Rawan Ghnemat

Cyrille Bertelle

Gérard H E Duchamp

Agent-Based Modeling Using Swarm Intelligence in Geographical Information Systems

In this paper swarm intelligence algorithms are presented to deal with dynamical and spatial Organization emergence. The goal is to model and simulate the development of spatial center and their dynamic interactions with the environment and the individuals; the swarm algorithms used are inspired from natural termite nest building and Ant culturing algorithm. Combination of decentralized approaches based on emergent clustering mixed with spatial multi criteria constraints or attractions developed , extension of termite nest building algorithms has been proposed to have multi center adaptive process. The modeling has been made using agent based modeling techniques and the simulation developed using Repast (REcursive Porous Agent Simulation Toolkit) and OpenMap as geographical information system (GIS) software, some simulations result are provided.

INTRODUCTION

Many natural and artificial systems have emergent properties based on spatial development. This spatial development is both the result of some mecanisms from the system behavior and the actor of the system formation by morphogenetic feedbacks [START_REF]Complex Systems and Self-Organization Modelling[END_REF]. Natural ecosystems or urban dynamics are typically based on systems which evolve mainly under the influence of such spatial emergent organizations. The goal of this paper is to study some models and algorithms able to describe such spatial selforganization processes, taking into account the complexity of the phenomena. Decision makers, like urban planners, need a better understanding of such a dynamics in order to adapt the infrastructure development according to the feed-back of citizen use. Simulation allows analysis and predictive studies.

Benenson [START_REF] Benenson | Modeling population dynamics in the city: from a regional to a multi-agent approach[END_REF] proposes to classify urban modelling in two classes: macro-modelling and individual-based modelling. The first one fails in accurate individual and emergent phenomena modelling. In the second one, cellular automata are often used to describe rule-based systems for diffusion phenomena description on regular grid space. The proposed model, based on swarm intelligence, allows to model dynamical emergence on more general spaces described by geographical information system.

In section 2, complex systems concepts are defined and their applications to urban dynamics understanding are described, then swarm intelligence algorithms are described as methodologies to implement the complex systems concepts, using distributed computations. In section 3, we propose some specific swarm intelligence methods based on ant systems in order to model the spatial organizations emergence and we give some experiments using RePast multi-agent platform mixed with a geographical information system.

MODELLING SPATIAL COMPLEXITY

Complex System Concepts

Complex system theory [START_REF] Le Moigne | La modélisation des systèmes complexes[END_REF] is based on the fact that for many applicative domains, we can find similar processes linking emergent global behavior and interaction network of constituents. The global behavior is generally not accessible using classical analytical methods.

In classical analytical methods, the global behavior of the system is the description of the equations. Simulations from these formulations, consist in obtaining the trajectories of the behavior predefined in the equation formulation.

In complex systems modelling, we have to model the constituents of the system and the interaction network or system which links these constituents, using a decentralized approach. So the global behavior of the system cannot be understood by the description of each constituent. In complex systems modelling, the global behavior is an emergent property from the interaction network or systems between its constituents and leads to the creation of an dynamical organization of these constituents This dynamical and emergent organization retro-acts on its own components. In many natural and artificial systems, the environement has some spatial effects which interact on the whole complexity of the phenomenon. This spatial environment can be modified by the system but he can also be the catalyst of its own evolution. Understanding and modelling the deep structural effect of the interaction between the systems and its spatial environment is the goal of the study presented in the sequel.

Application to urban dynamics

Social and human developments are typical complex systems. Urban development and dynamics are the perfect illustration of systems where spatial emergence, selforganization and structural interaction between the system and its components occur. In Figure 1, we concentrate on the emergence of organizational systems from geographical systems. The continuous dynamic development of the organization feed-back on the geographical system which contains the organization components and their environment. For the organization detection, we use swarm intelligence processes. We model the feed-back process of this emergent organization on the system constituents and its environment. To analyse or simulate urban dynamics, nowadays, we can use the great amount of geographical databases directly available for computational treatment within Geographical Information Systems. On the organizational level description, the new development of multiagent systems (MAS) allows nowadays to develop suitable models and efficient simulations.

The applications we focus on in the models that we will propose in the following concerns specifically the multi-center (or multi-organizational) phenomona inside urban development. As an artificial ecosystem, the city development has to deal with many challenges, specifically for sustainable development, mixing economical, social and environmental aspects. The decentralized methodology proposed in the following allows to deal with multi-criteria problems, leading to propose a decision making assistance, based on simulation analysis.

Cultural dynamics processes in urban area can be modelled using such a methodology. It is typically a multicriteria self-organization process where appears emergent usage of urban equipment. A modelling of these dynamics is presented latter in this paper.

Swarm Intelligence and Spatial Environment

Decentralized algorithms have been implemented for many years for various purposes. In this algorithm category, multi-agent systems can be considered as generic methods [START_REF]Multiagent Systems[END_REF]. We deal, in this paper, with agents based on reactive architecture and expected to be used inside numerous entity-based systems. The aim of programs using such architectures, is to deal with emergent organizations using specific algorithms called emergent computing algorithms. Swarm Intelligence is the terminology used to point out such reactive agent-based methods where each entity is built with the same basis of behavior, but reacts in an autonoumous way.

Based on swarm intelligence concepts, Ant systems [START_REF] Bonabeau | Swarm Intelligence, from natural to artificial systems[END_REF] propose some bio-inspired method from social insects mecanisms. For example, Ant Colony Optimization (ACO) methods is such a family of methods where the basic entities are virtual ants which cooperate to find the solution of graph-based problems, like network routing problems, for example. Using indirect communications, based on pheromon deposites over the environment (here a graph), the virtual ants react in elementary way by a probabilistic choice of path weighted with two coefficients, one comes from the problem heuristic and the other represents the pheromon rate deposit by all the ants until now. The feed-back process of the whole system over the entities is modelled by the pheromon action on the ants themselves.

Particule Swarm Optimization (PSO) is a metaheuristic method initially proposed by J. Kennedy and R. Ebenhart [START_REF] Kennedy | Particle Swarm Optimization[END_REF] which is also based on swarm intelligence concepts. This method is initialized with a virtual particle set which can move over the space of solutions corresponding to a specific optimization problem. The method can be considered as an extension of a bird flocking model, like the BOIDS simulation from C.W. Reynolds [START_REF] Reynolds | Flocks, Herds and Schools: a distributed behavioral model In Computer Graphics[END_REF]. In PSO algorithm, each virtual particle moves according to its current velocity, its best previous position and the best position obtained from the particles of its neighborhood. The feed-back process of the whole system over the entities is modelled by the storage of this two best positions as the result of communications between the system entities.

Other swarm optimization methods have been developped like Artificial Immune Systems [START_REF] De Castro | Artificial immune system: a new computational approach[END_REF] which is based on the metaphor of an immune system as a collective intelligence process. F. Schweitzer proposes also a generic method based on distributed agents, using approaches of statistical many-particle physics [START_REF] Schweitzer | Brownian Agents and Active Particles[END_REF].

The method proposed in this paper is based on Ant systems, allowing to deal with self-organization processes emerging from spatial constraints and attractive areas.

