Alain Kuzniak

Assia Nechache

Philippe R Richard

Chapter 3 The Theory of Mathematical Working Spaces in Brief

This chapter contains an overview of the main points of the theory of MWS in relation to frequently asked questions. The aim is to form the basis of an online glossary of the theory, to be completed according to the contributions of researchers.

Main purpose of the theory:

Describing, understanding and (trans)forming mathematical work in a school context.

Mathematical work

In the theory of MWS, mathematical work must be understood as an on-going intellectual human process of production, the orientation and finality of which are defined and supported by mathematics and more generally by mathematical culture.

Mathematical work is intended to perform a task, solve a problem or overcome an obstacle in relation to mathematics. It requires the sustained mobilization and development of material or intellectual resources derived from mathematical culture (1.3.1).

A reasonable goal of mathematics education is that, at the end of their studies, students should have a good idea and practice of mathematical work (7.2.2).

Purpose, processes and results

Mathematical work involves consideration of three aspects related to its performance and development (1.3.1):

The purpose of the work. This attribute helps us to distinguish the notion of 'work' from a simple activity. This is done by assigning a goal to an action, clarifying the stakes associated with it, and inscribing it in the long term by demonstrating its general relevance.

The processes of the work. These are related to the procedures and constraints of implementing given tasks. Appropriation of the goals of the task and compliance with the rules of functioning are important aspects for observing and characterizing the work.

The results of the work. Results should be valid and coherent within the mathematical domain in question.

Epistemological and cognitive aspects of mathematical work

Through the lens of MWS, the development by an individual, whether generic or not, of appropriate mathematical work is a gradual and progressively constructed process of bridging epistemological and cognitive planes.

The epistemological plane and its components (1.4.2.1)

The epistemological plane, organized according to purely mathematical criteria, has three interacting components: a set of concrete and tangible objects (representamen); a set of artefacts such as drawing tools or software; and a theoretical system of reference (the referential) based on definitions, properties and theorems.

Signs or representamen

The sign, or representamen, is "something which stands to somebody for something in some respect or capacity" (Peirce). Signs or representamen summarize the component using concrete and tangible symbols. Depending on the mathematical field at stake, the signs used can be geometrical images, algebraic symbols, graphics, or even tokens, models or photos in the case of problems involving modeling.

Artifacts

An artifact includes everything that has undergone a transformation, however small, of human origin. Its meaning is not limited to material objects and can include a strong symbolic dimension. In the theory of MWS, mathematical artifacts will generally be associated with material objects to avoid confusion with other components of the epistemological plane. These objects will be intimately linked to rules and techniques of construction or calculation (Euclidean division, "classical" constructions with ruler and compass, etc.). These techniques are based on algorithms, the validation and theoretical status of which are no longer problematic or are no longer questioned by their user.

Theoretical frame of reference or referential

The referential includes the set of properties, definitions, theorems and axioms and refers to the theoretical part of the mathematical work. This set is not only a collection of properties: it supports the deductive discourse of proof specific to mathematics, and this requires that it should be organized in a coherent way, well-adapted to the tasks the students are requested to solve and model. The referential is not absolute and determined in a unique way once and for all; its constitution and organization depend, in particular, on the mathematical paradigms and work targeted by the educational institutions or organizations.

The cognitive plane and cognitive processes (1.4.2.2)

The cognitive plane relates to the processes and procedures used by individuals in task-solving activities. It is structured around three cognitive processes: visualization, construction and proving.

Visualization

The process of visualization can be considered as the cognitive process associated with identifying and developing a set of treatments and transformations of the representamen or signs -which, in this respect, can very well be perceived as acoustic images in strictly oral contexts. It is thus possible to study the role of signs in different mathematical domains, such as mathematical analysis, which give rise to a specific work of visualization based on spreadsheet data or numerical writings. This visualization must be distinguished from the simple vision or perception of signs.

Construction

The process of construction is associated with actions triggered by the use of artifacts, actions which lead to tangible results such as figures, graphs or the results of a calculation based on algorithms or produced by a numerical machine. These results are concrete products of some mathematical entities which are generally not isolated and form configurations to be observed and explored (8.5.1).

Proving

The process of proof considered here relies on deductive and logical discourse and must be based on, or lead to, assertions with a clear theoretical status. In contrast, simple empirical validation is more likely to relate to construction or visualization, as discussed above. In addition to deductively organized arguments, these assertions may well be definitions, hypotheses, conjectures, or statements of counterexamples.

Mathematical Working Spaces -MWS

A Mathematical Working Space refers to an abstract structure organized in such a way as to account for mathematical work and enable individuals to conduct operations with problems in a specific domain (geometry, probability, etc.).

The theory of MWS aims to highlight the mathematical work proposed by educational institutions and organizations involved in the teaching and learning of mathematics.

The MWS diagram

The MWS diagram gives a general overview of the different relationships and interactions existing between the components of the epistemological plane and cognitive plane of the Mathematical Working Space. The diagram is closely associated with the development of the theory, of which it is in a way the emblem, but should not be confused with the theory as a whole.

Its use as a methodological tool requires selection of the most appropriate modes of representation to reflect the evolution of the work (3.8.3).

The three mathematical work geneses in the MWS

Development by an individual, whether generic or not, of appropriate mathematical work is a gradual process which is progressively developed, as a process of bridging the epistemological and cognitive planes. It evolves in accordance with different specific yet intertwined generative developments: semiotic, instrumental and discursive geneses.

Each of these geneses can be visualized through one of the dimensions of the MWS diagram.

