Cédric Bouhours
email: bouhours@irit.fr

Hervé Leblanc
email: leblanc@irit.fr

Christian Percebois
email: percebois@irit.fr

Thierry Millan
email: millan@irit.fr

W Class

R Class

Detection of Generic Micro-architectures on Models

Keywords: Pattern detection, Graph isomorphism, UML Model, I, Component Leaf Composite

Existing pattern detection methods generally use code information obtained during reengineering process. However, none of these methods exclusively works with design information. In this paper, we propose a novel pattern detection method based on structural properties of UML models. This technique allows the detection of any kind of generic micro-architecture, like design patterns or spoiled patterns. Since a generic architecture is context-free, the structure of the searched fragments depends on the use context. So, our technique uses a structural concordance paradigm to identify all possible instantiations of a generic micro-architecture. To increase the precision of the detection, authorized, prohibited, and optional relations can be directly precised into the micro-architecture model.

INTRODUCTION

Various works aim at identifying fragments representing correct, incorrect or incomplete instantiations of design patterns, in order to help the comprehension of existing designs and to provide a base for possible improvements [START_REF] Tsantalis | Design Pattern Detection Using Similarity Scoring[END_REF]. To identify characteristic fragments, it is necessary to parse models, to ensure that the execution time of the algorithm is adapted to consequent models and to recognize a form that is approximate or to supplement. This approximation is very problematic, because it introduces uncertainties into the research. In the case of design patterns, the designer adapts the pattern to his problem, obliging the detection methods to be able to detect every possible form [START_REF] Wenzel | Automatic detection of incomplete instances of structural patterns in UML class diagrams[END_REF]. To render possible these detections, some tools use source code to identify complete or distorted versions of design patterns [START_REF] Albin-Amiot | Instantiating and Detecting Design Patterns: Putting Bits and Pieces Together[END_REF]. The information extracted from the source code augments the precision of fragment intent and so the pattern detection.

However, during model-driven processes, the identification of patterns concerns the designer in order to target specific model fragments. For example, spoiled patterns allow the detection of fragments substitutable with design patterns before a coding stage [START_REF] Bouhours | Bad smells in design and design patterns[END_REF]. Thus, we have conceived a detection method based on UML structural properties of UML models. Thanks to this method, we are able to identify instantiations of generic micro-architectures, like design patterns or spoiled patterns. The first intent of our detection method concerns the detection of spoiled pattern, which we present in Section 2. Section 3 presents the model representation we use to formalize our detection technique, and takes a stand on our work in relation to existing graph matching problems. The remainder of the paper is composed by the techniques used to compute the detection (Section 4), and some validation tests in Section 5.

The paper ends with a discussion of related works and a conclusion, in Sections 6 and 7. The main contributions of this paper are the specification and the implementation of a generic UML graph matching method able to detect pattern instantiations whatever their form.

II. SPOILED PATTERN DETECTION

Choosing a good design pattern and ensuring the correct integration of the chosen pattern are non trivial for a designer who wants to use them. To help designers, we propose design inspection in order to detect "bad smells in design" and models reworking through use of design patterns. The automatic detection and the explanation of the misconceptions are performed thanks to spoiled patterns [START_REF] Bouhours | Bad smells in design and design patterns[END_REF].

If we consider that a design pattern is the optimal reusable micro-architecture for a type of problem, then for each design problem that is solvable with a design pattern, the optimal solution is the instantiation of the design pattern. Moreover, if we consider an alternative solution as a valid solution but with a different architecture compared to the optimal solution, then, an alternative solution is an inadequate solution for a given problem, and is substitutable with the instantiation of the concerned pattern. A spoiled Composite pattern is given in Fig. 1. Each spoiled pattern has a name that describes the misconception: here the development of the composition link on the composite participant of the pattern. So, there is not a maximal factorization of the composition which implies addition or removal of a leaf or a composite need code modification.

Structurally, a spoiled pattern is represented at the same level of granularity as a design pattern allowing us to identify them as design patterns. An alternative fragment is a model fragment such as its structural properties match with the structural properties of a spoiled pattern and whose intent conforms to the corresponding pattern. Then, after the detection, an alternative fragment can be considered as potential. The validation of its intent is assumed by the designer of the user model. More details on these concepts, especially the collect, the catalog, the use in a tooled design review activity, and the refactorings can be found in [START_REF] Bouhours | Detection, Explications et Restructuration de défauts de conception : les patrons abîmés[END_REF].

The static UML model in Fig. 2 represents a basic architecture for a file system management. Authors of this model are interested in the presentation of some object concepts: inheritance between classes (a uniform protocol for every FileSystemElement is encapsulated by a corresponding abstract class) and management of reference and delegation (there are composition links between container and components).

Nevertheless, this model contains a misconception. Although there is a uniform protocol owned by the class FileSystemElement, the composite links management along a hierarchical structure is duplicated. Indeed, the Directory class manages independently links on Files and Directories. {Directory, File, FileSystemElement} is an exact instantiation of the Composite spoiled pattern. It is easy for the designer to see that this fragment has the same intent as the Composite pattern and to consider it as a bad smell in design. Furthermore, when the authors have implemented this model, they realized that there were defects. They adapted their code to correct them, without changing the design model. Therefore, the fragment must be substituted with the instantiation of the composite pattern on the user model or context. During a process development, it is more interesting to detect bad smells in design before the coding stage. Indeed, the model correction is easier and uses less time if the code is not already written. So, we have conceived a detection technique working with design information, and without information issued from reverse engineering process. The existing techniques use code information issued from their own reversion methods.

As a spoiled pattern has the same abstraction level as a design pattern, we consider that they are both "pattern" and so "generic micro-architecture". Our detection technique is able to detect generic micro-architecture, and so, the remainder of this paper uses "pattern" term to mean "design pattern", "spoiled pattern" or "generic micro-architecture". For the sake of clarity, we use the Composite design pattern as an example.

III. GRAPH REPRESENTATION

We consider models described in UML 1.5 [START_REF]Unified Modeling Language[END_REF] according to the XMI standard [START_REF]XML Metadata Interchange[END_REF]. With this meta-model, models can be represented by directed graphs. A graph consists of typed nodes representing the classes and the relations between them. Arrows are used to indicate the direction of the relations between classes. In our case, we are interested by classes, associations, and generalizations only. There is a gap between the visualization and the internal representation of a UML model. Fig. 3 illustrates these two representations in UML 1.5 for a design pattern: in a class diagram and in a graph conforms to XMI format. In this example, the design pattern is a simple directed graph, with the vertices C, L and Co, respectively Component, Leaf and Composite. There are also sets of vertices {A}, {AE}, {G} and {S}, respectively Association, AssociationEnd, Generalization and Specialization of the UML meta-model.

