, a crisis response requires the collaboration of numerous, heterogeneous partners, under high stress and high time pressure [START_REF] Rosenthal | Crises and crisis management: Toward comprehensive government decision making[END_REF]. This paper unfolds the scenario of a 100-year flood provided by the ANR GéNéPi project3 . This project had enabled the interview of many practitioners often involved in crisis collaborations. The results, recorded in specifications [START_REF] Renou | Cahier des charges à l'origine du projet GéNéPi[END_REF], underlines the issues still faced by practitioners during crisis responses:

-Much of the information available is unclear, outdated or unreliable, and only the partners with high expertise can get by; -The diagnosis of the impacted territory and the analysis of the vulnerable assets at stake remains difficult; -Due to the number of partners involved, it's hard to take into account all possibilities of response process, and even harder to find the optimal response process.

The collaboration support system proposed in this paper can support them in dealing with :

-The issues due to the instability of the crisis, thanks to the COP that display the information contains in a model that can be continuously updated, as in [START_REF] Barthe-Delanoë | Event-driven agility of crisis management collaborative processes[END_REF]; -The issues faced during the understanding phase, thanks to the capacity of the COP to enhance collaborative awareness, as underlined by [START_REF] Björkbom | Localization Services for Online Common Operational Picture and Situation Awareness[END_REF][START_REF] Bunker | Repertoires of collaboration for common operating pictures of disasters and extreme events[END_REF]; -The issues due to information overload thanks to the collaboration and partners classifications, as described in Section 1.

In order to enable the collaboration support system to generate views of crisis situations, the classifications of collaboration stages and partners' roles dedicated to crisis collaboration still need to be defined.

The partners classification adapted to crisis collaboration support

In France, in case of a 100-year flood, the organization involves four different responsibility levels [START_REF] Dsc | Organisation de la réponse de sécurité civile[END_REF]:

-Local level; -County level; -Zonal level; -National level.
The hierarchies in place, corresponding to the crisis partners' roles, impose a dedicated information management. For example, a prefect (county level), aiming to communicate to the press, needs to know about the number of people without electricity supply in the county. Conversely, the power supplier (local level), aiming to ensure the continuity of their network, needs to know the exact locations of cut points on their network.

The collaboration classification adapted to crisis collaboration support

Like the collaboration stages proposed in this paper, several crisis collaboration phases has been defined over time. Among the first to distinct four phases were Uriel Rosenthal and Alexender Kouzmin [START_REF] Rosenthal | Crises and crisis management: Toward comprehensive government decision making[END_REF]: "Crises [...] may be considered in terms of circular processes involving mitigation and preparation, response as well as recovery and rehabilitation". Inside the response phase, a french official document [START_REF] Cedre | ORSEC Zonal et départemental, disposition spécifique POLMAR/Terre[END_REF] recognizes five more phases:

-The confirmation of the alert (Ca): "Is there a disaster? What is its scale?"; -The alert (Al): "What are the concerned organisations that will take part in the collaboration?"; -The characterisation of the crisis (Cc): "Where are the assets vulnerable to the consequences of this crisis?"; -The evaluation phase (Ev): "Where are the damaged assets? Where are the threatened assets?"; -The follow-up phase (F l) that consists of "thoughtful actions" to anticipate long-term consequences.

All the concepts from a meta-model dedicated to collaborative crisis management, as the one described in [START_REF] Benaben | A conceptual framework and a suite of tools to support crisis management[END_REF], can be linked to these crisis response phases. The obtained table (an extract is given in Table 4), along with the metamodel from [START_REF] Benaben | A conceptual framework and a suite of tools to support crisis management[END_REF], is used by the collaboration support system instead of the Table 2 suited to all kind of collaboration.

Rationally, the links (•) from the links of Table 3, can easily be applied to the crisis concepts of Table 4 because they all inherit from one concept of Table 3. For example, concerning a new event, useful during the evaluation phase of a crisis, are also useful during the monitoring phase of the collaboration, Table 4: Some crisis concepts from [START_REF] Lauras | Towards a better management of complex emergencies through crisis management meta-modelling[END_REF], labelled with the five crisis response phases from [START_REF] Cedre | ORSEC Zonal et départemental, disposition spécifique POLMAR/Terre[END_REF], according to their usefulness Crisis concepts from [START_REF] Lauras | Towards a better management of complex emergencies through crisis management meta-modelling[END_REF] Parent concept from [START_REF] Benaben | A conceptual framework and a suite of tools to support crisis management[END_REF] Ca

Al Cc Ev F l Danger Characteristic • ---• Actor Partner -• --- Good Environment component --• • - Event Fact ---• - Response Process ----•
because an event is considered as a fact.

Thanks to theses new crisis collaboration response phases, and new crisis partners' roles, a collaborative support system, as the one presented in Figure 2, can select the information to be displayed to its user according to their relevance, and therefore decrease information load of the partners involved in a crisis situation.

Conclusion

This paper offers to use a collaboration support system to display relevant information, via a Common Operational Picture (COP) based on a Geographical Information System (GIS) and describing the collaborative situation.

To further limit information overload and to take into account the different responsibility levels involved, the paper proposes two classifications, dedicated to collaborations:

-The collaboration classification to adapt the COP to the current collaboration stage: either the perception, the comprehension, the understanding, the convergence or the monitoring stage. -The partners classification to adapt the view to the goal of the current user.

It consists of three categories: communicating, open or federated partners.

To extend the proposed classifications, we have checked that these solutions, dedicated to collaborations, apply to crisis collaborations: collaboration in highly unstable environment, under high-stress and time pressure.

Fig. 1 :

 1 Fig. 1: A communication diagram, in Unified Modelling Language (UML) that illustrates how each partner of a collaboration can use the COP interface of the collaboration support system to enhance their collaborative awareness

  at least, one of the partner --• • • Performance indicator An indicator that measures the performance of one capability, given a goal ---• • Process A process that invokes the capabilities of some partner and order them according to flows of information, conditions and events ----• Fact An event witnessed by, at least, one of the partner ----•

  COP consisting of a view of the situation model, suited to the user's role and the current stage of

Table 2 :

 2 The collaboration classification by stages, defined in this paper

	Stage of a collaboration Definition of the paper
	Perception (P r)

Table 3 :

 3 The concepts of the meta-model[START_REF] Benaben | A conceptual framework and a suite of tools to support crisis management[END_REF], labelled with the five collaboration stages, according to their level of usefulness

	Concepts from [3]	Definition of this paper	P r Cp U d Cv M g
	Partner	A partner of the collaboration	• ----
	Environment Component Anything composing the environ-	• ----
		ment of the collaboration, that can
		be mapped in the COP

Project ANR-14-CE28-0029, web site

Acknowledgements & Funding sources

This work would have not been possible without the GéNéPi project research team, or the computer engineers team from the industrial engineering center of IMT Mines Albi. This work was supported by the french National Research Agency, through the GéNéPi project funding (program: Resilience and crisis management [DS0903] 2014; project ID: ANR-14-CE28-0029)


Jiating Luo

Van-Dung Pham

Cédric Killian

Daniel Chillet

Ian O'connor

Olivier Sentieys

Sébastien Le Beux

Run-Time management of energy-performance trade-off in Optical Network-on-Chip

Optical Network-on-Chip is a promising technology to leverage the interconnection bottleneck in manycore architectures. Indeed, this technology offers high bandwidth and low latency for communications between cores or clusters of cores. However, the high static power consumption of Optical NoCs (ONoCs) calls for reconfiguration of the interconnect in order to meet the performance requirements while minimizing the required energy. This paper addresses this challenge and proposes a method allowing to define, at design time, a set of ONoC execution modes to be loaded, at run-time, according to the applications' performance and energy requirements. The proposed methodology relies on a sequencer allowing to configure Optical Network Interfaces (ONI).