Top-down and bottom-up perspectives on geneses

Each of these geneses can be described according to two perspectives: one stance focuses on epistemological components (top-down perspective) and a second oriented towards cognitive processes (bottom-up perspective). The terms bottom-up and top-down are only relative to the position of the epistemological and cognitive planes in the MWS diagram.

The Semiotic genesis

The Semiotic genesis relates signs and representamen with visualization. It explains the dialectical relationship between syntactic and semantic perspectives on mathematical objects, represented and organized by semiotic systems. The semiotic genesis confers to the representamen their status of operational mathematical objects. In this way, it establishes the links between the function and structure of the expressed signs. The bottom-up perspective associated with this genesis can be considered as a process of decoding and interpreting signs from a given representamen through visualization. The top-down perspective, directed from visualization towards a representamen, can be evoked when a sign is produced or specified.

The Instrumental genesis

The instrumental genesis establishes the link between the artifacts and the construction processes contributing to the completion of mathematical work. The bottom-up perspective describes the actions by which the user appropriates the various techniques associated with the artifact. The bottom-down perspective relates to the adequate choice of the artifact required to perform the intended actions. In the theory of MWS, the two perspectives, epistemological and cognitive, are essential because if using the artifacts modifies the users' way of doing and thinking, in return, the mathematics they develop will also be transformed.

The Discursive genesis of proof

The discursive genesis of proof is the process by which the properties and results organized in the referential are actuated in order to be available for mathematical reasoning and discursive validations. By this we mean validations going beyond graphic, empirical or instrumented verification, but which could nevertheless be triggered by these. In this genesis, the bottom-up perspective is related to a deductive discourse of proof supported by properties structured into the referential. Conversely, the identification of properties and definitions to be included in the referential, possibly suggested by instrumental, computational or visual processes, relate to the top-down perspective.

Interactions between geneses and vertical planes

The vertical planes (named Sem-Ins, Ins-Dis and Sem-Dis) make interactions between the different geneses visible: in particular, they enable expression of the main coordination games involved in the mathematical work and the geneses that are temporarily subordinated to such work. ) is gained from looking at methodological investigation, but also at more specific and detailed studies of the proofs and validations used to solve tasks.

From a methodological perspective, recognition of active geneses and interactions between geneses is supported by the identification of tools (in the epistemological plane) and instruments (in the cognitive plane) used to perform the task (3.7).

Diversity of mathematical work associated with various MWS

The reference mathematical work in connection with the reference MWS

The reference mathematical work corresponds to the work normally expected by educational institutions and organizations in charge of teaching mathematics. The reference MWS is naturally defined according to mathematical criteria. But cultural, economic, social and political criteria also contribute towards shaping it, to such an extent that describing it requires thorough studies: of the intended curriculum, of treatises written by mathematicians, of essays and articles by mathematics educators. This aspect in turn supposes that epistemological vigilance is required. It means that the operating rules involved in this MWS enable knowledge to be organized into a well-defined and coherent mathematical field. The set of rules, procedures and questioning which help to define the reference work refers to the notion of paradigm (3.5). Pragmatically in educational contexts, the constitution of a MWS does not rely on a single paradigm, but rather on the interplay between different paradigms.

The personal mathematical work and personal MWS

The personal work and personal MWS will depend on the actors and subjects implicated in the work --a pupil or student, a teacher or teacher trainer. This MWS reflects the reality of the work of individuals who perform a task with their own knowledge and cognitive capacities. The personal work is very fluid and its evolution depends on the tasks the subjects are led to perform.

The idoine or suitable MWS

This MWS refers to an intermediate state of transmission and mediation of knowledge where tension exists between the teacher's expectations and the redefinition of the task and problems in order to enable students' personal work to progress. The MWS in question is designated as suitable or idoine MWS precisely to emphasize these transformations by which it is characterized in order to best adjust to the educational project and to the reality of the learners.

It can be seen as a connecting structure from reference MWS to personal MWS mediated by designer-actors (teacher or researcher) to help receptor-actors (students) build their personal mathematical work.

It is necessary to consider two states of this MWS: a potential state, which corresponds to what the teacher foresees, and an actual or effective state, which comprises the work produced in the classroom. These three MWS are not independent but are rather interrelated (4.2). Each requires a specific methodology which must be understood (Part II).

Mathematical domain and specific MWS

A mathematical domain should be organized into a consistent structure within mathematics. The best proof that a given segment of mathematics should be called a domain is that it is regarded as such by the community of mathematicians. In this way, a specific MWS is associated with each domain. Thus, we may speak of a MWSGeometry, MWSAlgebra, MWSAnalysis, MWSProbability, etc.

It is also worth reflecting upon the way new transversal themes -such as Algorithmics or Modeling -may be integrated into MWS, and what kind of reorganization of the framework is necessary to foster this integration.

Paradigms in the theory of MWS

A paradigm stands for the combination of beliefs, convictions, techniques, methods and values shared by a community. This community can be scientific, but in the theory of MWS it will be educational organizations engaged in mathematics education.

In the theory, different paradigms can exist and coexist. Identifying these paradigms and their interactions help us to understand and orient mathematical work.

Paradigms ensure the conformity of work processes and the accuracy of the results produced. Mathematical work is said to be compliant when the processes and procedures used are valid and consistent with the expectations present in the preferred paradigm(s) of work.

The search for paradigms associated with the MWS specific to a mathematical domain is an important focus for the theory of MWS.