We have separated the vertices in two different subsets: Vc containing all the classes of the model, and Vm containing all the meta-classes allowing the connections of the classes. The vertices of the set {AE}, for AssociationEnd, come from the meta-model and are used to connect Classifier to Association. These vertices are tagged by AssociationEnd meta-class attributes in order to characterize the extremity of associations. For example, for the vertex Co, the adjacent vertices are AE, G and S only, excluding A which is accessible from AE only. As we consider UML models as graphs, we can formulate our patterns identification problem as a problem of sub-graphs or directed sub-graphs identification in a graph. There are two main approaches in this domain. The first one is known as exact graph matching, which consists in finding exactly a given subgraph in a graph [START_REF] Ullmann | An Algorithm for Subgraph Isomorphism[END_REF] approach consists in identifying all the sub-graphs looking like more or less a given graph [START_REF] Tsantalis | Design Pattern Detection Using Similarity Scoring[END_REF] and is called inexact graph matching. Exact graph matching algorithms require the examination of all possible sub-graphs that have the same number of nodes and arrows with the source graph, which means a NPcomplete problem [START_REF] Ullmann | An Algorithm for Subgraph Isomorphism[END_REF]. For our problem, exact pattern matching algorithms are inefficient. As a pattern is a generative base of a family of specific designs, we do not exactly search the generic micro-architecture, but one of its instantiations, which are not known in advance.

Inexact graph matching algorithms are very useful when an isomorphism between two graphs cannot be found or is too strict for research. They find the best correspondence between two graphs.

For example, some algorithms calculate the distance between two graphs, expressed for instance in number of modifications to transform a graph to the compared graph [START_REF] Tsantalis | Design Pattern Detection Using Similarity Scoring[END_REF]. In the context of pattern detection, such algorithms are more interesting, because they are able to detect sub-graphs structurally close to the pattern. However, it is not sufficient, because a given design pattern may have several forms depending on the instantiation context.

Since we cannot use exact or inexact pattern matching, we have defined a detection method working by structural concordances. Thanks to structural properties allowing the structural detection of pattern, this technique is able to detect pattern instantiations, whatever their form, and taking into consideration authorized, prohibited or optional relations between classes, as described in section 3.3.

IV. SPECIFICATION OF THE DETECTION

A pattern is described with a set of structural properties allowing its structural description, and thus the detection of its instantiation in a model. We have decomposed the remarkable properties into two subsets: the local properties that characterize individually each class and the global properties which characterize the classes against each other depending on their inter-relations. This separation allows us to constitute different filters during the detection, through use of structural similarity comparisons. The result of the search is a set of fragments identified in the model analyzed.

A. Structural Concordance

The structural properties of a pattern enable us to detect fragments in models. Compared to graphs, they enable us to detect sub-graphs families, because they describe the patterns as well as the fragments they can generate. Our detection method uses the local and global structural properties to check the structural concordance of the fragments with the patterns.

Definition 1 presents in a formal way a model m. As seen previously, it is a directed graph with two sets of vertices Vc and Vm, respectively representing the model classes and the instances of meta-classes describing the relations between the classes.

(1)

Like for a model, we formally define a pattern in definition 2.

Each pattern has a unique reference participant which we note reference_dp. It represents a particular vertex of Vc dp that we detail in part B. This vertex is chosen by an oracle according to its structural complexity and its responsibilities on the problem to solve.

(2) Thus, we have two directed graphs where we search for combinations of occurrences of the first in the second. In order to avoid a combinatorial explosion of the research possibilities, and thus to limit the problem complexity, we do a first filtering of the sets of the vertices having the adequate local properties.

The first step consists in searching for all the vertices of graph m in accordance with the predicate 5 local_SPC. This predicate, meaning "structural properties concordance", allows to check if a vertex c of the graph m has, at least, the same adjacent vertices as a vertex p of dp. Thus, if c is local_SPC with p, the class corresponding to c has, at least, the same local structural properties as the participant of the pattern corresponding to the vertex p. The comparison between the adjacent vertices is done with an equivalence relation comparing the type and the attributes of the adjacent vertices, as definition 3 shows it, with a constraints relaxation presented in part 3.3.

(3)

By extension, we obtain the definition 4.

(4)

As the adjacent vertices of a vertex of Vc belong to Vm and as all the vertices Vm are strongly typed, the vertices of Vc can be filtered thanks to their local properties, as predicate 5 shows it.

(5)

In order to illustrate the predicate local_SPC, we search Composite design patterns in the model of Fig. 4, whose local properties of each participant are illustrated in Fig. 5. The result of the application of the predicate local_SPC is illustrated in Table 1. For more legibility, the models are represented in UML. After comparing all the vertices of Vc dp with all those of Vc m , i.e. all the participants of the pattern with all the classes of the model to analyze, we obtain a set of vertices having their adjacent vertices at least identical to those of the participants of the pattern. This first predicate is used as filter on the sets of the vertices of m.

Predicate 6 global_SPC allows to check the concordance of the global properties, i.e. if a subgraph sf of m is isomorphic to dp.

(6)

A sub-fragment global_SPC with dp has, by definition, the same number of vertices as the pattern. Although the instantiation of the pattern causes the multiplication of some vertices, all the combinations, such as each class represents a distinct participant, remain isomorphic with the pattern. Fig. 6 illustrates this isomorphism of the sub-fragments of the model presented in Fig. 4. Figure 6. Sub-fragments of Fig. 4 isomorphic with the pattern of Fig. 3 In the model of Fig. 4, according to the vertices identified as being local_SPC with the vertices of the pattern, we can build only three sub-fragments in conformity with global_SPC. For example, the combination class U, class W and class Y, is not a sub-fragment, because even if there is the same number of vertices as in the pattern and each vertex is local_SPC with a vertex different from the pattern, there is no isomorphism between this combination and the pattern.

Thus, the predicate global_SPC enables us to eliminate class W and class Z from the Leaf responsibilities, since it is not possible to build a combination of classes in conformity with the predicate global_SPC with one of these classes to the responsibilities of Leaf. Now, we have to build the complete fragments, i.e. to couple the sub-fragments which share the same vertices. A complete fragment cf is a subgraph of m including at least an isomorphic sub-fragment with the pattern and such as any graph induced by a combination of vertices referring once each participant of the pattern remains isomorphic with the pattern. Moreover, only one vertex of cf, that we name reference_class, is local_SPC with reference_sp, the reference vertex of the pattern. In the case of Fig. 6, if we

Class X Class U Class Z * Class Y Class W Class T
consider that Composite is the reference participant, we can regroup the sub-fragments {ClassY, ClassT, ClassW} and {ClassY, ClassV, ClassW} to form a complete fragment.

Thus, the predicate 7 complete_fragment allows to check if a fragment cf is a set of sub-fragments, each one isomorphic with dp.

(7)

The application 8 participant associates to each vertex of the fragment cf, a vertex of the pattern such as it is in local concordance and which it is connected in the same way to the reference participant.

In proceeding like that, we can build fragments representing all the possible instantiations of the pattern. Indeed, even if it is not possible to anticipate which form has a complete fragment, it is sure, whatever its form, that it is composed of isomorphic sub-fragments to the pattern, since all the possible complete fragments will always have their classes connected in the same way to the respective participants of the pattern. Fig. 7 presents two fragments with isomorphic subfragments to a pattern. The sub-fragments of the left fragment are {Composite, Component1, Leaf} and {Composite, Component2, Leaf}. Those of the right fragment are {Composite, Component1, Leaf1}, {Composite, Component2, Leaf1}, {Composite, Component1, Leaf2} and {Composite, Component2, Leaf2}. Thus, it is possible to notice that the two fragments presented are two different instantiations, but recognized as complete fragments.