I. INTRODUCTION

Multiprocessor System-on-Chip (MPSoC) are evolving towards the integration of hundreds of cores on a single chip. Designing an efficient interconnect for such complex architectures is challenging due to the ever growing data exchange between processors. Networks on Chip (NoCs) have been proposed to overcome these issues but they are now reaching their performance limits. Indeed, packetization and depacketization of data drastically impact NoC latency [START_REF] Owens | Research challenges for on-chip interconnection networks[END_REF] and the same bandwidth for all router ports is usually assumed while processors bandwidth requirements vary significantly [START_REF] Cilardo | Design automation for application-specific on-chip interconnects: A survey[END_REF]. NoC efficiency thus needs to be further improved in order to efficiently interconnect systems with still increasing number of cores and required bandwidth density [START_REF] Batten | Designing chip-level nanophotonic interconnection networks[END_REF].

To overcome the limitations of classical NoCs, nanophotonic interconnects seem a promising technology to replace support communications in MPSoCs. Indeed, Wavelength Division Multiplexing (WDM) allows the propagation of multiple signals simultaneously on a same waveguide [START_REF] Dutta | WDM technologies: optical networks[END_REF], thus leading to high aggregated bandwidth. However, WDM also leads to interchannel crosstalk noise [START_REF] Duong | Coherent and incoherent crosstalk noise analyses in interchip/intrachip optical interconnection networks[END_REF], which negatively impacts the Signal to Noise Ratio (SNR) and, therefore, the Bit Error Rate (BER). The BER to be reached depends on the application requirements and is closely related to the optical devices characteristics: it depends on the photodetector sensitivity [START_REF] Fusella | Lighting up on-chip communications with photonics: Design tradeoffs for optical noc architectures[END_REF], on the Microring Resonator transmission spectrum, and on the optical signal power emitted by the laser sources. The higher the number of signals propagating simultaneously, the higher the crosstalk and the higher the laser output power needed. This thus leads to the following conflicting objectives: high performance communications tend to rely on an exhaustive use of the available wavelengths while energy efficient communications involve a parsimonious use of signals which occupy distinct and separate wavelengths. Using an Optical NoC (ONoC) to support the communications of a given application is thus a tedious task, especially if performance, power and BER objectives are likely to evolve with the execution context.

Hence, for an application executed on a given architecture, several solutions allow implementing the communications, each one providing an optimal solution but with a unique trade-off between energy and performance. In the state of the art, ONoCs can be classified into two families regarding the communication allocation strategy: online or offline. However, to the best of our knowledge, there is no architecture allowing run-time adaptation of the ONoC to meet energy and performance application requirements for both types of allocations. In this paper, we propose a method allowing run-time adaptation of the ONoC according to energy and performance requirements of the executed applications. The method relies on an offline framework to generate a set of ONoC configurations to be stored in ONIs. An ONoC configuration sequencer, located in the center of the optical layer (Figure 1), then selects at runtime the configuration to be loaded according to the application needs.

Computing layer

IP core

Communication layer

The paper is organized as follow: Section II presents the proposed hardware manager allowing run-time adaptation of the ONoC. Section III describes the management of the configurations of the ONIs. Finally we conclude this paper in Section IV.

(c) (a) c 0à2 c 1à2 p 0 p 1 p 2 t 0 t 2 t 1 c 0à1 Q init Q A Q B Q C Q D Q init c 0à2 c 1à2 p 0 p 1 p 2 c 0à1 Q init Q A Q B Q C Q init t 2 t 1 t 0 (f) Energy Execution time … … HP: High Performance LP: Low Power Start C 0->1 & C 0->2 End C 0->1 Start C 1->2 End C 1->2 End C 0->2 Q init Q A Q B Q C Start C 1->2 Start C 0->1 & C 0->2 End C 0->1 End C 0->2 End C 1->2 Q A Q B Q D Q C Q init t 0 t 1 t 2 c 0à1 c 1à2 c 0à2 (b) (d) (e)

II. RUN-TIME ADAPTATION OF ONOC ENERGY AND PERFORMANCE

A. Considered architecture and applications

We consider a 3D ONoC, i.e. a 3D integrated circuit supporting mixed-technology with electronic-controlled Optical NoC. It combines the benefits of silicon photonics and 3D stacking to overcome traditional electrical network limitations. Photonic interconnect delivers high on-chip bandwidth and low latency while 3D stacking can reduce the interconnect distance. Figure 1 shows a ring-based 3D ONoC architecture composed of two layers interconnected by TSV: i) on the bottom, an electrical layer implementing processing cores and ii) on the top, an optical layer with optical routers and waveguides.

Following the layout defined in [START_REF] Wu | Suor: Sectioned undirectional optical ring for chip multiprocessor[END_REF], a centralized ONoC configuration sequencer driven by an Operating System allows run-time adaptation of the ONIs, as illustrated in red in Figure 1. The OS is responsible for deploying tasks among the cores and defining the strategy to adapt the ONoC to meet application requirements, such as energy consumption or execution time (called performance within this paper). The sequencer is responsible for reconfiguring the ONoC according to the mode selected by the OS.

Figure 2 illustrates the dependencies between the application tasks, the allocated bandwidth and the performance. Applications are represented as a Directed Acyclic Graph (DAG) as depicted in Figure 2-a, where each vertex t i represents a task and each directed edge (c i→ j ) represents a communication from t i to t j . In this example, we assume that tasks t 0 , t 1 , and t 2 are mapped onto different processors. Communications c 0→1 , c 0→2 , and c 1→2 are implemented using an ONoC which is configured according to execution performance and energy requirements.

As we consider WDM communication and laser power tuning, the energy consumption and the execution time of the application are linked to the resources allocated for each communication of the DAG. Indeed, it is possible to adapt the bandwidth for each communication, by varying the number of wavelengths, and to adapt the power laser to meet a targeted communication quality through the BER (not illustrated in this figure). Figures 2-c and 2-d are the communication graphs obtained considering i) the mapping on the processors and ii) the bandwidth allocated to each communication. While a complete communication task would consider all possible communication end-time sequence scenarios (for instance, c 1→2 can terminate before or after c 1→3 ), we consider a reduced graph for which communication times are estimated. It is worth mentioning that a reduced graph improves the scalability of our approach by simplifying the design of the controller and by reducing the memory footprint. Following the method detailed in Section III-A, each state of the communication graph is associated to a configuration Q.

Figures 2-e and2-f show the execution traces of the HP and LP solutions. We can clearly see that the HP solution reduces the execution time of the DAG by using more bandwidth for each communication compared to the LP solution.

B. Architecture of the sequencer

The proposed approach requires two hardware blocks, as depicted in Figure 3: an ONI manager, located in each ONI, in charge of activating the Lasers and MR, and an ONoC configuration sequencer, that is shared among all the ONIs, in charge of sequencing the configurations of the ONIs.