Mathematical Tasks

Mathematical tasks and the learner's work

In the theory of MWS, tasks must be interpreted in a broad and varied sense. They can thus be associated with traditional problems formulated with clear assumptions and known to be solvable in a predictable amount of time by students. They can also be given with relatively vague outlines as support for open and rich problematization or modeling situations. In this way, a range of tasks, from simple to rich tasks, exists. Performance of the tasks depends on their level of mathematical sophistication and on the degree of initiative the students are free to use.

Tasks are the essential units which trigger the subject's activity. More precisely, mathematical actions with well-determined goals are seen as the visible manifestation of a subject's cognitive activity (or mathematical thinking) when actively performing tasks. From analysis of the mathematical actions identified being performed by subjects, it becomes a matter of identifying their mathematical work with the idea of adapting or transforming it.

Task environment and mathematical work

A task is determined by its goals and the conditions necessary for its performance which can be individual or collective.

In mathematics education, tasks are conceived and implemented in various environments, which can relate to different didactical and theoretical traditions such as task design, lesson studies or didactical engineering.

The tasks within their environment and the mathematical actions performed form a set of observables which enable the mathematical work to be investigated. Studying these observables through the lens of the MWS enables generic forms of work in different mathematical domains to be characterized. In the long term, this characterization of forms serves as a solid basis for forming or transforming mathematical work to conform to the expectations of the given paradigm.

Tools and instruments of mathematical work

Solving a mathematical problem implies the use of different mathematical objects which can be considered as tools or instruments with which to perform the task.

In the theory, and whenever it is useful, the term tool is reserved for objects of the epistemological plane that have a specific potential use in the context of problem solving. Tools have a productive purpose for performing the task.

Use of the word instrument is prioritized as soon as a subject (pupil, student or teacher) has developed their use of a tool to effectively solve the proposed task. It supposes implementation of techniques by the active subject and thus concerns the cognitive aspects of the work.

In relation to the different components of the epistemological plane and the different geneses of mathematical work, it is possible to consider, a priori, three types of tools and instruments:

Semiotic tools are non-material tools -like diagrams, symbols or figuresthat may be used as instruments for operating on semiotic representations of mathematical objects.

Technological tools are artifacts -like drawing tools or routinized techniques -which are "to hand", based on algorithms or calculators with implemented calculation or drawing algorithms used as instruments for constructing mathematical objects such as figures, graphs, numbers, etc.

Theoretical tools are elements of the referential non-material tools like theorems which may be used as instruments for mathematical reasoning based on the logical and mathematical properties of mathematical objects.

3.8

The MWS diagram: a methodological tool to describe, characterize and transform mathematical work

Describing the circulation of work

The individuals involved in performing a task produce a series of actions oriented by the goals to be reached and triggered by knowledge. The set of dynamics thus created enables us to speak of a circulation of work.

The MWS diagram makes it possible to describe and visualize the evolution and dynamics of the various interactions between geneses during the performance of a task or set of tasks by individuals (Chapter 2.). This dynamic is called circulation of work.

Characterizing mathematical work

When the circulation of work engages all the geneses and the different planes of the diagram, the mathematical work will be considered complete.

The diagram can be used to demonstrate the completeness of the mathematical work. It also helps to identify the reasons for a blockage which may be due to confinements in a vertical plane or in a genesis. It provides an insight into the rebounds of this work to break through blockages and confinements.

(Trans)forming the circulation of work

By focusing on the various interactions which help or hinder mathematical work, the diagram can serve as a support for thinking about and rethinking the circulation of work (by tuning tasks and their implementation). In the long term, this can contribute towards forming or transforming the mathematical work.

3.9 Further aspects on specificity and originality of the theory

Epistemological anchoring

The theory of MWS is a didactic theory strongly supported by mathematical content. It achieves this by closely combining epistemological and cognitive aspects around the unifying notion of mathematical work.

A triadic approach

The synthetic and triadic vision of mathematical work gives it coherence and facilitates its analysis. In this conception of mathematical work, it is the arrangement and relationships between the different epistemological components and cognitive processes which ensure the coherent and complete development of mathematical work. The elaboration of the latter is seen as harmonious interactions of three types of genesis: semiotic, instrumental and discursive.

Processes and results

Work is based on a set of organized and finalized actions, which emphasizes the duality between, on the one hand, the processes of work and, on the other hand, the results of work.

Importance of the idea of 'idonéité'

The 'suitability' hallmark of a MWS is gradually constructed and adapted along the paths and vagaries of the teaching route. This explains our fondness for the French term 'idoine', the translation of which to 'suitable' in English may not fully convey the meaning suggested by Gonseth (1945Gonseth ( -1952, our translation), our translation):

A metaphorical way of presenting suitability in the theory of MWS is to think about it as resulting from recursive feedback loops, converging towards a stable state when one exists. Instead of trying to reduce the uncertainty related to any new experience, Gonseth proposes a recursive adaptation to it.

The link with learning theories

Accordingly, one of the main strengths of studies based on MWS is their investigation of the interactions between the cognitive and epistemological dimen-sions. The underlying learning model is not, as in many theories which emerged in the 1970s, the socioconstructivist model.

As far as the MWS are concerned, there is no prescribed learning model. Studies can look at other learning principles such as Socrates' maieutics or behaviorism, or even approaches which emphasize the imitation of models presented to students.

Link to institutional choices

The theory of MWS does not give priority to a determined institutional model with regards to the development of an epistemological basis, if only for the purpose of expecting -and this is no small matter -an approach which is globally coherent, in particular with respect to validation within a given paradigm or network of paradigms, meant to be consistent and rational. The notion of a paradigm is essential here and, in a certain way, mathematical work could be seen as a quest for harmonious articulation between coherent paradigms.