A particular case is presented in Fig. 8. The two sub-fragments of this model are isomorphic to the Composite pattern: {Composite1, Component, Leaf}, {Composite2, Component, Leaf}. However, if we analyze this case, we can wonder whether, except its structure, it constitutes a true complete fragment. Indeed, the composite participant are not linked between them. Thus, we propose that Fig. 8 presents two distinct fragments {Composite1, Component, Leaf} and {Composite2, Component, Leaf}. We do not authorize a fragment to have two classes having the Composite responsibilities. We named this additional characteristic the "reference participant", necessary to the representation of the results of detection and the limitation of the matching complexity.

B. Reference Participant

Each participant of a pattern has not the same importance in the intent aimed by the pattern. The fact that a design pattern is the best solution resides in its structural organization, obligatory support with any collaboration between objects. For the Composite pattern, take into consideration the UML models presented in Fig. 9 and try to answer the question: can these models be still regarded as instantiations of the Composite pattern? If we remove all the occurrences of the Leaf participant of the pattern, we do not loose the intent of the pattern, even if we lose the possibility of adding terminal elements in the hierarchical tree of composition. But, if we remove all the occurrences of the Composite participant, no more composition is possible. Indeed, it is Composite which completely manages the responsibilities for the composition of the objects, first intent of the pattern. Thus, this participant plays a dominant role in the pattern.

The reference participant depends on the pattern concerned. This participant is manually chosen with the heuristic evaluating its essentiality with the number of structural properties of each class.

C. Authorized, Prohibited or Optional Relations

Any class not having the responsibilities of the reference participant can be found multiplied in the fragment, whereas those marked reference are separate in different fragments. But what does occur if a class of the fragment is connected of more than another way that envisaged? Fig. 10 illustrates this case on a fragment. If we apply the method introduced previously, the fragment is detected, because it respects strictly the structural properties of the Composite pattern. However, the additional inheritance modifies the responsibilities of the classes concerned. A Leaf inheriting a Composite does not have a meaning in the hierarchical composition of objects, since a terminal object cannot be a specialized non-terminal object. Thus, we discriminate any fragment having connections invalidating the intent of a pattern.

To recognize which relations of the fragment are discriminating, we documented the pattern with information indicating which connections are optional, obligatory or prohibited. This documentation is done thanks to a UML profile allowing us to add specific information on relationships. Fig. 11 illustrates the Composite profiled pattern. Reported to the graph, we supplement the description of the pattern with a graph of the prohibited relations, comparable to the "Negative Conditions Application" (NAC) of graphs grammars [START_REF] Habel | Graph grammars with negative application conditions[END_REF]. If a connection appears in the graph of the pattern, it must obligatorily be present in the fragment; if it appears in the graph of the relations prohibited, it must absolutely miss in the fragment. If a connection is present in the fragment, but absent from the two other graphs, it is regarded as optional and neutral for detection. Thanks to this complement, we can now, besides detecting all possible forms of instantiation of the pattern, refuse certain forms which we suppose bad for the intent of the fragment.

D. Genericity of the Detection

Our generic detection algorithm is decomposed into three steps (local_SPC, closure form the reference participant, global_SPC). At the end of each step, a set of characteristic classes is selected from the set resulting from the preceding step. The first step consists in classifying all classifiers of the model to analyze according to local structural properties of each participant of the pattern. The second step consists in computing potential fragments according to the paths between the reference participant and the others participants of the pattern. The third step consists in verifying the global conformance on the potential fragments and to pinpoint complete fragments. Then the structure of the algorithm is flexible and does not depends on the structure of the pattern to retrieve. The genericity of our approach is provided by an automatic queries generator using profiled patterns. So, to detect a new pattern, it is sufficient to profile it and to generate its query [START_REF] Bouhours | Detection, Explications et Restructuration de défauts de conception : les patrons abîmés[END_REF].

We chose OCL to encode our detection queries. OCL is a language of constraints used to add semantics into UML models [START_REF]Object Constraint Language[END_REF]. The Neptune platform was developed by our team within the European Neptune project [START_REF]Method, Checking and Document generation for UML applications[END_REF]. The OCL interpreter proposed by the platform implements the standard OCL 2.0 [START_REF]Object Constraint Language[END_REF] and two extensions of OCL [START_REF] Millan | An OCL extension for checking and transforming UML Models[END_REF]. The second relates to the queries which can return a result of any type of the meta-model, which introduces the concept of view. We use this capacity to carry out our search for complete fragments in a model. Moreover, thanks to the navigational property of OCL, the generation of the queries consists in navigating in the meta-model of the pattern to detect. In following the three steps of the algorithm and in considering the pattern as a graph, the generator analyzes all the possible paths between each vertex and transforms them in an OCL query.

V. VALIDATION

In order to validate the detection algorithm we have sought models of real projects. We would like recall here that we search model fragments at design level without any information from the code. Unfortunately, we did not find industrial projects with exploitable models for our needs; the percentages of association links on inheritance links are too low. So, we have used the code of free projects to obtain reversed models. To do so, we used the Java reverse module of ArgoUML.

The problem with code reversion relates to associations between classes. It is very difficult for reverse softwares to know which variable must be regarded as attribute, association, composition, etc. However, the module of ArgoUML makes it possible to impose that all the attributes are transformed into associations. To take into consideration the parameterized genericity of the latest versions of Java, we have added in the ArgoUML module the capacity to convert these types into (1..n) associations. Thanks to this modification, we consider that the reversed models have a good abstraction level.

In order to make sure of the validity of our detection algorithm, we sought fragments corresponding to characteristic architectures. Fig. 12 presents the fragments that we have searched in nine models (ArgoUML, JUnit, JFreeChart, JabRef, Jena, AWT, JHotDraw, JRefactory, and Neptune). We can notice that these fragments are the Bridge pattern and its alternatives. In order to precise the detection, we have searched the same fragments twice: first, with information about authorized and prohibited relations, and second without any complementary information. For these examples, all supplementary relations between classes are prohibited.

Figure 12. The searched fragments

To validate the queries generated for these microarchitectures, we have executed them directly on the models allowing the generation of the queries. All queries are able to detect the model which has permitted their generation. So, we can say that we detect, at least, the minimal microarchitecture with exact instantiation.

The results obtained at the end of the detection show that all possible instantiations have been identified in the models, and no more. You can see the OCL queries generated corresponding to each pattern of Fig. 12 on [20]. It is interesting to notice that all fragments are identified twice: {SuperA, A, B, SuperB}, and {SuperB, B, A, SuperA}. The result is correct; each structure pattern is symmetric. Moreover, the researches with information about authorized and prohibited relations have retrieved fewer fragments than with the fragment without complementary information.

VI. RELATED WORKS

We present three generic detection techniques used in a similar context.

The first two concern approached identifications of design patterns for re-documentation, and the third concerns exact identification of design problems solvable by design patterns.