MULTI-CENTER ANT NEST BUILDING TO MODEL EMERGENT SPATIAL OR-GANIZATIONS

We will describe in this section, the general algorithm which is proposed to model emergent spatial organizations. This algorithm is based on the ant clustering. We introduce pheromon template to spatially control the clustering from local attraction. This method is a decentralized approach which allows to combinate multi-center and multi-criteria problems and we will show how we can apply it to model cultural dynamics in urban areas.

Ant clustering

Ant clustering algorithms are inspired by the corposes or larvea classification and aggregation that the ants colony are able to do in the real life. The ants are moving inside a closed area and are able to move some material which are randomly put on this area. After a while, and without any kind of centralized coordonation, the ants success to create some material clusters.

The algorithm is based on the following and very simple behavioral rules that each ant implements :

• When an ant is moving without carrying yet material and finds some material, the ant will take the material respecting the probability number :

P p = k 1 k 1 + f 2 (1)
where f is the material density that the ant perceives locally around itself and k 1 is the treshold. It is easy to check that if f << k 1 then P p is near the value 1 and if f >> k 1 then P p is near the value 0.

• When an ant is moving when carrying some material, the probability to deposit it is computed by :

P d = f k 2 + f 2 (2)
where f is still the material density that the ant perceives locally around itself and k 2 is another treshold. It is easy to check that if f << k 2 then P d is near the value 0 and if f >> k 2 then P d is near the value 1.

Spatial constraints using template

The ant clustering shows some spatial self-organizations but has the specificity to generate clusters at random places. According to the first random moves that the ants starts to do in the beginning of the algorithm, some material will initiate aggregation and the clustering processus will complete this aggregation from these initial random first aggregations. To simulate some urban dynamics, we need to introduce specific location with respect to city center for example or cultural equipments. The clustering here will represent the people usage of these centers or equipments and we need to introduce an attractive effect by using a pheromon template. This method follows the algorithm known as Ant Nest Building [START_REF] Bonabeau | Swarm Intelligence, from natural to artificial systems[END_REF]. In ant colonies, the center corresponds to the position of the queen which needs to build the nest and the ant colony moves around it to protect the nest by various material taken on the ground. The queen emits a pheromon which allows to attract the ants during their building. The ant has to deposit the material carried only if the pheromon quantity perceived belongs to a specific range. We use an attractive fonction called P t , corresponding to a pheromon template and represented by the part (a) of the Figure 2.

Using this template function, we remplace in the clustering algorithm, the two provious probabilities defined in equation [START_REF] Benenson | Modeling population dynamics in the city: from a regional to a multi-agent approach[END_REF] and equation (2) by In Figure 2, we show an implementation of this algorithm using the multi-agent platform called Repast [START_REF]Repast web site[END_REF]. The java version of this platform includes some packages allowing to interface with geographical databases and geographical information systems (GIS). The graphical output windows is made under OpenMap which is a GIS developped in Java. In this Figure, the materials moved by the ants are the small grey circles, the ants moving without material are the green circles, the ants carrying material are the red circles and the queen location is the yellow circle.

P ′ p = P p (1 -P t ) (3)
P ′ d = P d P t (4)

Multi-template modelling

The previous subsection describes one local attractive process characterized by the queen and its pheromon template emission. The advantage of this method is to be able to combine the solutions of multi-center and multi-criteria problems, using interactive processes, each Figure 3 shows the simulation with two queens and two pheromon templates. It is possible also for each queen to emit many different kinds of pheromons: we called them colored pheromons. Each colored pheromon will attract only the ants associated to its color.

Application to cultural equipment dynamics

The multi-template modelling can be used to model cultural equipment dynamics as described in Figure 4. On this Figure, we associate to each cultural center (cinema, theatre, ...) a queen. Each queen will emit many pheromon templates, each template is associated to a specific criterium (according to age, sex, ...). Initially, we put the material in the residential place. Each material has some characteristics, corresponding to the people living in this residential area. The simulation shows the spatial organization proces-

Conclusion and Perspectives

The paper develops some specific swarm intelligence algorithms based on ant colonies processes. Using such decentralized methods, we can model complex multi-center and multi-criteria self-organizations. Urban dynamics are one of the most relevant problems where these approaches can be efficient. This method can be used for any kind of urban infrastructure where the mechanism of using this infrastructure is complex like telecommunication infrastructures for example.

Figure 1 .

 1 Figure 1. Spatial organizations Complexity Description and the Conceptual Generic Model Based on Swarm Intelligence

Figure 2 .

 2 Figure 2. Ant nest building with one center using RePast MAS paltform over OpenMap GIS

Figure 3 .

 3 Figure 3. Multi-center ant nest buiding using RePast over OpenMap

Figure 4 .

 4 Figure 4. Cultural Equipment Dynamics Modelling


Leila Amgoud
email: amgoud@irit.fr

Five weaknesses of ASPIC +

This paper analyzes the ASPIC+ argumentation system. It shows that it is grounded on two monotonic logics which are not Tarskian ones. Moreover, the system suffers from five main problems: i) its logical formalism is ill-defined, ii) it may return undesirable results, iii) it builds on some counter-intuitive assumptions, iv) it violates some rationality postulates, and v) it allows counterintuitive instantiations.

Introduction

An argumentation system consists of a set of arguments and an attack relation among them. A semantics is applied for evaluating the arguments. It computes one (or more) set(s) of acceptable arguments, called extension(s). One of the most abstract systems is that proposed by Dung in [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF]. It leaves the definition of the notion of argument and that of attack relation completely unspecified. Thus, the system can be instantiated in different ways for reasoning about defeasible information stored in a knowledge base. An instantiation starts with an underlying logic which is a pair (L, CN). The part L represents the logical language in which the information of the knowledge base are encoded. It is thus, a set of well-formed formulas. The second component of a logic is its consequence operator CN. It represents the reasoning patterns that are used. In an argumentation system, CN is used for generating arguments from a knowledge base and also for defining attacks among arguments. It is worth mentioning that in almost all existing argumentation systems, the underlying logic (L, CN) is monotonic (see [START_REF] Hunter | Base logics in argumentation[END_REF]). This makes the construction of arguments monotonic, that is an argument remains an argument even when the knowledge base is extended with arbitrary information. However, the status of an argument may change. Consequently, the logic produced by the argumentation system is nonmonotonic. In sum, an argumentation system for defeasible reasoning is grounded on a monotonic logic and produces a nonmonotonic one.

Recently, Prakken proposed an instantiation of Dung's system [START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF], called ASPIC + . It is an extended version of ASPIC system which is developed in [START_REF] Amgoud | Towards a consensual formal model: inference part[END_REF]. ASPIC + takes as input an unspecified set of formulas, a contrariness function between formulas and two sets of rules: strict rules and defeasible ones. These rules may either represent knowledge (like penguins do not fly, generally birds fly) or inference rules (like modus ponens). From these inputs, arguments are built and attacks between arguments are specified. Thus, the only parameter which is somehow left unspecified is the underlying logic. In [START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF], it is claimed that this system satisfies the rationality postulates defined in [START_REF] Caminada | On the evaluation of argumentation formalisms[END_REF]. In [START_REF] Modgil | Revisiting preferences and argumentation[END_REF], the authors claimed that the logic underlying ASPIC + is too general that it captures even Tarskian monotonic logics [START_REF] Tarski | Logic, Semantics[END_REF]. It is sufficient to assume that strict rules conform to a Tarskian consequence operator (see the citation below):

"If the strict rules in a c-SAF conform to a Tarskian consequence operator, then it should be obvious to see that the cSAF is c-classical".