Points not addressed by the theory

-The impact on the development of the mental apparatus of a subject of information and retroactions induced by the environment.

-The nature of the black box in precise reference to the psychological constitution of the geneses.

-The existence and possible place in the brain of all kinds of schemas which can intervene in the mathematical work.

These are issues which the theory has neither the ambition nor the means to explore. The theory relies on the visible effects of a subject's actions through explicit observables in order to understand cognitive processes. In this sense, it can be seen as a materialist-type scientific approach.

Dialogue with other theories

The theory of MWS is not a holistic theory and has close links with theories which share its materialist approach on people's way of thinking when they are engaging in mathematics.

Moreover, the whole activity of teaching and learning mathematics is not limited to mathematical work. Many complementary and indispensable elements are external to this work yet participate in its undertaking in practice: affects and emotions, collaborative and collective work, language exchanges, cultural and social context, and so on.

These represent many points and perspectives which are not directly related to the core of the theory, but which can contribute to fruitful exchanges with specialists in these areas. These exchanges and interactions are part of the theoretical and methodological innovation which has characterized MWS research to date.

Fig. 3 . 1

 31 Fig. 3.1 The Mathematical Working Space diagram

Fig. 3 . 2

 32 Fig. 3.2 The three vertical planes in the MWS diagram A deep understanding of the different geneses and the three planes of interaction between these geneses ([Sem-Ins], [Ins-Dis] and [Sem-Dis]) is gained from looking at methodological investigation, but also at more specific and detailed studies of the proofs and validations used to solve tasks.From a methodological perspective, recognition of active geneses and interactions between geneses is supported by the identification of tools (in the epistemological plane) and instruments (in the cognitive plane) used to perform the task (3.7).


Thomas Heide Clausen

MANET Router Configuration Recommendations

Keywords: mobile network, ad hoc network, MANET, network architecture, configuration, routing, IP networks MANET Router Configuration Recommendations réseaux mobiles, réseaux ad-hoc, MANET, architecture de réseau, configuration, routage, réseaux IP

This memorandum describes a pragmatic set of configuration recommendations for MANETs, as well as provides a rationale for why these recommendations are sound. While there may be other equally valid ways of configuring a MANET, the recommendations in this memorandum have the merit of being supported by an existence proof (there're running networks in existence, configured according to these recommendations), and they require neither modifications to the IP stack nor to upper-layer protocols or applications.

Introduction

A MANET (Mobile Ad hoc NETwork) is, in academic circles, commonly described as a variation over the following: "a collection of independent nodes, communicating over a wireless, capacityrestricted medium, whereby they form a multi-hop connected network with no assumptions of an a-priori network infrastructure and with a highly dynamic topology."

While such a description may be extrapolated to interesting challenges for research and protocol design, such as fast-convergence routing algorithms and bandwidth-economic protocols, it does little to describe the exact morphology and IP configuration of the "independent nodes" from which the MANET is constructed. Consequently, it offers little guidance in how a "real-world" IP-based MANET can be configured and deployed.

MANET routing protocol specification as developed by the IETF, such as [START_REF] Clausen | RFC3626: Optimized Link State Routing Protocol[END_REF], [START_REF] Perkins | RFC3561: Ad-Hoc On Demand Distance Vector Routing[END_REF], [START_REF] Johnson | RFC4728: The Dynamic Source Routing Protocol (DSR) for Mobile Ad Hoc Networks for IPv4[END_REF] and [START_REF] Ogier | RFC3684: Topology Dissemination Based on Reverse-Path Forwarding[END_REF], stipulate internal processing of such "independent nodes" as well as specify the exact protocol message exchange between these, in order to allow construction of routing tables. As such, these specifications allow for operation of a MANET, and do provide indications as to under which assumptions such operation is assumed -but on the other hand do not have as vocation to provide explicit configuration and deployment guidelines for MANETs. MANET routing protocols explicitly assume that all MANET nodes in a MANET are already configured, i.e. that the network interfaces have appropriate IP addresses etc.

The purpose of this present memorandum is to remedy that by describing recommendations and considerations for configuring and deploying a MANET. These considerations are characterized by adhering strictly to IP -i.e. to require no modifications to neither the protocol stack nor to applications accessing the network -and to be applicable for, to the best of the authors knowledge, any MANET using any choice of routing protocol from among those developed for MANETs within the IETF.

Caveat lector: Other configurations and deployment approaches for MANETs are likely possible, and should not be disregarded or excluded. This memorandum, however, describes a set of configuration and deployment recommendations, which practical experience and real-world deployments have shown to be feasible. Also note that this memorandum describes only the end result -but not any process (automatic or otherwise). Hence, when the term "configuration" is employed, it is as a noun for "the end result".

Memorandum Outline

This memorandum is constituted of three parts.

Part I presents, without rationale, explanation or apologies, recommendations for the constitution and configuration of a MANET router and of a MANET. This should allow the reader to immediately understand how to -safely -configure and deploy an IP based MANET using any of the currently developed IETF MANET routing protocols. This, while maintaining compatibility with all "Internet applications" and with other protocols, and without requiring modifications to the IP stack.

Part II explains the background and rationale for the configuration recommendations presented in part I. For each configuration recommendation, a selection of justifications are presented and detailed. This should allow the reader to understand why the recommendations given in part I represent a reasonable and operational way of configuring a MANET.

Part III concludes this memorandum by summarizing the configuration recommendations and the characteristics of a MANET configured according to the recommendations given in this document.

INRIA

Part I

MANET Configuration Recommendations

In this first part, a small set of MANET configuration recommendations is given. As indicated in section 1, these recommendations constitutes an approach for which there is an existence proof that a MANET so configured is both correctly functioning and one in which "Internet applications" and other protocols operate unmodified without requiring modifications to the IP stack.