Patterns by comparison of similarities. [START_REF] Tsantalis | Design Pattern Detection Using Similarity Scoring[END_REF] proposes an algorithm for approximate pattern matching based on comparisons of similarities. This algorithm generates, from two graphs encoded by a matrix, an adjacency matrix representing the closeness of these two graphs. Matrix representations of relationships between artifact developments are used to compute effective approximations. Another way to represent UML models is to consider their visual forms as multi-graphs. Each type of relationship is represented by a different graph. A designer can choose on what information the similarity search is based (associations, generalizations, method specializations, method invocations, etc.). Thus, for a model and a pattern, the average of weighted matrices results for each significant input matrix (associations, generalizations, methods, etc.) is computable. The weighting is determined by the designer according to the importance it wishes to provide the various relationships of a static UML model. This weighted average is then the likeness of a fragment from a design pattern.

Patterns detection by fuzzy evaluation of UML models. Fujaba (From UML to Java And Back Again) is a tool that can generate Java code from a UML model and impact code changes on this model [START_REF] Fujaba | From UML to Java and Back Again[END_REF] [START_REF] Wenzel | Detection of Incomplete Patterns Using FUJABA Principles[END_REF]. A component of automatic detection of design patterns has been added. This component uses Abstract Syntax Graphs (ASG) [START_REF] Niere | Towards pattern-based design recovery[END_REF] to describe a model by eliminating most syntactic variants and formatting problems. Design patterns are decomposed into sub-patterns implemented as rules of graph transformation. However, if some sub-patterns are generic, eg. all possible ways to assign a value to an attribute in a method, the detection algorithm cannot detect patterns whose shape is not really the sub-assembly patterns provided. To overcome this limit, fuzzy evaluation mechanisms have been proposed by S. Wenzel [START_REF] Wenzel | Automatic detection of incomplete instances of structural patterns in UML class diagrams[END_REF]. They can detect patterns used differently than what is recommended, as well as incomplete patterns. To detect patterns in UML, it is necessary to describe a combination of roles in the UML sense. A role corresponds to a meta-class, which can be attached to OCL constraints [START_REF]Object Constraint Language[END_REF]. Constraints can describe the complex arrangements of certain patterns and clarify the internal organization of each role. Using this mode of representation, it is possible to detect design patterns in the same manner as a "cast of theater" [START_REF] Wenzel | Automatic detection of incomplete instances of structural patterns in UML class diagrams[END_REF]. The detection assigns a role in the pattern to some model elements. Each assignment is quantified by a value (0 to 100%) representing how the element can play the role. To get 100%, elements must have the same type as the role and respect each of the constraints described in the pattern. After the detection, the candidate fragments are presented to the designer. Implemented in a component of Fujaba, this technique detects target fragments "similar" to contextualization's design patterns.

Problems detection by constraint propagation. El-Boussaidi and Mili [START_REF] El-Boussaidi | Detecting Patterns of Poor Design Solutions Using Constraint Propagation[END_REF] proposes to detect fragments consistent with the meta-model of the problem of a pattern, and replace fragments by instantiation of the corresponding patterns. This detection technique reformulates the problem of homomorphism of graphs proposed by M. Rudolf [START_REF] Rudolf | Utilizing Constraint Satisfaction Techniques for Efficient Graph Pattern Matching[END_REF]. A CSP is defined by a finite set of variables in a domain, and a finite set of constraints specifying how values can be assigned to variables [START_REF] Bacchus | On the Conversion between Non-Binary and Binary Constraint Satisfaction Problems[END_REF]. CSPs are generally used to solve efficiently backtracking algorithms. When a value is assigned to a variable, all the constraints of this variable are propagated to other variables. To construct a CSP dedicated to pattern matching, it is necessary to work with two graphs, one to search (the source graph) and one in which research is conducted (the target graph) [START_REF] Rudolf | Utilizing Constraint Satisfaction Techniques for Efficient Graph Pattern Matching[END_REF]. Each vertex and each arrow of the source graph are associated with distinct variables. The domain of variable vertices and arrows correspond respectively to the set of vertices and arrows of the target graph. The constraints construction is done on the parameters compared to validate the research. H. Mili and G. El-Boussaidi [START_REF] Mili | Representing and Applying Design Patterns: What Is the Problem?[END_REF] defines design patterns as a triple (MP, MS, T) where MP is the problem solved by the pattern, MS is the solution to the problem, and T is the transformation that converts MP to MS. MP and MS are respectively the 2 summarizes the specifics of our problem with the related works. The last column refers to our approach. We guarantee a technique for early detection, even on large models. It is not pertinent to test all the meta-classes of the model to analyze, and it is better to make a quick filter to restrict the number of comparisons. Then we work directly and without pretreatment with the patterns encoded as models. Moreover, a spoiled pattern constitute a base generating a family of possible instantiations, and we identify accurately all the fragments of the same family.

VII. CONCLUSION

We have presented a complete method to detect generic micro-architectures on models. Then, a major issue of our work is the fact that we have reasoned at design level uniquely. That implies to use information present in models only and to define a re-documentation technique for retrieve patterns in a model. We have used standards dedicated to model engineering: UML profile and OCL queries. From a profiled model representing the structure of a pattern, we deduce automatically the OCL query that permit to retrieve all authorized instantiations of this pattern in a design model.

The detection is the core part of a tooling design review activity [START_REF] Bouhours | Bad smells in design and design patterns[END_REF]. We have implemented this activity into satellite software of the Neptune platform and we named it Triton [START_REF] Bouhours | Detection, Explications et Restructuration de défauts de conception : les patrons abîmés[END_REF]. As a code review permits to detect inconsistencies, no respect of coding rules, and bad smells in code before a production running, the design review permits to detect model fragments bad conceived and to refactor thanks to design patterns before a coding stage. The results of our queries are not too strict and not fuzzy; each detected fragment can be precisely built from a canonical form and by successive addition of participants. Therefore, we think that our detection method can be reused by the query part of any transformation model language.

However, the detection is based on structural properties only. For now, we have a catalog of spoiled structural design patterns. Dynamic views would be taking into consideration to detect behavioral design patterns and to precise some structural patterns by the detection of message exchange motifs.

Figure 1 .

 1 Figure 1. A spoiled pattern (development of the composition on Composite)

Figure 2 .

 2 Figure 2. A File System Management Design

Figure 3 .

 3 Figure 3. A UML model and its directed graph representation

Figure 4 .Figure 5 .

 45 Figure 4. A model example

Figure 7 .

 7 Figure 7. Two fragments with isomorphism sub-fragments to the Composite design pattern

Figure 8 .

 8 Figure 8. A particular case of a complete fragment

Figure 9 .

 9 Figure 9. The Composite pattern without Composite and without Leaf

Figure 10 .

 10 Figure 10. A model fragment with one supplementary relationship

Figure 11 .

 11 Figure 11. A graphical representation of the profiled pattern

  of the problem and solution. When a designer discovers a fragment of his model conforms to the metamodel of the problem, he has only to apply the transformation rules for modifying the fragment.

TABLE I .

 I RESULT OF THE PREDICATE LOCAL_SPC ON THE MODELThe classes marked in Table1validate predicate local_SPC with the corresponding participant. It is possible to notice that the classes have the same adjacent vertices as their participants, except for classes Y, W and Z which have more. For example, we can note that the class Y has three daughters. It is partly thanks to the fact that a vertex of Vc can have more adjacencies, that we can detect all the various possible pattern instantiations. Moreover, we can notice that class W and class Z validate the predicate on two different participants from the pattern, Leaf and Composite. Indeed, the local properties of Leaf are included in those of Composite. Without the global properties, we cannot differentiate the Composite classes from the Leaf classes yet.