Our aim in this paper is to investigate the underpinnings of this system, especially since there are no formal results proving some claims like the previous one on Tarski's logics. We study two main issues: the properties of ASPIC+ and the kind of monotonic logics underlying this system.

Regarding the first objective, we show that ASPIC+ suffers from serious technical problems. Namely, the basic concepts are ill-defined. This is particularly the case of the logical formalism on which the system is built. Consequently, counter-intuitive results may be given as outputs of the system, and the rationality postulates proposed in [START_REF] Caminada | On the evaluation of argumentation formalisms[END_REF] may be violated. We also show that the system is grounded on counter-intuitive assumptions leading thus to unjustified conclusions. The last weakness of the system consists of allowing counter-intuitive instantiations. Indeed, unlike what is claimed in [START_REF] Modgil | Revisiting preferences and argumentation[END_REF], the logical formalism of ASPIC+ is unable to capture classical logics. The main problem comes from the definition of consistency which is too poor.

In a second part of the paper, we investigate the monotonic logic underlying AS-PIC+. Prakken claims that strict and defeasible rules may play two roles: either they encode information of the knowledge base, in which case they are part of the language L, or they represent inference rules, in which case they are part of the consequence operator CN. In this paper, we define precisely the two corresponding logics, and show that they are monotonic but are not among the Tarskian ones. Moreover, we show that when rules encode information, then the system is fully instantiated apart from the kind of formulas that may be stored in L. This unique generality is fatal for the system since it may lead to irrational systems. When the rules are part of the consequence operator, the logic is far from capturing Tarski's logics [START_REF] Tarski | Logic, Semantics[END_REF], contrarily to what is claimed in [START_REF] Modgil | Revisiting preferences and argumentation[END_REF]. Indeed, not only it is unable to capture the monotonic logics that do not allow negation in their language, but it is also unable to capture even those which do have negation.

The paper is structured as follows: In section 2, we recall ASPIC + and show its limits. Section 3 recalls the monotonic logics of Tarski. In section 4, we discuss deeply the case where rules encode knowledge, while in section 5 we study the case where they encode inference rules. The last section is devoted to some concluding remarks.

APIC + argumentation framework

In [START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF], Prakken has proposed an instantiation of Dung's framework, called ASPIC + . It considers an abstract logical language L, that is a language which may be instantiated in different ways. It may for instance contain propositional formulas, etc. The only requirement on L is that it is equipped with a notion of contrariness, denoted by ¯.

Definition 1 (Contrariness). Let L be a logical language and ¯is a contrariness function from L to 2 L . For x, y ∈ L, if x ∈ ȳ, then if y / ∈ x then x is called a contrary of y, otherwise x and y are called contradictory.

Remark 1 It is worth mentioning that the above definition of contrariness does not capture the real intuition of contrary as discussed by several philosophers and logicians (e.g. [START_REF] Béziau | New light on the square of oppositions and its nameless corner[END_REF]). Indeed, a formula x is a contrary of y iff they cannot be both true but they can both be false. They are contradictory if the truth of one implies the falsity of the other and vise versa. Let us consider the following simple example:

Example 1 Assume that L = {>, ≥, <, ≤, =, =}. For instance, > = {<, ≤, =} and < = {>, ≥, =}. Note that > is the contrary of < and vise versa and <∈ > and >∈ <. According to Def. 1, > and < are contradictory while this is not the case.

In addition to the language L, two kinds of rules are assumed: strict rules and defeasible ones.

Definition 2 (Rules). Let x 1 , . . . , x n , x be elements of L. A strict rule is of the form x 1 , . . . , x n → x meaning that if x 1 , . . . , x n hold then without exception x holds. A defeasible rule is of the form x 1 , . . . , x n ⇒ x meaning that if x 1 , . . . , x n hold then presumably x holds. R s (resp. R d ) stands for the set of all strict (resp. defeasible) rules with R s ∩ R d = ∅.

Remark 2 In [START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF], no restrictions neither on the formulas of L nor on the rules are imposed. Consequently, the two connectors → and ⇒ may be used in elements of L. Thus, rules can be defined in several ways as shown in the next example. A notion of consistency is associated to this logical formalism as follows.

Definition 3 (Consistency

). A set X ⊆ L is consistent iff x, y ∈ X such that x ∈ ȳ, otherwise it is inconsistent.
This notion of consistency was used in ASPIC system [START_REF] Amgoud | Towards a consensual formal model: inference part[END_REF]. While it is sufficient in that system since the language L contains only literals, it is too poor when considering an arbitrary language. Indeed, it only captures binary conflicts between formulas. That is, it does not capture ternary or more conflicts. Let us consider the following example:

Example 3 Let L contain propositional formulas and let ¯stand for classical negation. The set X = {x, x → y, ¬y} is consistent wrt Definition 3 (whereas it is inconsistent wrt propositional logic).

Remark 3

This example shows clearly that this formalism cannot capture classical logics (e.g. propositional logic, first order logic, ...) contrarily to what is claimed in [START_REF] Modgil | Revisiting preferences and argumentation[END_REF]. For instance, even if the set of strict rules represent all the valid inference mechanisms of propositional logic, their notions of consistency are different as witnessed by the above example. One formalism declares the set X as consistent while the other declares it inconsistent.

In [START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF], arguments are built from a knowledge base K. It may contain four categories of information: axioms (which are certain information) (K n ), ordinary premises (K p ), assumptions (K a ) and issues (K i ). The set K n is assumed to be consistent. These subbases are disjoint and

K = K n ∪ K p ∪ K a ∪ K i .
Remark 4 In [START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF], it is claimed that strict and defeasible rules may encode respectively certain and defeasible information. In this case, it is clear that strict rules and axioms (elements of K n ) represent the same kind of information. Similarly, ordinary premises and defeasible rules refer to the same kind of information. Thus, the system allows some redundancies and this makes its instantiation by users unclear.

In what follows, for a given argument, the function Prem (resp. Conc, Sub, DefRules, TopRule) returns all the formulas of K which are involved in the argument (resp. the conclusion of the argument, its sub-arguments, the defeasible rules that are used, and the last rule used in the argument).

Definition 4 (Argument).