An Optical Network Interface, ONI (illustrated on the right part of Figure 3), integrates a receiver Rx and a transmitter Tx. The transmitter is composed of on-chip laser sources that can emit and inject optical signal at a specific wavelength into the waveguide. The power generated by the Lasers can be configured. In this example the Lasers can be configured independently among 4 levels of power. The data are directly transmitted from these lasers through current modulation (OOK) and each laser source can also be turned OFF for energy saving. As the lasers are sending data in serial and as a core sends data in parallel on N bits , a WDM stream serialization is required in order to perform the serialization and allocation of the serial data to the selected wavelengths (i.e. lasers). The receiver part includes wavelength-specific MR that can be turned ON or OFF to configure respectively drop or pass-through operations of the signals at a given wavelength. Signal dropped from a waveguide reaches a photodetector, where opto-electric data conversion generates an electrical signal suitable for the electronics part of the receiver. The considered architecture allows the reuse of wavelengths to realize multiple independent communications in a single waveguide. The receiver includes a WDM stream deserialization to perform serial to parallel conversion on the received data. Each ONI includes an ONI Manager in charge of configuring the receiver and the transmitter. It includes an ONI configurations memory which stores the configurations to apply on the MR and Lasers for different operating modes and at different steps composing a sequence of configurations.

Qinit

- - - - - - - - - - - - QA 1 1 - - 4 1 4 1 4 1 4 1 QB - - - - - - - - 4 1 4 1 QC - - - - - - - - 4 1 4 1 QD - - - - - - - - - - - - Balance Qinit - - - - - - - - - - - - QA 1 1 - - 2 1 2 1 4 1 -- QB - - - - - - - - 4 1 - - QC - - - - - - - - 4 1 - - QD - - - - - - - - - - - - Low Power Qinit - - - - - - - - - - - - QA 1 - - - 2 1 - - 4 1 - - QB - - - - - - - - 4 1 - - QC - - - - - - - - 4 1 - - QD - - - - - - - - - - - -
As mentioned above, the second hardware block required is the ONoC configuration sequencer in charge of synchronizing all the ONI Managers of the ONoC. It includes a state machine explained in Section III-B as a sequence diagram. This block takes as input decision from the Operating System in order to select the configuration mode satisfying application requirements. Moreover, it is connected to the ONIs through three signals. The first is a one-bit signal indicating the ONI to start exploiting the configuration associated to the current Step and Mode selected in the ONI configuration memory. The second is also a one-bit signal generated by the ONIs indicating that a step is over. The last signal bit-width depends on the number of Modes embedded in the ONI configuration memory and allows to select the mode to apply.

C. Run-Time management of energy-performance trade-off

Figure 4 illustrates the configuration of the ONIs for each step of the configuration sequence in the Low Power mode. In this mode, p 0 sends a control signal to the configuration manager when t 0 execution ends and when data involved in c 0→1 and c 0→2 are ready for transfer. The manager switches from configuration Q init to Q A : Tx 0 and Rx 1 are reconfigured with the values stored in the configuration memory. Two on-chip lasers are powered on to emit optical signals for purpose of reducing energy consumption (the laser output powers are not illustrated for the sake of clarity) as shown in Figure 4-a. For lowlatency purpose, the ONoC is reconfigured in parallel, which is implemented using dedicated electrical wires connecting the controller to the optical devices. Once the new configuration is loaded, the data transfer starts and, once it ends, another control signal is sent to the configuration manager to disable the communication channels. For this purpose, a new configuration (Q B as shown in Figure 4-b) is loaded i.e. the lasers and the MR involved (blue one) in the communication are turned OFF. Figure 4-c) illustrates communication c1 → 2 carried out by the controller. The reconfiguration of the interfaces to implement a new communication has no impact on the ongoing communications: in this example, green signal on p 0 to p 2 waveguide segment is used for c 0→2 while c 1→2 is carried out using blue wavelength (reconfiguration Q B → Q C ). Then c 0→2 ends and the green laser and MR are turned off as illustrated in Figure 4-d (Q D ). Finally, the resources are released and become available for the next iteration.

III. ONOC CONFIGURATION SEQUENCER

A. Generation of configuration

We consider the use of the framework presented in [START_REF] Luo | Offline optimization of wavelength allocation and laser power in nanophotonic interconnects[END_REF] to generate the ONoC configuration sequences. This flow takes as inputs an application mapped onto the 3D architecture illustrated in Figure 1. The application is modeled as a task graph characterized by task execution times, amount of data transmitted between tasks, and minimum BER to be reached. The design flow relies on device parameters since they impact the performance of optical communications. Instance of parameters are photodetectors sensitivity, waveguide losses and MR model. Regarding the laser, we take the data-rate, the efficiency, the maximum output power, and the number of power levels available into account. The aim of the flow is to optimize both power consumption and application execution time. For this purpose, our framework explores both device-level and systemlevel parameters. Based on a set of device and system input parameters, a multi-objective optimization is carried out using a genetic algorithm due to the two contradictory objectives. In the genetic algorithm, the ONoC configuration modes are represented by chromosomes and the genes encode both wavelength allocations and laser power levels. Finally the resulting ONoC configuration modes on a Pareto front from the multi-objectives optimization are thus reported, including low-power solutions, which tends to minimize the number of used wavelengths, and high-performance solutions, for which multiple wavelengths are allocated to shorten the communication time. It has to be noticed that the framework also optimized the selection of the wavelengths in order to reduce the crosstalk. Then the obtained trade-off configuration modes could be embedded in the system and be loaded on run-time according to the execution context (e.g. high performance and low power) which is out of the scope of this paper. Refer to [START_REF] Luo | Offline optimization of wavelength allocation and laser power in nanophotonic interconnects[END_REF] for more details on the generation of solutions.

Figure 5 illustrates individual coding and the corresponding ONoC configuration and how we generate the communication configuration that are embedded in each ONI.

As depicted in Figure 5-a, we assume three interfaces, four wavelengths (i.e. four lasers per interface) and four laser output power levels. Tasks t 0 , t 1 and t 2 are mapped on processor p 0 , p 1 , and p 2 , respectively, which leads to optical communications between p 0 and p 1 (c 0→1 ), p 0 and p 2 (c 0→2 ), p 1 and p 2 (c 1→2 ). The chromosome is divided into as many parts as there are communications (three in the example) [START_REF] Luo | Offline optimization of wavelength allocation and laser power in nanophotonic interconnects[END_REF]. Figure 5a shows one of the solution given by the framework. It is represented as a chromosome given the bandwidth and power allocation for each communication. The first gene of each chromosome part gives the selected laser output power level; it is an integer value corresponding to the laser configuration. The following genes correspond to the wavelengths utilization: value 0 or 1 indicates that a wavelength is used or unused, respectively. In the example, both c 0→1 and c 1→2 use λ 0 and λ 1 while c 0→2 is implemented using only λ 2 . Obviously, the more wavelengths are allocated for a given communication, the higher the bandwidth.

The configuration of each interface is obtained as follows, and is illustrated in Figure 5-b.

MR state and Laser power levels are extracted from the chromosome for each Tx and Rx of the ONIs. As we can see in this example, the Tx of p 0 will handle two communications that use different wavelengths and power levels.

Then, the configurations are merged and the results configuration of each Tx and Rx are obtained as shown in 5-c.