					Class
	local_SPC	R	T	U	V	W	X	Y	Z
	Composite					OK			OK
	Component						OK OK
	Leaf		OK OK OK OK			OK

TABLE II

 II

	.	SUMMARY OF APPROACHES
			[1] [14] [16] us
	Do not perform preprocessing on the model to analyse	OK	OK
	Do not use information from the code		OK
	Perform detection by successive steps	OK OK OK
	Limit the execution time			OK OK
	Limit the number of fragments identified		OK OK
	Do not degrade the consensus of the pattern OK OK	OK
	Detect all possible instantiations	OK OK	OK
	Table


R Bissacot
email: rodrigo.bissacot@gmail.com

E O Endo

A C D Van Enter

B Kimura

A Le Ny

W M Ruszel
email: w.m.ruszel@tudelft.nl

Dyson models under renormalization and in weak fields

Keywords: AMS 2000 subject classification: Primary-60K35 ; secondary-82B20 Long-range Ising models, hidden phase transitions, generalized Gibbs measures, slowly decaying correlated external fields

We consider one-dimensional long-range spin models (usually called Dyson models), consisting of Ising ferromagnets with a slowly decaying long-range pair potentials of the form 1 |i-j| α , mainly focusing on the range of slow decays 1 < α ≤ 2. We describe two recent results, one about renormalization and one about the effect of external fields at low temperature. The first result states that a decimated long-range Gibbs measure in one dimension becomes non-Gibbsian, in the same vein as comparable results in higher dimensions for short-range models. The second result addresses the behaviour of such models under inhomogeneous fields, in particular external fields which decay to zero polynomially as 1 (|i|+1) γ . We study how the critical decay power of the field, γ, for which the phase transition persists and the decay power α of the Dyson model compare, extending recent results for short-range models on lattices and on trees. We also briefly point out some analogies between these results.

Introduction

In this short review we investigate some properties of one-dimensional long-range spin models, also known as Dyson models. In his original work, Dyson [START_REF] Dyson | Existence of a Phase Transition in a One-Dimensional Ising Ferromagnet[END_REF] considered an Ising spin system with formal Hamiltonian given by H(ω) = -i>j J(|i -j|)ω i ω j and J(n) ≥ 0 for n ∈ N e.g. of the form J(n) = n -α . There is no phase transition for this model, if the series M 0 = ∞ n=1 J(n) is infinite, since then there is an infinite energy gap between the ground states and all other states, which yields that at all finite temperatures the system is expected to be ordered. Neither is there a transition if |{n : J(n) = 0}| < ∞, by [START_REF] Rushbrooke | On one-dimensional regular assemblies[END_REF], since then the system is disordered at all finite temperatures. See also [START_REF] Bricmont | On the Equivalence of Boundary Conditions[END_REF] and [START_REF] Cassandro | Renormalization group and analyticity in one dimension: A proof of Dobrushin's theorem[END_REF] for accessible proofs of different versions of this absence of a transition under conditions of sufficiently fast polynomial decay of J(n). Thus in particular, there is no phase transition for J(n) being of finite range, and neither for J(n) = n -α with α > 2. A conjecture due to Kac and Thompson [START_REF] Kac | Critical behaviour of several lattice models with long-range interaction[END_REF], early on, stated that there should be a phase transition for low enough temperatures if and only if α ∈ [START_REF] Aizenman | Discontinuity of the Magnetization in the One-Dimensional 1/ | xy | 2 Percolation, Ising and Potts Models[END_REF][START_REF] Bissacot | Phase transition in ferromagnetic Ising models with non-uniform external fields[END_REF]. Dyson proved a part of the Kac-Thompson conjecture, namely that for long-range models of the form n -α with α ∈ (1, 2) there is a phase transition. Note that for M 0 < ∞ the infinite-volume measure is well defined.

We will consider, analogously to Dyson, one-dimensional ferromagnetic models with slowly decaying pair interactions of the form J(|i -j|) = 1 |i-j| α , for appropriate values of the decay parameter, α ∈ (1, 2], which display a phase transition at low temperature. This makes Dyson models particularly interesting, because they thus can exhibit phase coexistence even in one dimension, which is very unusual. Varying this decay parameter plays a similar role as varying the dimension in short-range models. This can be done in a continuous manner, so one obtains analogues of well-defined models in continuously varying non-integer dimensions. This is a major reason why these models have attracted a lot of attention in the study of phase transitions and critical behaviour (see e.g. [START_REF] Cassandro | Geometry of Contours and Peierls Estimates in d = 1 Ising Models with Long Range Interactions[END_REF] and references therein).

In this paper, we first sketch the proof of the fact that, at low enough temperature, under a decimation transformation the low-temperature measures of the Dyson models are mapped to non-Gibbsian measures. Indeed, similar to what happens for short-range models in higher dimensions, in the phase transition region (1 < α ≤ 2 and low enough temperature), decimating the Gibbs measures to half the spins leads to non-Gibbsianness of the decimated measures. This is obtained by showing the alternating configuration to be a point of essential discontinuity for the (finite-volume) conditional probabilities of the decimated Gibbs measures.

Just as with external fields or boundary conditions, the configuration of renormalised spins, acting on the system of "hidden spins" which are to be integrated out, can prefer one of the phases, and there are choices where this preference depends only on spins far away. The renormalised spins can act as some kind of (possibly correlated) random field, acting on the other (hidden) spins.

We have extended our analysis to consider the effects of more general, possibly decaying, external fields on Dyson models and discuss how Dyson models in external fields decaying to zero as 1 (|i|+1) γ behave as regards phase coexistence. Again similarly to what happens in shortrange models, it appears that the existence of a plurality of Gibbs measures persists when the decay of the field is fast enough, whereas for slowly decaying fields we expect that there is only one Gibbs measure which survives, namely the one favoured by the field. What the appropriate decay parameter of the field, γ, which separates the two behaviours is, depends on the Dyson decay parameter α. This extends recent results on short-range models on either lattices or trees.

The review is organized as follows. In Section 2, we introduce notations and definitions of Gibbs measures and describe what is known about phase transitions in Dyson models. In Section 3, we introduce the decimation transformation -a renormalization transformation that keeps odd or even spins only -and sketch how to prove non-Gibbsianness at low temperature for the decimated Gibbs measures of the Dyson models. We show that, conditioned on the even spins to be alternating, a "hidden phase transition" occurs in the system of odd spins. In Section 4 we will discuss Dyson models in decaying fields.

2

Gibbs Measures and Dyson Models

Specifications and Measures

We refer to [START_REF] Van Enter | Decimation of the Dyson-Ising Ferromagnet[END_REF] and [START_REF] Bissacot | Phase transition of long-range Dyson models in inhomogeneous fields[END_REF] for proofs and more details on the general formalism considered here.

Dyson models are ferromagnetic Ising models with long-range pair-interactions in one dimension, possibly with an external field which we will take possibly inhomogeneous, random and/or correlated.