An argument A is:

-x if x ∈ K with Prem(A) = {x}, Conc(A) = x, Sub(A) = {A}, DefRules(A) = ∅, TopRule(A) = undefined. -A 1 , . . . , A n -→ x 1 (resp. A 1 , . . . , A n =⇒ x) if A 1 , . . . , A n , with n ≥ 0, are
arguments such that there exists a strict rule Conc(A 1 ), . . . ,

Conc(A n ) → x (resp. a defeasible rule Conc(A 1 ), . . . , Conc(A n ) ⇒ x). Prem(A) = Prem(A 1 ) ∪ . . . ∪ Prem(A n ), Conc(A) = x, Sub(A) = Sub(A 1 ) ∪ . . . ∪ Sub(A n ) ∪ {A}, DefRules(A) = DefRules(A 1 ) ∪ . . . ∪ DefRules(A n ) (resp. DefRules(A) = DefRules(A 1 ) ∪ . . . ∪ DefRules(A n ) ∪ {Conc(A 1 ), . . . Conc(A n ) ⇒ x}), TopRule(A) = Conc(A 1 ), . . ., Conc(A n ) → x (resp. TopRule(A) = Conc(A 1 ), . . ., Conc(A n ) ⇒ x). An argument A is strict if DefRules(A) = ∅, defeasible if DefRules(A) = ∅, firm if Prem(A) ⊆ K n , and plausible if Prem(A) K n .
Arguments may not have equal strengths in [START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF]. Thus, a partial preorder denoted by is available. For two arguments, A and B, the notation A B means that A is at least as good as B. The strict version of is denoted by . This preorder should satisfy two basic requirements: The first one ensures that firm and strict arguments are stronger than all the other arguments, while the second condition says that a strict rule cannot make an argument weaker or stronger.

Definition 5 (Admissible argument ordering).

Let A be a set of arguments. A partial preorder on A is an admissible argument ordering iff for any pair A, B ∈ A: R 1 : if A is firm and strict and B is defeasible or plausible, then A B. R 2 : if A = A 1 , . . . , A n → x, then ∀i = 1, n, A i A and for some i = 1, n, A A i .

Remark 5

The previous requirements reduce the generality of the preference relation. For instance, requirement R 1 is violated by the preference relation proposed in [START_REF] Bench-Capon | Persuasion in practical argument using value-based argumentation frameworks[END_REF]. In that paper, each argument promotes a value and the best argument is the one which promotes the most important value. Assume that there are two arguments A and B which promote respectively values v 1 and v 2 with v 1 being more important than v 2 . If we assume that A is defeasible and B is both strict and firm, then from requirement R 1 , B should be preferred to A. However, in [START_REF] Bench-Capon | Persuasion in practical argument using value-based argumentation frameworks[END_REF], A is strictly preferred to B.

Since information in a knowledge base K may be inconsistent, arguments may be conflicting too. In [START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF], Prakken has modeled three ways of attacking an argument: undermining its conclusion, or one of its premises or one of its defeasible rules. The following definition reports exactly Definition 6 (Attacks). Let A and B be two arguments.

-A undercuts B (on C) iff Conc(A) ∈ C with C ∈ Sub(B) and of the form C 1 , . . . , C n =⇒ x. -A rebuts B (on C) iff Conc(A) ∈ x for some C ∈ Sub(B) of the form C 1 , . . . , C n =⇒ x. A contrary-rebuts B iff Conc(A) is a contrary of x. -A undermines B iff Conc(A) ∈ x for some x ∈ Prem(B) \ K n . A contrary- undermines B iff Conc(A) is a contrary of x or if x ∈ K a .
Remark 6 Note that the first relation (undercut) applies the notion of contrariness on an argument. This is technically wrong since contrariness ¯is a function defined for formulas of the language L and arguments should not be elements of L. In informal discussions with Prakken, he claimed that each defeasible rule has a name which is a formula of L. Thus, the notation Conc(A) ∈ C means that the conclusion of argument A is the contrary of the name of the defeasible rule C 1 , . . . , C n =⇒ x. However, the above definition does not capture this intuition since the notion of name of a rule is not defined. Note that even if this intuition is correctly defined, ASPIC+ does not make any difference between formulas of L that encode information and formulas that encode names of rules. This may lead to undesirable results as shown on the following example.

Example 4 Let K n = {b}, R d = {b ⇒ f } where b stands for "Bird" and f for "Generally, birds fly". Assume that the name of the rule b ⇒ f is ¬f (nothing in the definitions forbids this assumption). Note also that in the formal definition, it is not specified in which set K j (j ∈ {n, p, i, a}) the names of rules should we stored. Assume that these names are not stored in the four bases. Thus, we get two arguments:

A: b B: b, b ⇒ f
The argument B is self-defeating (it undercuts itself). Thus, this system has one preferred extension {A}, and it returns a unique output which is b. The system is unable to conclude that this bird flies.

Let us now assume that the name of the rule is a certain information that is stored in K n , thus K n = {b, ¬f }. Note that this set contains two information of different nature. In this case, there is an additional argument C:

C: ¬f

It can be checked that C rebuts B. The system has one preferred extension {A, C} and returns the set {b, ¬f } as its output. This latter is certainly not intuitive.

Remark 7 Besides, in [START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF], the author claims that the three attack relations are syntactic and do not reflect any preference between arguments. However, in the definition of rebut, it is clear that an argument whose top rule is strict cannot be attacked by an argument with a defeasible top rule. Thus, the former is preferred to the latter. Moreover, this preference is not intuitive as shown by the following example.

Example 5 Assume an argumentation system with the two following arguments: A :⇒ p, p ⇒ q, q ⇒ r, r → x B :→ d, d → e, e → f, f ⇒ ¬x. According to the above definition, A rebuts B but B does not rebut A. Thus, A is assumed stronger than B. This is not intuitive since there is a lot of uncertainty on r. Works on non-monotonic reasoning would rather prefer B to A since defeasible rules violate transitivity and thus, p may not be r.

As in any preference-based argumentation system, in [START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF] preferences between arguments are used in order to decide which attacks result in defeats.

Definition 7 (Defeat).

Let A and B be two arguments.

-A successfully rebuts B if A rebuts B on C and either A contrary-rebuts C or not(C A). -A successfully undermines B if A undermines B on x and either A contraryundermines B or not(x A). -A defeats B iff no premise of A is an issue and A undercuts or successfully rebuts or successfully undermines B.

Remark 8 Note that the previous definition uses preferences only when the attack relation is symmetric. While this avoids the problem of conflicting extensions described in [START_REF] Amgoud | A new approach for preference-based argumentation frameworks[END_REF], it greatly restricts the use of preferences.

The instantiation of Dung's system is thus the pair (A s , defeat) where A s is the set of all arguments built using Definition 4 from (K, Cl(R s ), R d ). Cl(R s ) is the closure of strict rules under contraposition, that is Cl(R s ) contains all strict rules and all their contrapositions. A contraposition of a strict rule x 1 , . . . , x n → x is, for instance, the strict rule x 2 , . . . , x n , x → x1 , where x and x are contradictory. Recall that in [START_REF] Caminada | On the evaluation of argumentation formalisms[END_REF], contraposition was proposed for ensuring the rationality postulate on consistency.

Dung's acceptability semantics are applied for evaluating the arguments.

Definition 8 (Acceptability semantics). Let

B ⊆ A s , A ∈ A s . B is conflict-free iff A, B ∈ B s.t. A defeats B. B defends A iff ∀B ∈ A s if B defeats A, then ∃C ∈ B s.t. C defeats B.
B is admissible iff it is conflict-free and defends all its elements. B is a preferred extension iff it is a maximal (for set inclusion) admissible set.