Section 2 expands the notion of MANETs being constructed from "independent nodes" into the familiar terminology on the Internet and of IP networking of "Hosts" and "Routers". Section 3 describes address configuration of MANET router interfaces, and section 4 how prefixes delegated to a MANET router are used. Section 5 gives a few examples of common MANET configuration scenarios, with the intent of showing concrete instances of how the configuration approach is deployed.

Caveat lector: This part explains uniquely the HOW of configuration of a MANET router and a MANET. The WHY -the rationale and justification -is given in part II.

MANET "Node" Morphology -Hosts and Routers

Each "independent node" in a MANET can functionally be described according to figure 1, with the "MANET node being composed of a "MANET Router" with one or more "MANET interfaces" -i.e. the interface which is to be used for establishing connectivity between MANET routers -as well as zero or more interfaces towards other networks or hosts on classic IP links. ❼ MANET characteristics are "isolated behind an IP hop" from hosts or other networks;

❼ Applications on hosts and other networks can remain "MANET unaware".

A MANET, therefore, looks as depicted in figure 2: the inner white cloud represents where MANET routers have MANET interfaces and where only MANET aware protocols operate (e.g. MANET routing protocols) -and the outer gray cloud represents the non-MANET-aware part of the network, with hosts and other (non-MANET) networks. This is akin to, say, an OSPF network, which can be depicted similarly to figure 2: Within the inner wide cloud, OSPF routers with NBMA interfaces would be operating using protocol mechanisms suitably aware of NBMA characteristics, whereas the outer gray cloud would contain non-NBMA-aware parts of the network, such as hosts.

MANET Interface Configuration

Each MANET interface MUST be configured with an IP address, which is unique, at least within the MANET, and with a prefix length of /32 (IPv4) or /128 (IPv6). The following summarizes the configuration constraints for a MANET router.

MANET Interface Address Configuration:

❼ Each MANET interface on a MANET router MUST be configured with an IP address which is unique, at least within the MANET.

❼ Each MANET interface MUST be configured with a prefix length of /32 (for IPv4) or /128 (for IPv6).

Prefixes delegated to MANET Router

If a prefix is delegated to a MANET router, this prefix MUST NOT be assigned to any MANET interface of the router. Rather, such a prefix MAY be assigned to any non-MANET interface of the router (if appropriate).

Delegated Prefix Configuration:

❼ If a prefix is delegated to a MANET router, this prefix MUST NOT be configured on any MANET interface.

❼ If a prefix is delegated to a MANET router, other (non-MANET) interfaces MAY be configured with this prefix.

Example MANET Configurations

This section provides a set of illustrative examples of common MANET configurations, respecting the considerations given in section 3. Notice that this section does not attempt to exhaustively enumerate all possible MANET deployments or configurations.

In the examples below, IP addresses and prefixes are extracted from within the private address space 192.168/16. This is for illustrative purposes only. This memorandum does not take any position as to which address space is to be used for when configuring MANETs or other networks.

MANET Router and Single Host

Figure 3 illustrates a MANET node consisting of a MANET router with a single MANET interface and a single host. This is the typical case, for example, when one has a laptop, PDA or smartphone participating in the MANET: the "router" and the "host" are in that case logical components within the same device, and with a single physical interface.

The MANET node is assigned a single IPv4 address, in this case 192.168.1.1. This IP address is to identify both the MANET interface of the MANET router as well as the host. Logically, this is accomplished by configuring the interface of the MANET router as an "unnumbered interface", by assigning 192.168.1.1/32 to the MANET interface of the router. Traffic to the logical component that is the router will typically be addressed to a well-known INRIA multicast address, thus the router can distinguish between traffic to itself and traffic to the logical host. It is important to retain that the MANET interface is configured with a prefix length of /32, as this is an IPv4 network -had it been an IPv6 network, the required prefix length for the MANET interfaces would have been /128.

For administrative reasons, IP addresses assigned to MANET routers within the same MANET may be extracted from within a larger prefix, say, 192.168.0.0/16. Even so, each MANET interface configured with an IPv4 address MUST be configured with a prefix length of /32. A complete MANET so configured might look as depicted in figure 4.

MANET prefix: 192.168.0.0/16 The MANET interface is configured as an "unnumbered interface" by "borrowing" the address (192.168.1.1) from its interface towards the classic IP link -but is configured with the prefix length /32. The MANET interface is, specifically, not configured with a prefix length of /24, even though that prefix is delegated to the MANET router, as the MANET interface is not able to reach all other interfaces with addresses from within 192.168.1.0./24 in a single IP hop. Traffic to the MANET interface with the IP address 192.168.1.1 can be distinguished from traffic to the non-MANET interface with the same address since traffic destined to the MANET interface of the router typically will be addressed to a well-known multicast address.

INRIA

MANET Router with two MANET Interfaces and Attached Network

Similar to figure 5, figure 6 illustrates the configuration of a MANET router with two MANET interfaces. In this case, the attached network and one of the MANET interfaces is configured exactly as in figure 5, whereas the second MANET interface is configured using an otherwise unused IP address -in this case 192.168. Using IP addresses from within the prefix delegated to the MANET router for configuring MANET interfaces on that router ensures that these IP addresses are unique, at least within the MANET -assuming, of course, that the prefix delegation mechanism does not delegate the same prefix to multiple MANET routers within the same network.