We study these models within a more general class of lattice (spin) models with Gibbs measures on infinite-volume product configuration spaces (Ω, F, ρ) = (E Z , E ⊗Z , µ ⊗Z o ), the singlesite state space being the Ising space E = {-1, +1}, with the a priori counting measure µ 0 = 1 2 δ -1 + 1 2 δ +1 . We denote by S the set of the finite subsets of Z and, for any Λ ∈ S, write (Ω Λ , F Λ , ρ Λ ) for the finite-volume configuration space (E Λ , E ⊗Λ , µ ⊗Λ o ) -and extend afterwards the notations when considering infinite subsets S ⊂ Z and (restricted) infinite-volume configuration spaces (Ω S , F S , µ S ) ∋ σ S .

Microscopic states or configurations, denoted by σ, ω, η, τ, , etc., are elements of Ω equipped with the product topology of the discrete topology on E for which these configurations are close when they coincide on large finite regions Λ (the larger the region, the closer). For ω ∈ Ω, a neighborhood base is provided by

N L (ω) = σ ∈ Ω : σ Λ L = ω Λ L , σ Λ c L arbitrary , L ∈ N, Λ L := [-L, +L] ∈ S.
For any integers N > L, we shall also consider particular open subsets of neighborhoods

N + N,L (ω) = σ ∈ N L (ω) : σ Λ N \Λ L = + Λ N \Λ L , σ arbitrary otherwise resp. N - N,L (ω) .
We denote by C(Ω) the set of continuous (quasilocal) functions on Ω, characterized by

f ∈ C(Ω) ⇐⇒ lim Λ↑S sup σ,ω:σ Λ =ω Λ | f (ω) -f (σ) |= 0. (2.1)
Monotonicity for functions and measures concerns the natural partial (FKG) order "≤ ", which we have on our Ising spin systems : σ ≤ ω if and only if σ i ≤ ω i for all i ∈ Z. Its maximal and minimal elements are the configurations + and -, and this order extends to functions: f : Ω -→ R is called monotone increasing when σ ≤ ω implies f (σ) ≤ f (ω). For measures, we write µ ≤ ν if and only if µ[f ] ≤ ν[f ] for all f monotone increasing1 .

Macroscopic states are represented by probability measures on (Ω, F, ρ), whose main description -at least in mathematical statistical mechanics -is in terms of consistent systems of regular versions of finite-volume conditional probabilities with prescribed boundary conditions, within the so-called DLR formalism [START_REF] Dobrushin | The Description of a Random Field by Means of Conditional Probabilities and Conditions of its Regularity[END_REF][START_REF] Lanford | Observables at Infinity and States with Short Range Correlations in Statistical Mechanics[END_REF]. To do so, one introduces families of probability kernels that are natural candidates to represent such versions of conditional probabilities.

Definition 1 (Specification

) : A specification γ = γ Λ Λ∈S on (Ω, F) is a family of probability kernels γ Λ : Ω Λ × F Λ c -→ [0, 1]; (ω, A) -→ γ Λ (A | ω) s.t. for all Λ ∈ S: 1. (Properness) For all ω ∈ Ω, γ Λ (B|ω) = 1 B (ω) when B ∈ F Λ c . 2. (Finite-volume consistency) For all Λ ⊂ Λ ′ ∈ S, γ Λ ′ γ Λ = γ Λ ′ where ∀A ∈ F, ∀ω ∈ Ω, (γ Λ ′ γ Λ )(A|ω) = Ω γ Λ (A|σ)γ Λ ′ (dσ|ω).
(

These kernels also act on functions and on measures:

for all f ∈ C(Ω) or µ ∈ M + 1 , γ Λ f (ω) := Ω f (σ)γ Λ (dσ|ω) = γ Λ [f |ω] and µγ Λ [f ] := Ω (γ Λ f )(ω)dµ(ω) = Ω γ Λ [f |ω]µ(dω).

Definition 2 (DLR measures) :

A probability measure µ on (Ω, F) is said to be consistent with a specification γ (or specified by γ) when for all A ∈ F and Λ ∈ S

µ[A|F Λ c ](ω) = γ Λ (A|ω), µ-a.e. ω. (2.3)
We denote by G(γ) the set of measures consistent with γ.

A specification γ is said to be quasilocal when for any local function f , the image γ Λ f should be a continuous function of the boundary condition :

γ quasilocal ⇐⇒ γ Λ f ∈ C(Ω) for any f local (or any f in C(Ω)).
(2.4)

A measure is said to be quasilocal when it is specified by a quasilocal specification.

A particularly important subclass of quasilocal measures consists of the Gibbs measures with (formal) Hamiltonian H defined via a potential Φ, a family Φ = (Φ A ) A∈S of local functions Φ A ∈ F A . The contributions of spins in finite sets A to the total energy define the finite-volume Hamiltonians with free boundary conditions

∀Λ ∈ S, H Λ (ω) = A⊂Λ Φ A (ω), ∀ω ∈ Ω.
(2.5)

To define Gibbs measures, we require for Φ that it is Uniformly Absolutely Convergent (UAC), i.e. that A∋i sup ω |Φ A (ω)| < ∞, ∀i ∈ Z. One then can give sense to the Hamiltonian at volume Λ ∈ S with boundary condition ω defined for all σ, ω ∈ Ω as

H Φ Λ (σ|ω) := A∩Λ =∅ Φ A (σ Λ ω Λ c )(< ∞).
The Gibbs specification at inverse temperature β > 0 is then defined by

γ βΦ Λ (σ | ω) = 1 Z βΦ Λ (ω) e -βH Φ Λ (σ|ω) (ρ Λ ⊗ δ ω Λ c )(dσ) (2.6)
where the partition function Z βΦ Λ (ω) is an important normalizing constant. Due to the, in fact rather strong, UAC condition, these specifications are quasilocal. It appears that the converse is also true up to a non-nullness condition2 (see e.g. [START_REF] Georgii | Gibbs Measures and Phase Transitions[END_REF][START_REF] Fernández | Gibbsianness and non-Gibbsianness in Lattice random fields[END_REF][START_REF] Kozlov | Gibbs Description of a System of Random Variables[END_REF][START_REF] Sullivan | Potentials for Almost Markovian Random Fields[END_REF][START_REF] Ny | Introduction to Generalized Gibbs measures[END_REF]) and one can take :

Definition 3 (Gibbs measures) : µ ∈ M + 1 is a Gibbs measure iff µ ∈ G(γ)
, where γ is a non-null and quasilocal specification.

Quasilocality, called Almost Markovianness in [START_REF] Sullivan | Potentials for Almost Markovian Random Fields[END_REF], is a natural way to extend the global (two-sided) Markov property. When µ ∈ G(γ) is quasilocal, then for any f local and Λ ∈ S, the conditional expectations of f w.r.t. the outside of Λ are µ-a.s. given by γ Λ f , by (2.2), and each conditional probability has a version which itself is a continuous function of the boundary condition, so one gets for any ω

lim ∆↑Z sup ω 1 ,ω 2 ∈Ω µ f |F Λ c (ω ∆ ω 1 ∆ c ) -µ f |F Λ c (ω ∆ ω 2 ∆ c ) = 0 (2.7)
Thus, for Gibbs measures the conditional probabilities always have continuous versions, or equivalently there is no point of essential discontinuity. Those are configurations which are points of discontinuity for ALL versions of the conditional probability. In particular one cannot make conditional probabilities continuous by redefining them on a measure-zero set if such points exist. In the generalized Gibbsian framework, one also says that such a configuration is a bad configuration for the considered measure, see e.g. [START_REF] Ny | Introduction to Generalized Gibbs measures[END_REF]. The existence of such bad configurations implies non-Gibbsianness of the associated measures.