In [START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF], it was shown under several conditions, that ASPIC+ satisfies the rationality postulates defined in [START_REF] Caminada | On the evaluation of argumentation formalisms[END_REF], namely closure under strict rules and direct and indirect consistency. Unfortunately, due to the described limits of this system, counter-examples can be provided. For instance, the following example shows that an instance of ASPIC+ that violate closure and indirect consistency.

Example 6 Let R d = {⇒ a, ⇒ b, ⇒ x, ⇒ z, a ⇒ (x → y), b ⇒ (z → ¬y)},
and let all the other sets be empty and contrariness stand for classical negation. The following arguments can be built:

A: ⇒ a B: ⇒ b C: ⇒ x D: ⇒ z E: ⇒ a, a ⇒ (x → y) F: ⇒ b, b ⇒ (z → ¬y)
It can be checked that this system has one preferred extension: {A, B, C, D, E, F } which is not closed under strict rules. Note that in this example strict rules are derived from defeasible ones. Moreover, a, b, x, z, x → y, z → ¬y are outputs of the system. It can be checked that it is indirectly inconsistent.

Another source of problems for ASPIC + is the abstract aspect of the language L and its poor notion of consistency. The following example shows that some instantiations may lead to undesirable results, namely inconsistent ones.

Example 7 Assume that L contains propositional formulas, thus contrariness encodes classical negation. Assume also that all the sets are empty except R d = {⇒ x, ⇒ ¬x ∨ y, ⇒ ¬y}. Here the rules encode information. Only the following three arguments can be built: A 1 :⇒ x, A 2 :⇒ ¬x ∨ y, A 3 :⇒ ¬y. Note that the three arguments are not attacking each other. Thus, the set {A 1 , A 2 , A 3 } is an admissible/preferred set. This set supports three conclusions which are inconsistent (according to the semantics given to contrariness and to formulas in L). Note that this example would return this undesirable result even if the definition of consistency was more general than the one given in Definition 3.

The previous example reveals another problem with ASPIC + system. Its result is not closed under classical logic. Indeed, the three conclusions x, ¬x ∨ y, and ¬y are inferred while y which follows from the two first ones is not deduced. In works on nonmonotonic reasoning, namely the seminal paper [START_REF] Kraus | Nonmonotonic reasoning, preferential models and cumulative logics[END_REF], it is argued that one should accept as plausible consequences all that is logically implied by other plausible consequences. This property is known as right weakining.

Tarski's monotonic logics

Before studying the underlying logic of ASPIC+, let us first recall the abstract logic (L, CN) as defined by Tarski [START_REF] Tarski | Logic, Semantics[END_REF]. While there is no requirement on the language L, the consequence operator CN should satisfy the following basic axioms.

1. X ⊆ CN(X) (Expansion) 2. CN(CN(X)) = CN(X) (Idempotence) 3. CN(X) = Y ⊆ f X CN(Y ) 2 (Finiteness) 4. CN({x}) = L for some x ∈ L (Absurdity) 5. CN(∅) = L (Coherence)
Once (L, CN) is fixed, a notion of consistency arises as follows:

Definition 9 (Consistency). Let X ⊆ L. X is consistent w.r.t. the logic (L, CN) iff CN(X) = L. It is inconsistent otherwise.
Almost all well-known monotonic logics (classical logic, intuitionistic logic, modal logic, etc.) are special cases of the above notion of an abstract logic. The following logic for representing the color and the size of objects is another Tarskian logic.

Example 8 Let L = L col ∪ L size ∪ L err with L col = {white, yellow, red, orange, blue, black}, L size = {tiny, small, big, huge}, and L err = {⊥}. The consequence operator captures the fact that if two different colors or two different sizes are present in the description of an object, then information concerning that object is inconsistent. We define CN as follows: for all X ⊆ L,

CN(X) =        L if (∃x, y ∈ X s.t. x = y and ({x, y} ⊆ L col or {x, y} ⊆ L size )) or if (⊥ ∈ X) X else For example, CN(∅) = ∅, CN({red, big}) = {red, big}, CN({red, blue, big}) = CN({⊥}) = L.
The set {red, big} is consistent, while {red, blue, big} is inconsistent. Note that this logic does not need any connector of negation.

Rules as object level language

As said before, the (strict and defeasible) rules in ASPIC+ may either encode information or reasoning patterns. In this section, we investigate the first case. Our aim is to study the properties of the corresponding logic, denoted by (L o , CN o ).

The language L o is composed of the logical formulas of the set L (in [START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF]). Note that in [START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF] no particular requirement is made on L, neither on the kind of connectors that are used nor on the way of defining the formulas. However, as said before, L is equipped with the contrariness function ¯. The language L o contains also two kinds of information: strict rules (elements of R s ) encoding certain knowledge like 'penguins do not fly' and defeasible rules (elements of R d ) encoding defeasible information like 'generally birds fly'. Note that in ASPIC system [START_REF] Amgoud | Towards a consensual formal model: inference part[END_REF], the same language is considered with the difference that L contains only literals. Thus,

L o = L ∪ R s ∪ R d with L ∩ (R s ∪ R d ) = ∅.
An important question now is what are the contents of the bases K n , K p , K a and K i in this case. Since K n contains axioms, or undefeasible information, this may be represented by strict rules since these latter encode certain information. Similarly, since ordinary premises are defeasible, then they should be represented by defeasible rules. Otherwise, the language would be redundant and ambiguous. In sum, K = K a ∪ K i .

When strict and defeasible rules encode knowledge, the consequence operator of the logic used in ASPIC + is not specified. The only indication can be found in Definition 4 of the notion of argument. A possible CN o would be the following:

-x i ∈ X ∩ L Example 9 Let X = {x, x → y, t ⇒ z}, CN o (X) = {x, y}. Property 1 Let X ⊆ L o . -CN o (X) ⊆ L -CN o (∅) = cl(R s ∪ R d ) where cl(R s ∪ R d )
is the smallest set such that: -For all

• if → x ∈ R s (resp. ⇒ x ∈ R d ), then x ∈ cl(R s ∪ R d ) • if x 1 , . . . , x n → x ∈ R s (resp. x 1 , . . . , x n ⇒ x ∈ R d ) and {x 1 , . . . , x n } ⊆ cl(R s ∪ R d ), then x ∈ cl(R s ∪ R d )
X ⊆ L o s.t. either X ∩ R s = ∅ or X ∩ R d = ∅, it holds that X ⊆ CN o (X). -There is no x ∈ L o s.t. CN o ({x}) = L o .
The previous result shows that the logic (L o , CN o ) is not a Tarskian one since CN o violates the key axioms proposed in [START_REF] Tarski | Logic, Semantics[END_REF]. Moreover, the notion of consistency given in Definition 3 is weaker than that proposed in [START_REF] Tarski | Logic, Semantics[END_REF]. According to Tarski, a set X ⊆ L is consistent iff CN(X) = L. Thus, this notion captures not only binary minimal conflicts (as with Definition 3), but also ternary or more ones.

In the previous section, we have seen how strict and defeasible rules are used for encoding certain and defeasible information. The second way of using these rules is as inference rules. In [START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF], it is argued that strict rules may represent classical reasoning patterns, like modus ponens whereas defeasible rules may capture argument schemes. In this section, we study the resulting logic denoted by (L i , CN i ).