Finally the configuration can be applied to the ONIs as illustrated in 5-d. First, optical channels are open by switching ON the MRs involved in communications (i.e. MRs localized in the transmitter Tx of the source processor and the receiver Rx of the destination). Then, the power of the optical signals propagating through the channels is defined according to the selected lasers power level. In the example, at the transmitter of p 0 interface, three MRs are turned ON to implement communications c 0→1 (λ 0 and λ 1 ) and c 0→2 (λ 2 ). On the receiver side, the MRs corresponding to λ 0 and λ 1 are set to the ON state in p 1 interface, while the MR corresponding to λ 1 remains OFF to let the signal reaching p 2 interface were it will be dropped. In the chromosome part dedicated to c 0→1 , the laser output power level is set to 2: in p 0 interface, the lasers emitting at wavelength λ 0 and λ 1 are set to 50% of the maximum power. In the same interface, laser at λ 2 is set to 100% to match value 4 in corresponding gene for c 0→2 . Regarding the communication through the ONoC, the OS is only in charge of defining the mode (energy-performance trade-off) of the communication. It is possible to change the mode at the beginning of a new sequence. Regarding the sequencing of steps within a sequence, the ONoC configuration sequencer sends the order to apply the ONI configuration for Tx Rx by using a one-bit signal to the ONI manager. The ONI Manager reads the associated memory containing the Tx and Rx configurations, hence the communication can start.

B. ONoC sequencing illustration

When the communications are over for a processor, or when no communication are necessary in a step, the ONI manager associated to a core sends back a one-bit signal to the ONoC configuration sequencer. When all the communications are completed, the ONoC configuration sequencer indicates to all the ONI to go through the next step. Hence, the aforementioned sequencing is repeated.

IV. RESULT

In this section we show the energy versus performance trade-offs available in an ONoC. We consider the architecture in Fig. 1 in which each ONI is connected to a cluster of four electrically connected cores. We assume two waveguides, eight wavelengths per waveguide, five configurable levels of electrical power for the Lasers: [START_REF] Cilardo | Design automation for application-specific on-chip interconnects: A survey[END_REF][START_REF] Dutta | WDM technologies: optical networks[END_REF][START_REF] Fusella | Lighting up on-chip communications with photonics: Design tradeoffs for optical noc architectures[END_REF][START_REF] Luo | Offline optimization of wavelength allocation and laser power in nanophotonic interconnects[END_REF][START_REF] Kennedy | Laser pooling: Static and dynamic laser power allocation for on-chip optical interconnects[END_REF] mW. The Table I summarizes the technological parameters used to generate the ONoC configurations with the framework proposed in [START_REF] Luo | Offline optimization of wavelength allocation and laser power in nanophotonic interconnects[END_REF]. Regarding the applications, we use a random task graph generator that provides applications including from 52 to 107 tasks and from 80 to 158 communications. The task execution time values are randomly selected between [100, 1000] cc and the communication volumes are randomly selected within [100, 1000] bytes range. The targeted BER is 10 -9 and each task is randomly mapped on a dedicated core. As we assume shared memory within a same cluster, no latency is assumed for intracluster communications.

Table II summarize the characteristics of the task graphs and the total energy and execution time in clock cycles (cc) required. Since the generation of the ONoC configuration leads to a Pareto front with several solutions, we only show the solutions with i) the lowest energy consumption (denoted Low Power in the table) and i) the lowest execution time (High Perf.). As shown in Table II, the solutions offer, on average, 44% energy variation and 71% execution time variation tradeoffs. From these solutions, the designers can implement in the configuration memory of the ONI manager several solutions that provide intermediate trade-offs. These results demonstrate the efficiency of flexible wavelengths allocation and laser output power tuning to adapt the nanophotonic interconnects according to application requirements. V. CONCLUSION In this paper we presented a method to offer dynamicity to ONoC in terms of energy versus performance trade-offs.

Fig. 1 :

 1 Fig. 1: Considered 3D ONoC architecture with the configuration sequencer located in the center of optical layer.

Fig. 2 :

 2 Fig. 2: Energy-performance trade-off: a) Application represented as a DAG, b) Energy versus Performance plot highlighting High Performance (HP) and Low Power (LP) modes, c-d) ONoC configuration sequence for HP mode and the resulting execution traces, e-f) ONoC configuration sequence of LP mode and the resulting traces.

Figure 2 -

 2 b illustrates energy versus execution time plot and shows a Pareto front of communication solutions for the architecture. This figure also highlights HP (High Performance) and LP (Low Power) modes on the Pareto front, which allows to maximize the performance and energy efficiency respectively.

  MR Plvl MR Plvl MR Plvl MR Plvl MR High Performance

Fig. 4 :

 4 Figure 4 illustrates the configuration of the ONIs for each step of the configuration sequence in the Low Power mode. In this mode, p 0 sends a control signal to the configuration manager when t 0 execution ends and when data involved in c 0→1 and c 0→2 are ready for transfer. The manager switches from configuration Q init to Q A : Tx 0 and Rx 1 are reconfigured with the values stored in the configuration memory. Two on-chip lasers are powered on to emit optical signals for purpose of reducing energy consumption (the laser output powers are not illustrated for the sake of clarity) as shown in Figure4-a. For lowlatency purpose, the ONoC is reconfigured in parallel, which is implemented using dedicated electrical wires connecting the controller to the optical devices. Once the new configuration is loaded, the data transfer starts and, once it ends, another control signal is sent to the configuration manager to disable the communication channels. For this purpose, a new configuration (Q B as shown in Figure4-b) is loaded i.e. the lasers and the MR involved (blue one) in the communication are turned OFF. Figure4-c) illustrates communication c1 → 2 carried out by the controller. The reconfiguration of the interfaces to implement a new communication has no impact on the ongoing communications: in this example, green signal on p 0 to p 2 waveguide segment is used for c 0→2 while c 1→2 is carried out using blue wavelength (reconfiguration Q B → Q C ). Then c 0→2 ends and the green laser and MR are turned off as illustrated in Figure4-d)

4 •Fig. 5 :

 45 Fig. 5: Generation of the configurations: a) DAG description, architectural assumptions and the results of the offline optimization framework, b) Extraction of each ONI configuration, c) Result of the ONI configurations, d) Configuration of each ONI with respect to the extraction configurations.

Figure 6

 6 Figure6illustrates the relation between the Operating System (OS), the ONoC configuration Sequence, and the ONIs.Regarding the communication through the ONoC, the OS is only in charge of defining the mode (energy-performance trade-off) of the communication. It is possible to change the mode at the beginning of a new sequence. Regarding the sequencing of steps within a sequence, the ONoC configuration sequencer sends the order to apply the ONI configuration for Tx Rx by using a one-bit signal to the ONI manager. The ONI Manager reads the associated memory containing the Tx and Rx configurations, hence the communication can start.When the communications are over for a processor, or when no communication are necessary in a step, the ONI manager associated to a core sends back a one-bit signal to the ONoC configuration sequencer. When all the communications are completed, the ONoC configuration sequencer indicates to all the ONI to go through the next step. Hence, the aforementioned sequencing is repeated.

Fig. 6 :

 6 Fig. 6: Sequence diagram detailing the interactions between the Operating Software, the ONoC configuration Sequencer, and ONIs.

TABLE I :

 I Technological parameters.

ACKNOWLEDGEMENT

This work has received a French Government support granted to the COMIN Labs excellence laboratory and managed by the National Research Agency in the "Investing for the Future" program under reference ANR-10-LABX-07-01. J. Luo is supported by China Scholarship Council (CSC).

The hardware blocks in charge of the run-time management of the ONoC have been presented and the method generates the allocation solution through an offline optimizing framework. The ONoC dynamicity is exploited by the Operating System regarding the application requirements. An off line analyze of the application mapped on the architecture allows to extract all the possible configurations to support the communications Based on this analyze, a sequence of communication configurations is then controlled step by step by the Operating System to ensure the current requirements.