Dyson models: Ferromagnets in One Dimension

Definition 4 (Dyson models) : Let β > 0 be the inverse temperature and consider 1 < α ≤ 2. We call a Dyson model with decay parameter α the Gibbs specification (2.6) with pair-potential Φ D defined for all ω ∈ Ω by

Φ D A (ω) = - 1 |i -j| α ω i ω j when A = {i, j} ⊂ Z, and Φ D A ≡ 0 otherwise. (2.8)
We shall also consider Dyson models with non-zero magnetic fields h = (h i ) i ∈Z acting as an extra self-interaction part

Φ D A (ω) = -h i ω i when A = {i} ⊂ Z
We first use that as a consequence of FKG property [START_REF] Fortuin | Correlation Inequalities on some Partially Ordered Sets[END_REF][START_REF] Hulse | On the Ergodic Properties of Gibbs States for Attractive Specifications[END_REF], the Dyson specification is monotonicity-preserving 3 , which implies that the weak limits obtained by using as boundary conditions the maximal and minimal elements of the order ≤ are well defined and are the extremal elements of G(γ D ).

Proposition 1 [START_REF] Fernández | Global specifications and non-quasilocality of projections of Gibbs measures[END_REF][START_REF] Hulse | On the Ergodic Properties of Gibbs States for Attractive Specifications[END_REF][START_REF] Lebowitz | Coexistence of Phases for Ising Ferromagnet[END_REF]: For α > 1 ( and not only for α ∈ (1, 2]), the weak limits

µ -(•) := lim Λ↑Z γ D Λ (•|-) and µ + (•) := lim Λ↑Z γ D Λ (•|+) (2.9)
are well-defined, translation-invariant and extremal elements of G(γ D ). For any f bounded increasing, any other measure µ ∈ G(γ D ) satisfies

µ -[f ] ≤ µ[f ] ≤ µ + [f ]. (2.10)
Moreover, µ -and µ + are respectively left-continuous and right-continuous.

While µ -and µ + coincide at high temperatures, and at all temperatures when there is fast decay, α > 2, one main peculiarity of this one-dimensional model is thus that when the range is long enough (1 < α ≤ 2), it is possible to recover low-temperature behaviours usually associated to higher dimensions for the standard Ising model, in particular phase transitions can occur. For more details on the history of the proofs, one can consult [START_REF] Van Enter | Decimation of the Dyson-Ising Ferromagnet[END_REF] and references therein or below.

Proposition 2 [START_REF] Dyson | An Ising ferromagnet with discontinuous long-range order[END_REF][START_REF] Fröhlich | The Phase Transition in the One-Dimensional Ising Model with 1/r 2 interaction energy[END_REF][START_REF] Ruelle | On the use of "small external fields" in the problem of symmetry breakdown in statistical mechanics[END_REF][START_REF] Georgii | Gibbs Measures and Phase Transitions[END_REF][START_REF] Friedli | Equilibrium Statistical Mechanics : a Concrete Mathematical Introduction[END_REF][START_REF] Fröhlich | Phase Transitions and Reflection Positivity. I. General Theory and Long Range Lattice Models[END_REF][START_REF] Aizenman | Discontinuity of the Magnetization in the One-Dimensional 1/ | xy | 2 Percolation, Ising and Potts Models[END_REF][START_REF] Cassandro | Geometry of Contours and Peierls Estimates in d = 1 Ising Models with Long Range Interactions[END_REF][START_REF] Littin | Quasiadditive estimates on the Hamiltonian for the One-dimensional Long-Range Ising Model[END_REF][START_REF] Johansson | Condensation of a One-Dimensional Lattice Gas[END_REF].

The Dyson model with potential (2.8), for 1 < α ≤ 2, exhibits a phase transition at low temperature:

∃β D c > 0, such that β > β D c =⇒ µ -= µ + and G(γ D ) = [µ -, µ + ]
where the extremal measures µ + and µ -are translation-invariant. They have in particular opposite magnetisations µ + [σ 0 ] = -µ -[σ 0 ] = M 0 (β, α) > 0 at low temperature. Moreover, the Dyson model in a non-zero homogeneous field h has a unique Gibbs measure.

We remark that the infinite-volume limit of a state (or a magnetisation) in which there is a + (resp. -)-measure or a Dyson model in a field h > 0 (resp. h < 0) outside some interval is the same as that obtained from + (resp. -)-boundary conditions (independent of the magnitude of h). This can be e.g. seen by an extension of the arguments of [START_REF] Lebowitz | Analytic and clustering properties of thermodynamic functions and distribution functions for classical lattice and continuum systems[END_REF], see also [START_REF] Lebowitz | Thermodynamic Limit of the Free Energy and Correlation Functions of Spins Systems[END_REF]. Notice that taking the +-measure of the zero-field Dyson model outside a finite volume enforces this same measure inside (even before taking the limit); adding a field makes it it more positive, and taking the thermodynamic limit then recovers the same measure again.

The case of α = 2 is more complicated to analyse, and richer in its behaviour, than the other ones. There exists a hybrid transition (the "Thouless effect"), as the magnetisation is discontinuous while the energy density is continuous at the transition point. Moreover, there is second transition below this transition temperature. In the intermediate phase there is a positive magnetisation with non-summable covariance, while at very low temperatures the covariance decays at the same rate as the interaction, which is summable. For these results, see [START_REF] Aizenman | Discontinuity of the Magnetization in the One-Dimensional 1/ | xy | 2 Percolation, Ising and Potts Models[END_REF][START_REF] Imbrie | Decay of Correlations in One-Dimensional Ising Model with J ij =| ij | -2[END_REF][START_REF] Imbrie | An Intermediate Phase with Slow Decay of Correlations in One-Dimensional 1/ | xy | 2 Percolation, Ising and Potts Models[END_REF], and also the more recent description in [START_REF] Littin | Quasiadditive estimates on the Hamiltonian for the One-dimensional Long-Range Ising Model[END_REF].

Decimation

We first apply a decimation transformation to the lattice 2Z. Similarly to what was discussed in [START_REF] Van Enter | Regularity Properties and Pathologies of Position-Space R.G. Transformations: Scope and Limitations of Gibbsian Theory[END_REF], to analyse whether the transformed measure is a Gibbs measure, and in particular to show that it is non-Gibbsian, we have to show that conditioned on a particular configuration of the transformed spins, the "hidden spins" display a phase transition. If we choose this particular configuration to be the alternating one, each hidden spin feels opposite terms from the left and the right side, coming from all odd distances. Thus the conditioned model is a Dyson model in zero field, at a reduced temperature. As such it has phase transition. To translate this hidden phase transition into nonlocality of the "visible" transformed spins follows straightforwardly the arguments of [START_REF] Van Enter | Regularity Properties and Pathologies of Position-Space R.G. Transformations: Scope and Limitations of Gibbsian Theory[END_REF]. See [START_REF] Van Enter | Decimation of the Dyson-Ising Ferromagnet[END_REF] for the details. We make use of the fact that one can define global specifications, so there are no measurability problems due to global conditioning.