Let us start by defining the logical language L i . In this case, it is exactly the set L, that is L i = L. Thus, the only requirement on L i is that it has a contrariness function ¯. It is worth mentioning that the distinction made in [START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF] between the four bases K n , K p , K i , K a is meaningful. Thus, arguments are built from these bases.

Let us now define the consequence operator CN i . It is similar to CN o , except that strict and defeasible rules express inference schemas.

(∅) = cl(R s ∪ R d ) where cl(R s ∪ R d ) is the smallest set such that: • if → x ∈ R s (resp. ⇒ x ∈ R d ), then x ∈ cl(R s ∪ R d ) • if x 1 , . . . , x n → x ∈ R s (resp. x 1 , . . . , x n ⇒ x ∈ R d ) and {x 1 , . . . , x n } ⊆ cl(R s ∪ R d ), then x ∈ cl(R s ∪ R d ) -CN i (∅) = ∅ iff → x ∈ R s and ⇒ x ∈ R d for any x ∈ L Example 11 Let R s = {x, y → z; → x} and R d = {⇒ y}. cl(R s ∪R d ) = {x, y, z}.
The previous property shows that the coherence axiom of Tarski may be violated by CN i . It is namely the case when cl(R s ∪ R d ) = L. CN i does not guarantee neither the absurdity axiom. Indeed, there is no x ∈ L i such that CN i ({x}) = L i . In case strict rules encode propositional logic, then such formula exists. However, we can build other logics which do not offer such possibility. Let us consider the following logic which expresses the links between the six symbols of comparisons described in Example 1.

Example 1 (Continued): Assume that L i = {>, ≥, <, ≤, =, =, ≥ ∧ ≤, > ∨ <}, R d = ∅, and R s = {> → ≥, < → ≤, = → ≥ ∧ ≤, = → > ∨ <}. Note that there is no element in L i that has the whole set L i as a set of consequences.

As a consequence, the logic (L i , CN i ) is not a Tarskian one since it violates the coherence and absurdity axioms. Note that this result holds even when CN i encodes exactly the classical inference . The reason in this case is due to the poor definition of consistency. As shown is Example 3, in propositional logic the set {x, x → y, ¬y} is inconsistent while it is consistent according to Definition 3.

It is also worth mentioning that there is another family of Tarskian logics that cannot be captured by the monotonic logic (L i , CN i ). It is the family of logics whose language does not allow negation or contrariness like the one given in Example 8.

Conclusion

This paper investigated ASPIC+ argumentation system. It shows that this system suffers from the following drawbacks: i) The logical formalism is ill-defined, ii) The system may thus lead to undesirable results, iii) The system is grounded on several assumptions which may appear either counter-intuitive like the one on rebut, or restrictive like the one on the preference relation between arguments. iv) The system may violate the rationality postulates on closure and indirect consistency, v) The system returns results which may not be closed under classical logic. Thus, it violates the right weakening axiom [START_REF] Kraus | Nonmonotonic reasoning, preferential models and cumulative logics[END_REF]. vi) The system may not support intuitive instantiations due to its poor notion of consistency. Contrarily to what is claimed in [START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF][START_REF] Modgil | Revisiting preferences and argumentation[END_REF], the underlying logics of AS-PIC+ cannot encode the Tarskian ones: neither the ones which make use of a notion of negation nor the ones which do not.

Example 2

 2 The rule x → (y → z) is strict whereas the rules a ⇒ (b → c), (a → b) ⇒ (x → y) are defeasible.

  Definition 10 (Consequence operator). CN o is a function from 2 Lo to 2 Lo s.t. for all X ⊆ L o , x ∈ CN o (X) iff there exists a sequence x 1 , . . . , x n s.t. 1. x is x n , and 2. for each x i ∈ {x 1 , . . . , x n }, -∃y 1 , . . . , y j → x i ∈ X ∩ R s (resp. ∃y 1 , . . . , y j ⇒ x i ∈ X ∩ R d ) s.t. {y 1 , . . . , y j } ⊆ {x 1 , . . . , x i-1 }, or

  Example 10 Let X = {x, z → y, → z}, CN o (∅) = {z, y}. Now that the logic (L o , CN o ) is defined, let us see whether it is a Tarskian one. Proposition 1. Let (L o , CN o ) be as defined above. CN o is monotonic, satisfies idempotence, and finiteness axioms. The next result shows that CN o violates expansion and absurdity axioms.Proposition 2. Let (L o , CN o ) be as defined above.

  Definition 11 (Consequence operator). CN i is a function from 2 Li to 2 Li s.t. for all X ⊆ L i , x ∈ CN i (X) iff there exists a sequence x 1 , . . . , x n s.t. 1. x is x n , and 2. for each x i ∈ {x 1 , . . . , x Proposition 3. The logic (L i , CN i ) is monotonic. It satisfies expansion, idempotence, and finiteness.

	Property 2
	-CN i

n }, -∃y 1 , . . . , y j → x i ∈ R s (resp. ∃y 1 , . . . , y j ⇒ x i ∈ R d ) s.t. {y 1 , . . . , y j } ⊆ {x 1 , . . . , x i-1 }, or x i ∈ X

The symbols -→ and =⇒ are used to denote arguments. In[START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF], arguments are denoted by → and ⇒ while these latter are used for defining rules.

Y ⊆ f X means that Y is a finite subset of X.


Rawan Ghnemat
email: rawan.ghnemat@litislab.eu

Cyrille Bertelle
email: cyrille.bertelle@litislab.eu

Gérard H E Duchamp

MANAGING SPATIAL SELF-ORGANIZATION VIA COLLECTIVE BEHAVIORS

Keywords: swarm intelligence, complex systems, self-organization, ant systems, Schelling model, spatial organization

Spatial self-organizations appear in many natural and artificial systems. Spatial systems creation and development, called morphogenesis, is the subject of many research studies since many years (1). Fractal computation approach is, for exemple, one of the methods proposed to deal with such studies. But, even if this method is able to describe unlimited local formations on multi-scale descriptions, the formation process itself is described in a global way. The goal of this paper is to introduce the distributed and decentralized computing as a general methodology to propose emergent spatial formation, able to deal with local perturbations and with non homegeneous formation rules. We propose a study case on Schelling Model dealing with interacting population over an environment based on a regular grid.

SPATIAL MORPHOLOGY MODELLING ON THE EDGE OF COMPLEXITY

The study of spatial morphology is a major aspect of the understanding of many phenomena for natural or artificial systems. Living systems or social systems, for example, are systems where the spatial formation has a high meaning and modifies deeply by itself the system evolution. The system evolution leads to modify itself the spatial formations by feed-back processes.

Spatial morphology models can be classified by many criteria. Some of these models are static (finding the optimal shape of some problem) or dynamic (morphogenesis, for example). When the models involve dynamical processes, these dynamics can be expressed in a global way, like we do using partial differential equations: the objective of the system description consists in describing the different phenomena involved (diffusion, transport, ...).

Spatial morphology systems can be involved inside a multi-scale processus, giving some specific properties to this multi-scale formation, like the development of important exchange area. Fractal systems are well-known to model these multi-scale systems like fractal shape of plants.