Da-Jung Cho
email: dajungcho@yonsei.ac.kr

Yo-Sub Han

Kai Salomaa
email: ksalomaa@cs.queensu.ca

Taylor J Smith
email: tsmith@cs.queensu.ca

Site-Directed Insertion: Decision Problems, Maximality and Minimality

Site-directed insertion is an overlapping insertion operation that can be viewed as analogous to the overlap assembly or chop operations that concatenate strings by overlapping a suffix and a prefix of the argument strings. We consider decision problems and language equations involving site-directed insertion. By relying on the tools provided by semantic shuffle on trajectories we show that one variable equations involving site-directed insertion and regular constants can be solved. We consider also maximal and minimal variants of the site-directed insertion operation.

Introduction

Site-directed mutagenesis is one of the most important techniques for generating mutations on specific sites of DNA using polymerase chain reaction (PCR) based methods [START_REF] Reikofski | Polymerase chain reaction (PCR) techniques for sitedirected mutagenesis[END_REF]. The algorithmic applications of mutagenesis have been considered e.g. by Franco and Manca [START_REF] Franco | Algorithmic applications of XPCR[END_REF]. Contextual insertion/deletion systems in the study of molecular computing have been used, e.g. by Kari and Thierrin [START_REF] Kari | Contextual insertions/deletions and computability[END_REF], Daley et al. [START_REF] Daley | Circular contextual insertions/deletions with applications to biomolecular computation[END_REF] and Enaganti et al. [START_REF] Enaganti | A formal language model of DNA polymerase enzymatic activity[END_REF].

Site-directed insertion (SDI) of a string y into a string x involves matching an outfix of y with a substring of x and inserting the "middle part" of y not belonging to the outfix into x. Site-directed insertion has earlier been considered under the name outfix-guided insertion [START_REF] Cho | Outfix-guided insertion[END_REF]. The operation is an overlapping variant of the insertion operation in the same sense as the overlap assembly, a.k.a. chop operation, is a variant of string concatenation [START_REF] Csuhaj-Varju | Self-assembly of string and languages. Theoret[END_REF][START_REF] Enaganti | On the overlap assembly of strings and languages[END_REF][START_REF] Holzer | Descriptional complexity of chop operations on unary and finite languages[END_REF][START_REF] Holzer | The chop of languages[END_REF].

The maximal (respectively, minimal) SDI of a string y into a string x requires that, at the chosen location of x, the operation matches a maximal (respectively, minimal) outfix of y with a substring of x. This is analogous to the maximal and minimal chop operations studied by Holzer et al. [START_REF] Holzer | The chop of languages[END_REF].

Site-directed insertion can be represented as a semantic shuffle on trajectories (SST). Shuffle on trajectories was introduced by Mateescu et al. [START_REF] Mateescu | Shuffle on trajectories: Syntactic constraints[END_REF] and the extension to SST is due to Domaratzki [START_REF] Domaratzki | Semantic shuffle on and deletion along trajectories[END_REF]. Further extensions of the shuffle-ontrajectories operation have been studied by Domaratzki et al. [START_REF] Domaratzki | Interpreted trajectories[END_REF].

Here we study decision problems and language equations involving sitedirected insertion and its maximal and minimal variants. The representation of SDI as a semantic shuffle on a regular set of trajectories gives regularity preserving left-and right-inverses of the operation. By the general results of Kari [START_REF] Kari | On language equations with invertible operations[END_REF] on the decidability of equations, translated for SST by Domaratzki [START_REF] Domaratzki | Semantic shuffle on and deletion along trajectories[END_REF], this makes it possible to decide linear equations involving SDI where the constants are regular languages.

The maximal and minimal SDI operations do not, in general, preserve regularity. This means that the operations cannot be represented by SST [START_REF] Domaratzki | Semantic shuffle on and deletion along trajectories[END_REF] (on a regular set of trajectories) and the above tools are not available to deal with language equations. We show that for maximal and minimal SDI certain independence properties related to coding applications [START_REF] Jürgensen | Codes. In: Handbook of Formal Languages[END_REF] can be decided in a polynomial time. The decidability of whether a regular language is closed under max/min SDI remains open.

In the last section we give a tight bound for the nondeterministic state complexity of alphabetic SDI, where the matching outfix must consist of a prefix and suffix of length exactly one. An upper bound for the state complexity of the general site-directed insertion is known but it remains open whether the bound is optimal.

Preliminaries

We assume the reader to be familiar with the basics of finite automata, regular languages and context-free languages [START_REF] Shallit | A Second Course in Formal Languages and Automata Theory[END_REF]. Here we briefly recall some notation.

Let Σ be an alphabet and w ∈ Σ * . If we can write w = xyz we say that the pair (x, z) is an outfix of w. The outfix (x, z) is a nontrivial outfix of w if x = ε and z = ε. For L ⊆ Σ * , L = Σ * -L is the complement of L.

A nondeterministic finite automaton (NFA) is a tuple A = (Σ, Q, δ, q 0 , F ) where Σ is the input alphabet, Q is the finite set of states, δ : Q × Σ → 2 Q is the transition function, q 0 ∈ Q is the initial state and F ⊆ Q is the set of final states. In the usual way δ is extended as a function Q × Σ * → 2 Q and the language accepted by A is L(A) = {w ∈ Σ * | δ(q 0 , w) ∩ F = ∅}. The automaton A is a deterministic finite automaton (DFA) if |δ(q, a)| ≤ 1 for all q ∈ Q and a ∈ Σ. It is well known that the deterministic and nondeterministic finite automata recognize the class of regular languages.

The so called fooling set lemma gives a technique for establishing lower bounds for the size of NFAs: Lemma 1 (Birget 1992 [1]). Let L ⊆ Σ * be a regular language. Suppose that there exists a set P = {(x i , w i ) | 1 ≤ i ≤ n} of pairs of strings such that: (i) x i w i ∈ L for 1 ≤ i ≤ n, and, (ii) if i = j, then x i w j ∈ L or x j w i ∈ L for 1 ≤ i, j ≤ n. Then, any minimal NFA for L has at least n states.

Finally we recall some notions concerning operations on languages and language equations. Let be a binary operation on languages, and L, R are languages over an alphabet Σ.

(i) The language L is closed under if L L ⊆ L. (ii) The language L is -free with respect to R if L R = ∅. (iii) The language L is -independent with respect to R if (L Σ + ) ∩ R = ∅. (iv) A solution for an equation X L = R (respectively, L X = R) is a language S ⊆ Σ * such that S L = R (respectively, L S = R).
The -freeness and independence properties can be related to coding applications, where it might be desirable that we cannot produce new strings by applying an operation, such as site-directed insertion, to strings of the language. Domaratzki [START_REF] Domaratzki | Trajectory-based codes[END_REF] defines trajectory-based codes analogously with (iii). As we will see, languages that are site-directed insertion independent with respect to themselves have a definition closely resembling outfix-codes of index one [START_REF] Jürgensen | Codes. In: Handbook of Formal Languages[END_REF].

3 Site-Directed Insertion

The site-directed insertion is a partially overlapping insertion operation analogously as the overlap-assembly (or self-assembly) [START_REF] Csuhaj-Varju | Self-assembly of string and languages. Theoret[END_REF][START_REF] Enaganti | On the overlap assembly of strings and languages[END_REF] models an overlapping concatenation of strings. The overlapping concatenation operation is also called the chop operation [START_REF] Holzer | The chop of languages[END_REF].