Notice that the same logic applies in this scenario as in the one in figure 5: the second MANET interface could assume any IP address from within 192.168.1.0/24, even one already used by a host such as 192.168.1.2, as long as it is not used on another MANET interface and as long as the second MANET interface is also configured with a prefix length of /32. Traffic to the MANET interface with the IP address 192.168.1.2 can be distinguished from traffic to the non-MANET interface with the same address since traffic destined to the MANET interface of the router typically will be addressed to a well-known multicast address.

Part II

Rationale

Part I describes three recommendations when configuring MANET routers and MANET interfaces on MANET routers:

❼ Each MANET interface on a MANET router MUST be configured with an IP address which is unique, at least within the MANET;

❼ MANET interfaces MUST be configured with prefix lengths of /32 (IPv4) or /128 (IPv6);

❼ MANET routers MUST isolate MANET characteristics from non-MANET interfaces.

This second part will provide a rationale for these configuration recommendations by presenting the reasoning behind each of these -as well as a selection of consequences from configuring a MANET router without following these recommendations.

In the descriptions in the following sections, IP addresses and prefixes are extracted from within the private address space 192.168/16. This is for illustrative purposes only. This memorandum not take any position as to which address space is to be used for when configuring MANETs or other networks.

INRIA 6 Unique MANET Interface Addresses

MANET interfaces are typically not attached to point-to-point links, but are rather of a "semi-broadcast nature" such as indicated in figure 7. Hence, a packet transmitted on a MANET interface may be received by any number of MANET interfaces, including the transmitting MANET interface itself. In order for MANET routers to be able to forward data packet not destined to themselves or to directly attached non-MANET hosts or networks, they need to be able to uniquely identify the "next hop" for a packet. Symmetrically, in order for a MANET router to discriminate between incoming data packets for it to forward (either within the MANET or to an attached host or network) or to discard, it needs to be able to identify if indeed it is the "next hop" for a received data packet. In order to accomplish this, MANET interfaces which are in direct L2 communication with each other need to be configured with unique IP addresses. Some MANET routing protocols employ flooding mechanisms (also commonly denoted "optimized flooding") in order to reduce the overhead involved in topology dissemination throughout the network. Examples include [START_REF] Clausen | RFC3626: Optimized Link State Routing Protocol[END_REF] and [START_REF] Clausen | RFC5449: OSPF Multipoint Relay (MPR) Extension for Ad Hoc Networks[END_REF]. Such flooding mechanisms are based on each node selecting a subset of its neighbors as "relays", with only these relays taking part in flooding operations. Such relay selection algorithm in general require the ability to uniquely identify each MANET interface up to two hops away from the source; such is the case for [START_REF] Clausen | RFC3626: Optimized Link State Routing Protocol[END_REF] and [START_REF] Clausen | RFC5449: OSPF Multipoint Relay (MPR) Extension for Ad Hoc Networks[END_REF].

Other flooding algorithms, studied for MANETs, perform relay selection requiring the ability to uniquely identify each MANET interface further away -such is the case with various non-connected dominating set flooding approaches, currently a subject of research.

By configuring MANET interfaces such that each has a MANET-wide unique address, it is ensured that both the flooding mechanisms from the current crop of MANET routing protocols, as well as potential future developments, are supported.

/32 and /128 Prefix Lengths

In early MANET deployments, a common occurrence was to configure a MANET as "a subnet" and configure it as indicated in figure 8 An early, and often suggested, solution was to "treat the symptoms rather than cure the disease" by disabling ICMP redirects on MANET routers -i.e. to require modifications to the IP stack operation in order that it can be supporting MANETs.

INRIA

The ICMP redirect is intended to be used to inform a source to send packets using an alternative, more direct, route -e.g. if a source, s wishes to send a data packet to a destination node d via the path s-R1-R2-d, and if R1 knows that a direct path s-R2-d is available, then the ICMP redirect from R1 will inform s of this route.

Two interfaces, configured with addresses from within the same prefix, and with same prefix length are defined to be reachable from each other within a single IP hop. More precisely, it is assumed that all interfaces with IP addresses within that subnet prefix and configured with the same prefix length are directly reachable from each other without forwarding by an intermediate router. Hence, a way for R1 to know that a direct path may exist between h and R2 is if h, R1 and R2 are configured with IP addresses from within the same subnet prefix and within the same subnet prefix.

Returning to the MANET scenario in figure 8, with all MANET interfaces being configured with the same subnet prefix and the same prefix length, it follows from the discussion above that all these MANET interfaces are expected to be directly reachable from each other within a single IP hop. When, in this configuration, the router 192.168.1.2 is requested to forward a data packet from 192. Rather than "treating the symptoms" and disabling ICMP redirects, requiring /32 and /128 prefix lengths on MANET interfaces "cures the disease". An interface so configured not make any assumptions about other interfaces being within a single IP hop, and so will not generate ICMP redirects when forwarding traffic.

An argument could be made, suggesting that for each set of MANET interfaces which are all directly reachable in one IP hop from each other, a shared "subnet prefix" can be assigned. Considering the network from figure 8, this could allow two "subnets" to be formed, as indicated in figure 9, with the MANET interface of the center router being configured with two IP addresses (or, being configured with two virtual interfaces, each with one IP address), one in each of the two subnets.