Even if these fractal geometries are able to model multi-scale descriptions, they are generally completly deterministic and they are not suitable to describe geometrical evolutions or to integrate local disturbations. For this purpose, we need to change the model concept and go from global deterministic models to decentralized approaches where the whole system is only known (or emerge) by the interaction system of behavior population.

SCHELLING MODEL EXTENSION IN OR-DER TO MODEL SELF-ORGANIZATION BY MEANING OF MULTI-CRITERIA SYSTEM SEPARATION

Thomas Schelling's city segregation model illustrates how spatial organizations can emerge from local rules, concerning the spatial distribution of people which belong to different classes. In this model, people can move, depending on their own satisfaction to have neighbours of their own class. Based on this model, a city can be highly segregated even if people have only a mild preference for living among people similar to them.

In this model, each person is an agent placed on a 2D grid (in his original presentation, a chessboard was used by Thomas Schelling). Each case can be considered like a house where the agent lives. Each agent cares about the class of his immediate neighbours who are the occupants of the abutting squares of the chessboard. Each agent has a maximum of eight possible neighbours. He computes the rate of the neighbours of its own class from its eight possible neighbours. Each agent has a tolerence rate determining whether he is happy or not at his current house location. If the rate of the neighbours of its own class is under this tolerence rate, he decides to move to live in another free place in the 2D grid. The exact degree of segregation which emerges in the city depends strongly on the specification of the agents tolerance rate. It is noticeable that, under some rule specifications, Schelling's city can transit from a highly integrated state to a highly segregated state in response to a small local disturbance. We can observe some bifurcation phenomena which lead to chain reaction of displacements.

In figure 1, we show an implementation of this algorithm using the multi-agent platform called Repast [START_REF]Repast web site[END_REF]. Here, we show the "classical" and original problem, modelling the segregation phenomenon with two population classes, described by red and blue squares. Both, In figure 2, we detail the impact of tolerence rate on the segregation result. In this figure, we extend the original problem based on 2 population classes to 5 classes population. Part (a) describes the initial distribution according to a whole population density equal to 0.625. Part (b) describes the stable population distribution for a similar tolerance rate for each agent, equal to 0.375, corresponding to 3 neighbours on 8. This value is a discrete bifurcation point from where all small additional value leads to a very different distribution. To illustrate this phenomenon, part (c) describes the population distribution for a tolerance rate greater than 0.375, corresponding to 4 neighbours. The final population distribution is completly different than the one in part (b). Figure 3 describe some singular formation which can appears in very few cases, when we go over the bifurcation point which leads to no global clustering formation, except if some very small cluster kernels appear according to stochatic move spatial conjonction.

CONCLUSION

In this paper, we discuss about the properties of decentralized methods to model the spatial self-organization. These methods are based on the description of the whole system by the interaction systems of individual behaviors. We present some experiments on Schelling's model where auto-organization emerge from local rules. This model is able to take into account multi-criteria : each population class could be understand as the characteristic of some specific criteria. And the processus leads to simulate the emergent system morphology in order to define its criteria-specific sub-population separation (as an extension of segregation problem toward multicomponent morphology development).

Figure 1 :

 1 Figure 1: Standard 2 populations seggregation Schelling model on RePast

  (a) Initial situation (b) Final Stable situation for a tolerence rate = 0.375, corresponding to 3 neighbours over 8 (c) Final Stable situation for a tolerence rate greater than 0.375 and corresponding to 4 neighbours.

Figure 2 :Figure 3 :

 23 Figure 2: 5 populations seggregation Schelling model on RePast with density = 0.625 and with 2 values of tolerance rate


Rawan Ghnemat
email: rawan.ghnemat@litislab.eu

Françoise Lucchini
email: francoise.lucchini@univ-rouen.fr

Cyrille Bertelle
email: cyrille.bertelle@litislab.eu

URBAN CULTURAL DYNAMICS MODELLING USING SWARM INTELLIGENCE AND GEOGRAPHICAL INFORMATION SYSTEMS

Keywords: swarm intelligence, complex systems, urban dynamics, geographical information systems, cultural equipment, self-organization, ant systems, spatial organization

This paper deals with the understanding of collective cultural shape dynamics within urban area. The cultural facilities inside the city are various and follow complex spatial mecanisms. Both geographical aspects and social factors are major in these dynamics, leading to specific equipment. The complexity of these phenomena which are the basis of the cultural development, needs some specific modelling technics. In a first step, we propose some analysis and description of the cultural activity distribution based on a specific geographical information system (GIS). A case study is developped for the french city of Rouen in Normandy. In a second step, and respecting the complexity of the studied systems, we propose to use swarm intelligence models over this GIS to model the urban cultural dynamics. The goal is a better understanding of the relevent interactions within multi-criteria interactions.

The first methodology consists to generate a simulation where all the components, behaviors and rules-based, interact over a environment, perceiving and acting on it. The environment evolving is the support of emergent properties. The cellular automata modeling deals with this kind of simulations. The basic definition of cellular automata for urban or regional modeling, for instance, consists in the decomposition of the city, region or any geographical area in a lattice of cells. Each cell is in some state which belongs to a finite set S. At each time step, the cells change its own state according to some transition rules based on its previous state and its neighbor cells. Many works based on cellular automata, have been developped for geographical systems and urban dynamics (2; 9; 15). To deal with some urban dynamics, we need an important extension to cellular automata where we have to represent individual moving, for instance. The mixing of spatial data and cellular automata with autonomous entities, like agents, is here needed [START_REF] Daude | Introduction à la modélisation multi-agents des systèmes complexes en géographie[END_REF].

The second methodology to deal with emergent description in micro-modelling, consists to complete the previous approach based on simulation, by introducing some computational processes which are able to detect emergent systems or organizations. The final goal of this method is then to be able to re-introduce these emergent systems or organizations inside the simulation and manage their evolutions and their interactions with the components of the system. The re-integration of the emergent systems, during the simulation, can be explicitely expressed like in the multiscale fluid flow simulation proposed by P. Tranouez (25) or it can be implicitly expressed using an adaptive process as we will describe in the following.

Cultural equipment development within urban dynamics must be understanding through two aspects: the specific mecanisms which control the creation of cultural infrastructures and the use and attraction phenomena of these cultural equipments to follow their development. Emergent systems and organizations simulation must be implement to analysis the cultural dynamics and we will propose swarm intelligence technologies to model these phenomena.

ANALYSIS OF CULTURAL ACTIVITY DE-VELOPMENT BASED ON GIS -A CASE STUDY FOR THE FRENCH URBAN AREA OF ROUEN

Our goal is to study the diffusion of cultural facilities and propagation during time, inside the urban area of Rouen. A contructive approach is proposed in order to understand the spatial dynamics of cultural infrastructures development. We want to better understand what are the relevent interactions within heterogeneous phenomena.

Data Support

We design the cultural mechanism in a social and territorial context. The cultural activities and equipment are particularly all services present in French city [START_REF] Lucchini | La culture au service des villes, collection Villes[END_REF], where the majority of the population is located in, considering the general urbanization of the French Society. Like others French cities, Rouen proposes a large diversity of cultural activities in the hopes of educating, entertaining and satisfying inhabitants, but also for self-promotion.