The site-directed insertion (SDI) of a string y into a string x is defined as

x sdi ← y = {x 1 uzvx 2 | x = x 1 uvx 2 , y = uzv, u = , v = }.
The above definition requires that the pair (u, v) is a nontrivial outfix of the string y and uv is a substring of x. If y = uzv is inserted into x by matching the outfix with a substring uv of x, we say that (u, v) is an insertion guide for the operation. Note that a previous paper [START_REF] Cho | Outfix-guided insertion[END_REF] uses the name "outfix-guided insertion" for the same operation. The site-directed insertion operation is extended in the usual way for languages by setting

L 1 sdi ← L 2 = wi∈Li,i=1,2 w 1 sdi ← w 2 .
We recall that regular languages are closed under site-directed insertion.

Proposition 1 ([2]

). If A and B are NFAs with m and n states, respectively, the language L(A) sdi ← L(B) has an NFA with 3mn + 2m states.

A simpler form of the overlap-assembly operation requires the overlapping part of the strings to consist of a single letter. This operation is called "chop" by Holzer and Jacobi [START_REF] Holzer | Descriptional complexity of chop operations on unary and finite languages[END_REF] but the later definition of the chop-operation [START_REF] Holzer | The chop of languages[END_REF] coincides with general overlap-assembly [START_REF] Enaganti | On the overlap assembly of strings and languages[END_REF]. Analogously we define alphabetic site-directed insertion by requiring that the overlapping prefix and suffix of the inserted string each consist of a single letter.

The alphabetic site-directed insertion of a string y into a string x is

x a-sdi ← y = {x 1 azbx 2 | x = x 1 abx 2 , y = azb, a, b ∈ Σ, x 1 , x 2 , z ∈ Σ * }.
Note that the alphabetic site-directed insertion will have different closure properties than the standard site-directed insertion. For example, it is not difficult to see that the context-free languages are closed under alphabetic sitedirected insertion, while the context-free languages are not closed under general site-directed insertion [START_REF] Cho | Outfix-guided insertion[END_REF].

Decision problems

For a regular language L, it is decidable whether L is closed under site-directed insertion. The algorithm relies on the construction of Proposition 1 and operates in polynomial time when L is specified by a DFA [START_REF] Cho | Outfix-guided insertion[END_REF]. Deciding whether a contextfree language is closed under site-directed insertion is undecidable [START_REF] Cho | Outfix-guided insertion[END_REF].

A language L is sdi ←-free, or SDI-free, with respect to R if no string of R can be site-directed inserted into a string of L, that is, if L sdi ← R = ∅.
The language L is SDI-independent with respect to R if site-directed inserting a non-empty string into L cannot produce a string of R. Note that L being SDI-independent with respect to itself resembles the notion of L being an outfix-code of index one [START_REF] Jürgensen | Codes. In: Handbook of Formal Languages[END_REF] with the difference that we require the outfix to be nontrivial. For example, {ab, b} is SDI-independent but it is not an outfix-code of index one.

Theorem 1. For NFAs A and B we can decide in polynomial time whether (i) L(A) is SDI-free (or SDI-independent) with respect to L(B).

(ii) L(A) is alphabetic SDI-free (or alphabetic SDI-independent) with respect to L(B).

For context-free languages deciding SDI-freeness and SDI-independence is undecidable.

Proposition 2. For context-free languages L 1 and L 2 it is undecidable whether

(i) L 1 is SDI-free with respect to L 2 , (ii) L 1 is SDI-independent with respect to L 2 .
For dealing with language equations we express the site-directed insertion operation as a semantic shuffle on a set of trajectories (SST) due to Domaratzki [START_REF] Domaratzki | Semantic shuffle on and deletion along trajectories[END_REF]. The semantic shuffle extends the (syntactic) shuffle on trajectories originally defined by Mateescu et al. [START_REF] Mateescu | Shuffle on trajectories: Syntactic constraints[END_REF]. We use a simplified definition of SST that does not allow content restriction [START_REF] Domaratzki | Semantic shuffle on and deletion along trajectories[END_REF].

The trajectory alphabet is Γ = {0, 1, σ} and a trajectory is a string over Γ . The semantic shuffle of x, y ∈ Σ * on a trajectory t ∈ Γ * , denoted by x t y, is defined as follows.

If x = y = ε, then x t y = ε if t = ε and is undefined otherwise. If x = ax , a ∈ Σ, y = ε and t = ct , c ∈ Γ , then x t ε = a(x t ε) if c = 0, ∅, otherwise. If x = ε, y = by , b ∈ Σ, and t = ct , c ∈ Γ , then ε t y = b(ε t y ) if c = 1, ∅, otherwise.
In the case where all the strings are nonempty, for x = ax , y = by , a, b ∈ Σ, and t = ct , c ∈ Γ , we define

x t y =          a(x t y) if c = 0, b(x t y ) if c = 1, a(x t y ) if a = b and c = σ, ∅, otherwise.
Intuitively, the trajectory t is a sequence of instructions that guide the shuffle of strings and y: 0 selects the next symbol of x, 1 the next symbol of y (these are as in the original definition of syntactic shuffle [START_REF] Mateescu | Shuffle on trajectories: Syntactic constraints[END_REF]) and σ represents synchronized insertion where the next symbols of the argument strings must coincide. For x, y ∈ Σ * and t ∈ Γ * , x t y either consists of a unique string or is undefined. For T ⊆ Γ * , x T y = t∈T x t y and the operation is extended in the natural way for languages over Σ.

Directly from the definition it follows that the SDI and alphabetic SDI operations can be represented as semantic shuffle on a regular set of trajectories. Proposition 3. Let T sdi = 0 * σ + 1 * σ + 0 * and T a-sdi = 0 * σ1 * σ0 * . Then, for any languages L 1 and L 2 ,

L 1 sdi ← L 2 = L 1 T sdi L 2 , and, L 1 a-sdi ← L 2 = L 1 T a-sdi L 2 .
Now using the strong decidability results of Domaratzki [START_REF] Domaratzki | Semantic shuffle on and deletion along trajectories[END_REF] we can effectively decide linear language equations involving site-directed insertion where the constants are regular languages. The representation of SDI using SST guarantees the existence of regularity preserving left-and right-inverse of the operation. This makes it possible to use the results of Kari [START_REF] Kari | On language equations with invertible operations[END_REF] to decide existence of solutions to linear equations where the constants are regular languages. The maximal solutions to the equations are represented using semantic deletion along trajectories [START_REF] Domaratzki | Semantic shuffle on and deletion along trajectories[END_REF]. For the deletion operation we consider a trajectory alphabet ∆ = {i, d, σ}. Intuitively, a trajectory t ∈ ∆ * guides the deletion of a string y from x as follows: a symbol i (insertion) indicates that we output the next symbol of x, a symbol d (deletion) indicates that the next symbol of y must match the next symbol of x and nothing is produced in the output and a symbol σ (synchronization) indicates that the next symbols of x and y must match and this symbol is placed in the output. The result of deleting y from x along trajectory t is denoted x ; t y and the operation is extended in the natural way for sets of trajectories and for languages.