While this could be a valid configuration, considering that the "M" in MANET is for "mobile", the network topology can be expected to change dynamically and frequently. A network so configured would, therefore, require constant "subnet formation" and "interface reconfiguration", in order to maintain the configuration valid. It would require (complex) mechanisms for tracking the topology dynamics and would entail potentially frequent changes to MANET interface address configurations. Such frequent changes of interface configurations would likely also adversely affect e.g. relay selection mechanisms such as those used for flooding in [START_REF] Clausen | RFC3626: Optimized Link State Routing Protocol[END_REF] and [START_REF] Clausen | RFC5449: OSPF Multipoint Relay (MPR) Extension for Ad Hoc Networks[END_REF]: from the point of view of a router R, even if the topology within a 2 hop radius from R did not physically change (and so, no relay set recalculations Figure 9: MANET Interfaces configured with a shared /24 subnet prefixes among neighbors which can all communicate directly in one IP hop. Notice that some MANET interfaces will be required to have multiple IP addresses, and as the network topology changes, also change their MANET interface configuration. would be needed), the logical configuration of interface addresses on a node 2 hops away from R might change (e.g. due to physical topology changes to the neighborhood of that node, i.e. 3 hops away from R) in order to preserve the ability of having "shared prefixes" among neighboring MANET interfaces. This would then potentially require relay set recalculations by R and additional signaling to this effect by a routing protocol on R -even though topological changes were 3 hops away from R and such recalculations and signaling would not be necessary in case MANET interface addresses were kept stable.

Finally, as the only application(s) to be exposed to MANET interfaces are MANET routing protocols and other protocols explicitly MANET aware and designed for management of a MANET infrastructure, it is highly questionable if there would be any benefits from introducing the complexities necessary for maintaining such dynamically changing "subnets" among MANET interfaces.

Configuring MANET interfaces with a /32 or /128 prefix length is, therefore, a safe approach which entails no additional complicated mechanisms or signaling, and which requires no changes to the IP stack.

INRIA

IP Hop Isolation

Upper-layer applications and protocols are designed with a set of (often not explicitly expressed) assumptions as to the nature and characteristics of the underlying network communications fabric. This includes, but is not limited to, assumptions on (i) the relationship between a "subnet" and the ability for interfaces configured within the same subnet to all be able to communicate with each other directly, i.e. in one IP hop (as evoked in section 6 and section 7), (ii) that communication between interfaces on the same link is symmetric (if interface a can receive packets from interface b, then interface b can receive packets from interface a), (iii) that link-local multicast (i.e. with TTL or hop-limit equal to 1) is supported and is well defined (e.g. that the recipients of a link local multicast from interface a is identical to those of interface b, assuming that interfaces a and b are on the same link). Essentially, upper-layer applications and protocols assume an underlying link with characteristics similar to those of an Ethernet.

Over MANET interfaces, these assumptions may not hold true. Consider for example the network in figure 10. If interface a makes a link local multicast transmission, then this is received by the interface b -whereas if interface b makes a link local multicast transmission, then this is received by the interfaces a and c. There are, essentially, four potential ways of addressing this problem: requiring all upperlayer applications and protocols to become "MANET aware", inventing mechanisms for presenting a MANET as-if it was an Ethernet, pushing the problem to layer 2, or encapsulating any MANET specific behavior in a way such that it is only exposed to explicitly MANET aware applications and protocols.

Requiring all upper-layer applications and protocols to be reworked such that they be MANET aware is impossible, due to the sheer volume of such, is undesirable as a tenant of the Internet is to provide an abstraction for applications from the interconnect characteristics, and would likely, not be without technical problems of its own -for example, section 7 exposes the complexity with respect to the single problem of correctly ensuring the relationship between "subnet" and which MANET interfaces are able to directly communicate. Pushing problem to layer 2, and requiring a layer 2 to present an "Ethernet-like" medium would for a given layer 2 solve the problem -but do away with the ability to deploy heterogeneous MANETs, and would be akin to e.g. requiring OSPF to do away with all but one interface type and require all layer 2's over which OSPF is expected to run to adhere to this single remaining interface type.

The last way of addressing this problem is to encapsulate MANET specific behaviors in a way such that it only is exposed to explicitly MANET aware applications and protocols. This approach has several merits, not the least of which is that this is exactly the way in which existing routing protocols are deployed: interfaces between routers, and their characteristics, are exposed only to applications and protocols which are explicitly aware of these interfaces and their characteristics, with all other applications and protocols being isolated from these by being connected via an "Ethernet-like" link to the router -i.e. by virtue of being isolated behind an IP hop (the router).

For example, applications on hosts are exposed to an "Ethernet-like" link with other hosts and potentially routers -and are blissfully unaware whether these routers are connected using point-to-point links, NBMA links, broadcast links etc. For these reasons, configuring a MANET such that the MANET router is isolating the MANET interfaces and their characteristics from hosts and other (non-MANET) networks by an IP hop, and requiring that on MANET routers, only explicitly MANET aware applications are exposed to MANET interfaces, is a safe and simple approach that entails no additional complex mechanisms and signaling. This also requires no changes to neither the IP stack nor the upper-layer applications and protocols operating on these hosts and (non-MANET) networks.

INRIA

Part III

Summary and Characteristics

This memorandum proposes recommendations as to how MANETs can be configured and deployed, specifically, it provides the following three sets of recommendations:

MANET Interface Address Configuration:

❼ Each MANET interface on a MANET router MUST be configured with an IP address which is unique, at least within the MANET.

❼ Each MANET interface MUST be configured with a prefix length of /32 (for IPv4) or /128 (for IPv6).

Delegated Prefix Configuration:

❼ If a prefix is delegated to a MANET router, this prefix MUST NOT be configured on any MANET interfaces.

❼ If a prefix is delegated to a MANET router, other (non-MANET) interfaces MAY be configured with this prefix.