The data from the city of Rouen concern urban environment and public or private cultural equipments within wide vision of culture. There are with academic sites (museums, operas, theatres, libraries, ...) and popular ones (musical pubs, concert places, festivals, cinemas, ...). These data have been obtained from the agglomeration population composed of 400,000 inhabitants. We plan to study the various logic of geographical repartition and the motivation to create and build cultural equipment, including strategies for municipalities or commercial exploitations.

At this local level we wish understand two mechanisms. The first one is the conditions that conduct people or private and public institutions to create new cultural structures and how the new locations are chosen. Second one is to better know the circulation of the inhabitants during time in these cultural places which have different levels of attraction. Because, in addition to the problem of geographical distances, other parameters play an equal role in accessing culture: social, educational and economic selection constitute another restraint for the cultural practices of citizens [START_REF] Bourdieu | L'amour de l'art, les muses d'art europen et leur public[END_REF].

First Graphical Analysis using GIS

Urban environment is based on land use of Rouen from topographical maps from IGN. Various GIS have been developped from these maps, mixing communication networks and cultural edifices (see on figure 1) or mixing land use and cultural equipments (see the figure 2). The mixing with the municipality administration and service attractivity is also plan to be studied.

Processes Analysis and Formalization

Cultural equipment repartition within city does not follow stochastic distribution [START_REF] Unwin | Introductory Spatial Analysis[END_REF]. This implies that the geography of these activities call on other dynamics than that of chance. This repartition is very different from sports equipment development, for example, as we can see on figure 3. The sports development follow the public school development that the cultural equipment does not follow.

These observations lead us to better understand what are the mechanisms involved in cultural phenomena. We have to study first the specific spatial constraints : river separation for the city of Rouen, industrial and housing proximity and municipalities service location. Spatial constraints are so a major aspect within the development dynamics, including heterogeneous factors. To model such complexity, we propose a swarm intelligence method base on ant systems as detailed in the next section.

MIXING GIS AND SWARM INTELLIGENCE FOR THE DYNAMICS SIMULATION

We will describe in this section, the general algorithm which is proposed to model emergent spatial organizations within cultural dynamics. This algorithm is based on the ant clustering. We introduce pheromon template to spatially control the clustering from local attraction. This method is a decentralized approach which allows to combinate multi-center and multi-criteria problems.

Ant clustering

Ant clustering algorithms are inspired by the corposes or larvea classification and aggregation that the ants colony are able to do in the real life. The ants are moving inside a closed area and are able to move some material which are randomly put on this area. After a while, and without any kind of centralized coordonation, the ants success to create some material clusters.

The algorithm is based on the following and very simple behavioral rules that each ant implements :

• When an ant is moving without carrying yet material and find some material, the ant will take the material respecting the probability number :

P p = k 1 k 1 + f 2 (1)
where f is the material density that the ant perceives locally around itself and k 1 is the treshold. It is easy to check that if f << k 1 then P p is near the value 1 and if f >> k 1 then P p is near the value 0.

• When an ant is moving when carrying some material, the probability to deposite it is computed by :

P d = f k 2 + f 2 (2)
where f is still the material density that the ant perceives locally around itself and k 2 is another treshold. It is easy to check that if f << k 2 then P d is near the value 0 and if f >> k 2 then P d is near the value 1.

Spatial constraints using template

The ant clustering shows some spatial self-organizations but has the specificity to generate clusters at random places. According to the first random moves that the ants start to do in the beginning of the algorithm, some material will initiate aggregation and the clustering processus will complete this aggregation from these initial random first aggregations. To simulate some urban dynamics, we need to introduce specific location with respect to city center for example or cultural equipments. The clustering here will represent the people use of these centers or equipments and we need to introduce an attractive effect by using a pheromon template. This method follow the algorithm known as Ant Nest Building (4). In ant colonies, the center corresponds to the position of the queen which needs to build the nest and the ant colony moves around it to protect the nest by various material taken on the ground. The queen emits a pheromon which allows to attract the ants during their building. The ant has to deposite the material carried only if the pheromon quantity perceived belongs to a specific range. We use an attractive fonction called P t , corresponding to a pheromon template and represented by the part (a) of the figure 4.

Using this template function, we remplace in the clustering algorithm, the two provious probabilities defined in equation [START_REF] Benenson | Geosimulation -Automata-based modeling of urban phenomena[END_REF] and equation ( 2) by

P ′ p = P p (1 -P t ) (3)
P ′ d = P d P t (4)
In figure 4, we show an implementation of this algorithm using the multi-agent platform called Repast [START_REF]Repast web site[END_REF]. The java version of this platform includes some packages allowing to interface with geographical database and geographical information systems (GIS). The graphical out-put windows is made under OpenMap which is a GIS developped in Java. In figure, the materials moved by the ants are the small grey circles, the ant moving without material are the green circles, the ant carrying material are the red circles and the queen location is the yellow circle.

Multi-template modelling

The previous subsection describes one local attractive process characterized by the queen and its pheromon template emission. The advantage of this method is to be able to combine the solutions of multi-center and multi-criteria problems, using interactive processes, each one is represented by a queen and its pheromon template.

On the figure [START_REF] Bourdieu | L'amour de l'art, les muses d'art europen et leur public[END_REF], we can see a simulation with two queens and two pheromon templates. It is possible also for each queen to emit many different kinds of pheromons : we called them colored pheromons. Each colored pheromon will attract only the ants associated to its color.

Application to cultural equipment dynamics

The multi-template modelling can be used to model cultural equipment dynamics as described in the figure [START_REF] De Castro | Artificial immune system: a new computational approach[END_REF].

On this figure, we associate to each cultural center (cinema, theatre, ...) a queen. Each queen will emit many pheromon templates, each template is associated to a specific criterium (according to age, sex, ...). Initially, we put the material in the residential place. Each material has some characteristics, corresponding to the people living in this residential area. The simulation shows the self-organization processus as the result of the set of the attractive effect of all the center and all the templates.

CONCLUSION AND PERSPECTIVES

We study in this paper the cultural equipment dynamics within urban area with a specific case study concerning the french city of Rouen. Geographical Information Systems have been developped from large database concerning Rouen Agglomeration and allow to highlight the complexity of this problem: the link between communication networks or land use with cultural development.

To model the complexity of cultural center attractivity and use, we propose a swarm intelligence method using a bio-inpired model called ant nest building. This model allows to deal with spatial self-organization dynamics using multi-center and multi-criteria aspects. Implementation on Repast platform with GIS mixing is proposed as the basis or the development framework. Specific results on Rouen Agglomeration are still in progress and are expected to give relevent analysis to better un-derstand these specific cultural dynamics.

Figure 1 :

 1 Figure 1: Communication networks and cultural equipment development mixing

Figure 3 :Figure 2 :

 32 Figure 3: Sports equipement and cultural equipment development within Rouen city agglomeration

Figure 4 :

 4 Figure 4: Ant nest building with one center using RePast MAS paltform over OpenMap GIS

Figure 5 :

 5 Figure 5: Multi-center ant nest buiding using RePast over OpenMap