We can express the left-and right-inverse (as defined in [START_REF] Kari | On language equations with invertible operations[END_REF][START_REF] Domaratzki | Semantic shuffle on and deletion along trajectories[END_REF]) of SDI using semantic deletion along trajectories, and these relations are used to express solutions for linear language equations. Given a binary operation 3 on strings, let 3 rev be the operation defined by x3 rev y = y3x for all x, y ∈ Σ * . Using Theorems 6.4 and 6.5 of [START_REF] Domaratzki | Semantic shuffle on and deletion along trajectories[END_REF] we obtain: Theorem 2. Let L, R ⊆ Σ * be regular languages. Then for each of the following equations it is decidable whether a solution exists: (a)

X sdi ← L = R, (b) L sdi ← X = R, (c) X a-sdi ← L = R, (d) L a-sdi ← X = R. Define T 1 = i * σ + d * σ + i * , T a 1 = i * σd * σi * , T 2 = d * σ + i * σ + d * ,
S b = L(; T2 ) rev R, (c) S c = R ; T a 1 L, (d) S d = L(; T a 2 ) rev R.
The above result does not give a polynomial time decision algorithm, even in the case where the languages L and R are given by DFA's. Semantic shuffle on and deletion along regular sets of trajectories preserve regularity but the operations are inherently nondeterministic and complementation blows up the size of an NFA. Note that deleting an individual string y from a string x along trajectory t is deterministic, but the automaton construction for the result of the operation on two DFA languages is nondeterministic. An explicit construction of an NFA for the syntactic shuffle of two regular languages is given in [START_REF] Mateescu | Shuffle on trajectories: Syntactic constraints[END_REF].

The known trajectory based methods for two variable equations [START_REF] Domaratzki | Semantic shuffle on and deletion along trajectories[END_REF] do not allow the trajectories to use the synchronizing symbol σ that is needed to represent the overlap of SDI. However, if we are just interested to know whether a solution exists (as opposed to finding maximal solutions), it is easy to verify that an equation X 1 sdi ← X 2 = R has a solution if and only if all strings of R have length at least two.

Maximal and minimal site-directed insertion

Holzer et al. [START_REF] Holzer | The chop of languages[END_REF] define two deterministic variants of the chop operation. The max-chop (respectively, min-chop) of strings x and y chooses the non-empty suffix of x overlapping with y to be as long (respectively, as short) as possible.

By a maximal site-directed insertion of string y into a string x we mean, roughly speaking, an insertion where neither the overlapping prefix nor the overlapping suffix can be properly extended. The operation is not deterministic because y could be inserted in different positions in x. At a specific position in x, a string y can be maximally (respectively, minimally) inserted in at most one way.

Formally, the maximal site-directed insertion (max-SDI) of a string y into string x is defined as follows:

x max-sdi ← y = { x 1 uzvx 2 | x = x 1 uvx 2 , y = uzv, u = , v = ,
and there exist no suffix x 1 of x 1 and prefix x 2 of x 2 such that x 1 x 2 = and y = x 1 uz vx 2 , z ∈ Σ * } Equivalently the maximal SDI of x and y is

x max-sdi ← y = { x 1 uzvx 2 | x = x 1 uvx 2 , y = uzv, u = = v, no suffix of x 1 u
of length greater than |u| is a prefix of uz and no prefix of vx 2 of length greater than |v| is a suffix of zv }. For example also the string abacbabab is obtained by site-directed inserting y = acbab into x = ababab. In this operation the prefix a of y is matched with the 3rd symbol of x and the suffix b of y is matched with the 4th symbol of x. However, this operation does not satisfy the maximality condition because after the 3rd symbol of x we can match a longer suffix of y.

The minimal site-directed insertion (min-SDI) operation is defined as follows:

x min-sdi ← y = { x 1 uzvx 2 | x = x 1 uvx 2 , y = uzv, u = , v = ,
no proper nonempty suffix of u is a prefix of u, and no proper nonempty prefix of v is a suffix of v }.

Note that in the definition of min-SDI, u and v are unbordered words. Figure 1 (b) illustrates the defininition of minimal SDI. The alphabetic SDI can be viewed as an "extreme" case of minimal SDI: if the first and last letter of y coincide with a substring of x of length two, then the alphabetic and minimal site-directed insertion of y in that position coincide.

If x and y are unary strings with |x|, |y| ≥ 2, then x min-sdi ← y is the unary string of length |y| + |x| -2 and the operation is undefined for |x| < 2 or |y| < 2.

Note that while the maximal or minimal SDI is considerably more restricted than the unrestricted SDI operation, if a string y can be site-directed inserted to a string x, it can be also maximally or minimally inserted at the same position. The result of an alphabetic insertion is always a minimal insertion. These observations are formalized in the next lemma.

Lemma 2. Let x, y ∈ Σ * . (i) x max-sdi ← y ⊆ x sdi ← y and x a-sdi ← y ⊆ x min-sdi ← y ⊆ x sdi ← y. (ii) x sdi ← y = ∅ iff x max-sdi ← y = ∅ iff x min-sdi ← y = ∅. (iii) It is possible that x min-sdi ← y = ∅ and x a-sdi ← y = ∅.
Since the max-chop and min-chop operations do not preserve regularity [START_REF] Holzer | The chop of languages[END_REF], it can be expected that the same holds for maximal and minimal SDI. The proof of the following proposition is inspired by Theorem 3 of [START_REF] Holzer | The chop of languages[END_REF].

Proposition 4. The maximal and minimal site-directed insertion do not preserve regularity.

Proof. Let Σ = {a, b, $, %} and choose

L 1 = ba + ba + $, L 2 = ba + ba + %$ We claim that (L 1 max-sdi ← L 2 ) ∩ (ba + ) 3 %$ = {ba m ba n ba k %$ | m = n or k < n, m, n, k ≥ 1}
We denote the right side of the equation by L result which is clearly nonregular. Since the strings of L 2 contain the marker % that does not occur in strings of L 1 , when inserting a string y ∈ L 2 into a string of L 1 the overlapping suffix of y must consist exactly of the last symbol $. Consider x = ba i ba j $ ∈ L 1 and y = ba r ba s %$ ∈ L 2 . In order for the resulting string to have three symbols b, a prefix of ba r must overlap with ba j , that is, j ≤ r. In order for the overlap to be maximal we must have r = i or s < j. These relations guarantee that the unique string in

x max-sdi ← y is in L result .
For the converse inclusion we note that, for m

= n or k < n, ba m ba n ba k %$ ∈ ba m ba n $ max-sdi ← ba n ba k %$.
For non-closure under min-SDI we claim that

(L 1 min-sdi ← L 2 ) ∩ (ba + ) 2 %$ = {ba m ba n %$ | n > m ≥ 1} = def L result .
Consider x = ba i ba j $ ∈ L 1 and y = ba r ba s %$ ∈ L 2 . In order for the result of site-directed insertion of y into x to have two b's, ba i ba j must be a prefix of ba r ba s , that is, i = r and j ≤ s. For the site-directed insertion to be minimal, no proper non-empty prefix of ba i ba j can be its suffix, that is i < j. These relations guarantee that the minimal SDI of x and y is in L result .

Conversely, for n > m, ba m ba n %$ ∈ ba m ba n $ min-sdi ← ba m ba n %$. In fact, extending the max-chop and min-chop constructions from Theorem 3 of [START_REF] Holzer | The chop of languages[END_REF] it would be possible to show that there exist regular languages L 1 and L 2

such that L 1 max-sdi ← L 2 (or L 1 min-sdi ← L 2 ) is not context-free.
The maximal or minimal site-directed insertion of a finite language into a regular language (and vice versa) is regular.

Proposition 5. Let R be a regular language and L a finite language. Then the languages R

max-sdi ← L, R min-sdi ← L, L max-sdi ← R, and L min-sdi ← R are regular.
Proof. We show that R max-sdi ← L is regular. The other cases are very similar. Since

R max-sdi ← L = y∈L R max-sdi ← y
and regular languages are closed under finite union, it is sufficient to consider the case where L consists of one string y.