MANET Interface Characteristics Encapsulation:

❼ MANET Interface characteristics are exposed only to explicitly MANETaware protocols and applications, such as MANET routing protocols;

❼ MANET Interfaces are not exposed to applications and upper-layer protocols, i.e.;

❼ MANET Routers provide connectivity to hosts and other (non-MANET) networks via links with "Ethernet-like" link characteristics, and;

❼ MANET Routers provides connectivity from hosts and other (non-MANET) networks via IP routing, i.e. MANET interface characteristics are "hidden behind an IP hop" from applications and upper-layer protocols.

A MANET configured according to these recommendations has the following key characteristics:

MANET Characteristics:

❼ It requires no modifications to the IP stack on hosts and non-MANET aware networks;

❼ It requires no modifications to the IP stack on MANET routers;

❼ It requires no modifications to upper-layer applications and protocols on hosts and (non-MANET) routers;

❼ It supports all MANET routing protocols, as currently developed within the IETF.

INRIA

Figure 1 :

 1 Figure 1: MANET "node" morphology

Figure 2 :

 2 Figure 2: A MANET with routers (R) and hosts (H)

Figure 3 :

 3 Figure 3: MANET router (R) and a single internal host (H)

Figure 4 :

 4 Figure 4: A correctly configured MANET. IP addresses within the routers are those assigned to the MANET interface. Notice that while all such IP addresses are extracted from 192.168.0.0/16, all MANET interfaces are configured with prefix lengths of /32.

Figure 5 :

 5 Figure 5: MANET router (R) with an attached network and a delegated subnet prefix.Notice that the prefix 192.168.1.0/24 is assigned to the non-MANET link, where interfaces are configured with that prefix, e.g. 192.168.1.3/24. The MANET interface "borrows" the IP address from that link, however is configured with a prefix length of /32.

Figure 6 :

 6 Figure 6: MANET router (R) with an attached network and a delegated subnet prefix and two MANET interfaces. Notice that the prefix 192.168.1.0/24 is assigned to the non-MANET link, where interfaces are configured with that prefix, e.g. 192.168.1.3/24. The MANET interface "borrows" the IP address from that link, however is configured with a prefix length of /32. The second MANET interface is configured using an unused IP address from within the same subnet prefix, in this case 192.168.1.42 -also configured with a prefix length of /32

Figure 7 :

 7 Figure 7: Example of "semi-broadcast" MANET interface characteristics. The gray circles indicate the transmission range for each MANET interface, i.e. the range within which a transmitted packet can be successfully received and decoded.

Figure 8 :

 8 Figure 8: Misconfigured MANET: MANET Interfaces configured with a shared /24 prefix, causing the central router to produce an ICMP redirect when forwarding packets from 192.126.1.1 to 192.168.1.3. If, for example, a data packet was transmitted by the MANET router 192.168.1.1 to be received by 192.168.1.3, then this would -with reference to figure 8 -have to be forwarded by the MANET router 192.168.1.2. With the MANET interfaces in this MANET being configured with the subnet prefix 192.168.1 and the prefix length /24, it was observed that this produced ICMP redirects by 192.168.1.2.

  168.1.1 to 192.168.1.3, it is expected that it generates an ICMP redirect to 192.168.1.1 suggesting that a direct path exists from 192.168.1.1 to 192.168.1.3 -as this is what the configuration suggests.

Figure 10 :

 10 Figure 10: Link Local Multicast over MANET interfaces: without forwarding, a transmission from the manet interface a reaches a different set of interfaces than those for a transmission from interface b (or interface c).

  5.2 MANET Router and Attached NetworkFigure5illustrates a MANET router with a single MANET interface and an interface towards a classic IP link such as an Ethernet. The MANET router is delegated the IPv4 prefix 192.168.1.0/24. That prefix is assigned to the non-MANET link, on which interfaces are assigned addresses such as 192.168.1.1/24, 192.168.1.2/24, 192.168.1.3/24 etc. Note that the interfaces on that classic IP link are configured with a prefix length of /24, indicating that the interfaces with addresses from within that IP prefix are all reachable within a single IP hop.

  1.42 -from within the 192.168.1.0/24 prefix. This interface is also configured with a prefix length of /32.

R H H H 192.168.1.0/24 192.168.1.1/24 192.168.1.2/24 192.168.1.3/24 192.168.1.4/24 192.168.1.5/24 H 192.168.1.1/32 192.168.1.42/32

  : the MANET would have a subnet prefix, say 192.168.1.0/24 and MANET interfaces would be configured with this subnet prefix, e.g. 192.168.1.3/24.

	192.168.1.1	192.168.1.2	192.168.1.3
	/24	/24	/24

Data packet: dest 3, next-hop 2 ICMP Redirect

RR n➦ 6852

Acknowledgements

The considerations presented in this memorandum are the authors condensation of years of experience within and outside the IETF, from studying, building, testing and deploying MANETs.

The author would like to express his profound thanks to (in no specific order) Dave Thaler (Microsoft), Thomas Narten (IBM), Jari Arkko (Ericsson), Mark Townsley (Cisco), Chris Dearlove (BAE Systems), Justin Dean (NRL), Joe Macker (NRL), Ian Chakeres (CenGen), Charles E. Perkins (WiChorus), Ulrich Herberg (LIX), Emmanuel Baccelli (INRIA), Henning Rogge (FGAN) and Ryuji Wakikawa (Toyota) for having given their valuable time to in the (at time vivid) discussions with the author, and forming the foundation for writing this present memorandum.

Ulrich Herberg (LIX) and Emmanuel Baccelli (INRIA), both, provided valuable proofreading of this memorandum in its final stages. Emmanuel also provided the French translation of the abstract.