Let A be an NFA for R and y ∈ Σ * . We outline how an NFA B can recognize L(A) max-sdi ← y. On an input w, B nondeterministically guesses a decomposition w = x 1 y 1 y 2 y 3 x 2 where x 1 y 1 y 3 x 2 ∈ L(A), y 1 y 2 y 3 = y and y 1 , y 3 = ε. When reading the prefix x 1 y 1 , B simulates a computation of A ending in a state q, then skips the substring y 2 , and continues simulation of A from state q on the suffix y 3 x 2 .

The above checks that the input is in L(A) sdi ← y and, additionally, B needs to verify that the insertion is maximal. This is possible because B is looking for maximal insertions of the one fixed string y.

(i) When processing the prefix x 1 , the state of B remembers the last |y| -1 symbols scanned. When the computation nondeterministically guesses the substrings y 1 , y 2 , y 3 , it can then check that for no nonempty suffix x 1 of x 1 , x 1 y 1 is a prefix of y 1 y 2 . If this condition does not hold, the corresponding transition is undefined.

(ii) Similarly, when processing the (nondeterministically selected) suffix x 2 of the input, B remembers the first |y| -1 symbols and is able to check that for no nonempty prefix x 2 of x 2 , y 3 x 2 is a suffix of y 2 y 3 .

If the checks in both (i) and (ii) are successful and at the end the simulation of A ends with a final state, this means that the decomposition x 1 y 1 y 2 y 3 x 2 gives a maximal site-directed insertion of y into a string of L(A).

Decision problems for maximal/minimal SDI

From Proposition 4 we know that the maximal or minimal SDI of regular languages need not be regular. However, for regular languages L As we have seen, the maximal and minimal SDI operations are often more difficult to handle than the unrestricted SDI. Using Lemma 2 (ii) we note that deciding maximal (or minimal) SDI-freeness is the same as deciding SDI-freeness and by Theorem 1 we have: Corollary 1. For NFAs A and B we can decide in polynomial time whether or not L(A) is maximal SDI-free (respectively, minimal SDI-free) with respect to L(B).

Also, deciding whether regular languages are max-SDI (or min-SDI) independent can be done in polynomial time.

Theorem 3. For NFAs A and B, we can decide in polynomial time whether or not L(A) is maximal SDI-independent (respectively, minimal SDI-independent) with respect to L(B).

Proof. Let Σ be the underlying alphabet of A and B. We verify that L(A)

←

x 1 y 1 y 2 y 3 x 2 , where the latter insertion uses the outfix (x 1 y 1 , y 3 x 2 ) as insertion guide. The insertion is maximal because the outfix cannot be expanded. In the same way we see that L(A) min-sdi ← Σ + = L(A) sdi ← Σ + . Now the claim follows by Theorem 1. Since the max-SDI and min-SDI operations do not preserve regularity there is no straightforward algorithm to decide whether a regular language is closed under maximal SDI or under minimal SDI. We conjecture that the problem is decidable.

Problem 1. Is there an algorithm that for a given regular language L decides whether or not

L max-sdi ← L ⊆ L (respectively, L min-sdi ← L ⊆ L)?
Using Proposition 5 we can decide closure of a regular language under max/min-SDI with a finite language.

Corollary 2. Given a regular language R and a finite language F we can decide whether or not Recall that the NFA stores in the state a sequence of symbols having length of the inserted string. Strictly speaking, the proof of Proposition 5 assumes that F consists of a single string, but a similar construction works for a finite language. When c F is a constant, the size of B is polynomial in m and we can decide in polynomial time whether or not L(B) ∩ L(A) = ∅.

(i) R max-sdi ← F ⊆ R, (ii) R min-sdi ← F ⊆ R. If R is
The max-SDI and min-SDI operations do not preserve regularity and, consequently, they cannot be represented using semantic shuffle on trajectories. Thus, the tools developed in Section 6 of [START_REF] Domaratzki | Semantic shuffle on and deletion along trajectories[END_REF] to deal with language equations are not available and it remains open whether we can solve language equations involving max-SDI or min-SDI. 5 Nondeterministic state complexity

The site-directed insertion (SDI) operation preserves regularity [START_REF] Cho | Outfix-guided insertion[END_REF] (Proposition 1 above) and the construction can be modified to show that also alphabetic SDI preserves regularity. To conclude, we consider the nondeterministic state complexity of these operations. The upper bound is the same as the bound for the nondeterministic state complexity of ordinary insertion [START_REF] Han | State complexity of insertion[END_REF], however, the construction used for Lemma 4 is not the same. Using Lemma 1 (the fooling set lemma [START_REF] Birget | Intersection and union of regular languages and state complexity[END_REF]) we can establish a matching lower bound. Lemma 5. For m, n ∈ N, there exist regular languages L 1 and L 2 over a binary alphabet having NFAs with m and n states, respectively, such that any NFA for L 1 a-sdi ← L 2 needs at least mn + 2m states.

The above lemmas establish the precise nondeterministic state complexity of alphabetic SDI. Likely the bound cannot be improved but we do not have a proof for the lower bound.

Problem 3. What is the nondeterministic state complexity of site-directed insertion?

and T a 2 =

 2 d * σi * σd * . If a solution exists, a superset of all solutions is, respectively, for the different cases: (a) S a = R ; T1 L, (b)

Fig. 1 .Example 1 .

 11 Fig.1. Insertion of y into x depicted at top left is not maximal when x and y have decompositions as depicted at bottom left.

  1 and L 2 we can decide membership in L 1 max-sdi ← L 2 (or L 1 min-sdi ← L 2 ) in polynomial time. Lemma 3. For DFAs A and B and w ∈ Σ * we can decide in time O(n 6 ) whether w ∈ L(A) max-sdi ← L(B), or whether w ∈ L(A) min-sdi ← L(B).

Σ

  + = L(A) sdi ← Σ + . The inclusion from left to right holds by Lemma 2 (i). Conversely, suppose w ∈ L(A) sdi ← y 1 y 2 y 3 , where w = x 1 y 1 y 2 y 3 x 2 , y 1 , y 3 = ε, x 1 y 1 y 3 x 2 ∈ L(A). Then w ∈ L(A) max-sdi

  specified by a DFA and the length of the longest string in F is bounded by a constant, the algorithm works in polynomial time. Proof. By Proposition 5 the languages R max = R max-sdi ← F and R min = R min-sdi ← F are effectively regular. Suppose R = L(A) where A is a DFA with m states and underlying alphabet Σ and the length of the longest string in F is c F . The NFA B constructed in the proof of Proposition 5 for R max (or R min ) has O(m • |Σ| c F ) states.

Problem 2 .

 2 Let L and R be regular languages. Is it decidable whether the equationX max-sdi ← L = R (respectively, L max-sdi ← X = R, X min-sdi ← L = R, L min-sdi ← X = R) has a solution?

Lemma 4 .

 4 For NFAs M and N having, respectively, m and n states, the language L(M ) a-sdi ← L(N ) can be recognized by an NFA with mn + 2m states.

Corollary 3 .

 3 The worst case nondeterministic state complexity of the alphabetic site-directed insertion of an n-state NFA language into an m-state NFA language is mn+2m. The lower bound can be reached by languages over a binary alphabet.It is less obvious what is the precise nondeterministic state complexity of the general SDI. IfA has m states and B has n states, Proposition 1 gives an upper bound 3mn + 2m for the nondeterministic state complexity of L(A) sdi ← L(B).

Cho and Han were supported by the Basic Science Research Program through NRF (2015R1D1A1A01060097) and the International Research & Development Program of NRF (2017K1A3A1A12024971). Salomaa and Smith were supported by Natural Sciences and Engineering Research Council of Canada Grant OGP0147224.