We start from µ + (2.9), the +-phase of a Dyson model without external field in the phase transition region and apply the decimation transformation

T : (Ω, F) -→ (Ω ′ , F ′ ) = (Ω, F); ω -→ ω ′ = (ω ′ i ) i∈Z , with ω ′ i = ω 2i (3.11)
Denote ν + := T µ + the decimated +-phase, formally defined as an image measure via

∀A ′ ∈ F ′ , ν + (A ′ ) = µ + (T -1 A ′ ) = µ + (A) where A = T -1 A ′ = ω : ω ′ = T (ω) ∈ A ′ .
We study the continuity of conditional expectations under decimated Dyson Gibbs measures of the spin at the origin when the outside is fixed in some special configuration ω ′ alt . By definition,

ν

+ [σ ′ 0 |F {0} c ](ω ′ ) = µ + [σ 0 |F S c ](ω), ν + -a.s. (3.12)
where S c = (2Z) ∩ {0} c , i.e. with S = (2Z) c ∪ {0} is not finite: the conditioning is not on the complement of a finite set, and although the extension of the DLR equation to infinite sets is direct in case of uniqueness of the DLR-measure for a given specification [START_REF] Fernández | Global specifications and non-quasilocality of projections of Gibbs measures[END_REF][START_REF] Föllmer | On the Global Markov Property[END_REF][START_REF] Goldstein | Remarks on the Global Markov Property[END_REF], it can be more problematic otherwise: it is valid for finite sets only and measurability problems might arise in case of phase transitions when one wants to extend them to infinite sets. Nevertheless, beyond the uniqueness case, such an extension was made possible by Fernández and Pfister [START_REF] Fernández | Global specifications and non-quasilocality of projections of Gibbs measures[END_REF] in the case of attractive models. As we will make essential use of it, we describe it now in our particular case. The concept they introduced is that of a global specification, and this is in fact a central tool in some of our arguments.

Definition 5 (Global specification [19]) :

A global specification Γ on Z is a family of probability kernels Γ = (Γ S ) S⊂Z on (Ω S , F S c ) such that for any S subset of Z:

1. Γ S (•|ω
) is a probability measure on (Ω, F) for all ω ∈ Ω.

Γ S (A|•

) is F S c -measurable for all A ∈ F.

Γ

S (B|ω) = 1 B (ω) when B ∈ F S c . 4. For all S 1 ⊂ S 2 ⊂ Z, Γ S 2 Γ S 1 = Γ S 2
where the product of kernels is made as in (2.2).

Similarly to the consistency with a (local) specification, one introduces the compatibility of measures with a global specification.

Definition 6 Let Γ be a global specification. We write µ ∈ G(Γ), or say that µ ∈ M + 1 is Γ-compatible, if for all A ∈ F and any S ⊂ Z, µ[A|F S c ](ω) = Γ S (A|ω), µ-a.e. ω.

(3.13) Note, by considering S = Z, that G(Γ) contains at most one element.

In the case considered here, we get a global specification Γ + such that µ + ∈ G(Γ + ), with S = (2Z) c ∪ {0} consisting of the odd integers plus the origin. Hence S = (2Z) c ∪ {0} and (3.12) yields for ν + -a.e. all ω ′ ∈ N Λ ′ (ω ′ alt ) and ω ∈ T -1 {ω ′ }:

ν + [σ ′ 0 |F {0} c ](ω ′ ) = Γ + S [σ 0 |ω] µ + -a.e.(ω). (3.14)
to eventually get (see [START_REF] Fernández | Global specifications and non-quasilocality of projections of Gibbs measures[END_REF][START_REF] Van Enter | Decimation of the Dyson-Ising Ferromagnet[END_REF]) an expression of the latter in terms of a constrained measure µ +,ω (2Z) c ∪{0} , with ω ∈ T -1 {ω ′ } so that we get for any

ω ′ ∈ N Λ ′ (ω ′ alt ), ν + [σ ′ 0 |F {0} c ](ω ′ ) = µ +,ω (2Z) c ∪{0} ⊗ δ ω 2Z∩{0} c [σ 0 ].
Thanks to monotonicity-preservation, the constrained measure is explicitly built as the weak limit obtained by +-boundary conditions fixed after a freezing the constraint to be ω on the even sites :

∀ω ′ ∈ N Λ ′ (ω ′ alt ), ∀ω ∈ T -1 {ω ′ }, µ +,ω (2Z) c ∪{0} (•) = lim I∈S,I↑(2Z) c ∪{0} γ D I (• | + (2Z) c ∪{0}) ω 2Z∩{0} c ). (3.15)
Observe that when a phase transition holds for the Dyson specification -at low enough T for 1 < α ≤ 2 -the same is true for the constrained specification with alternating constraint (although at a lower T). This phase transition then implies non-Gibbsianness of ν + (and for all other Gibbs measures of the model, see [START_REF] Van Enter | Decimation of the Dyson-Ising Ferromagnet[END_REF]).

Theorem 1 ( [START_REF] Van Enter | Decimation of the Dyson-Ising Ferromagnet[END_REF]) : Let α ∈ (1, 2], let µ be a Gibbs measure for the interaction given by (2.8) and let the transformation T be defined by (3.11). Then for low temperatures, β > 2 α β D c , the decimated measure ν = T • µ is non-Gibbs.

Sketch of Proof:

The main idea is to prove that the alternating configuration is an essential point of discontinuity for the decimated conditional expectations. As already observed, because any non-fixed site at all odd distances has a positive and a negative spin whose influences cancel, conditioning by this alternating configuration yields a constrained model that is again a Dyson model at zero field, but at a temperature which is higher by 2 α . This again has a low-temperature transition in our range of decays 1 < α ≤ 2. The coupling constants are indeed multiplied by a factor 2 -α , due to only even distances occurring between interacting (hidden) spins.

To prove non-Gibbsianness in [START_REF] Van Enter | Decimation of the Dyson-Ising Ferromagnet[END_REF], we essentially follow the proof strategy sketched in [START_REF] Van Enter | Regularity Properties and Pathologies of Position-Space R.G. Transformations: Scope and Limitations of Gibbsian Theory[END_REF], by showing that within a neighborhood N L (ω ′ alt ), there exists two subneighborhoods N ± N,L of positive measure on which the conditional magnetizations defined on N N , L ±

M + = M + (ω) = µ +,ω + (2Z) c ∪{0} [σ 0 ] and M -= M -(ω) = µ +,ω - (2Z) c ∪{0} [σ 0 ]. (3.16)
differ significantly.

The role of the "annulus" where configurations are constrained to be either + oris played by two large intervals [-N, -L -1] and [L + 1, N ]. Due to the long range of the interaction, their might be a direct influence from the boundary beyond the annulus, to the central interval.

To avoid effects from this influence, we take N much larger than L. An argument based on "equivalence of boundary conditions" as in e.g. [START_REF] Bricmont | On the Equivalence of Boundary Conditions[END_REF] , under a choice N = L 1 α-1 then implies that (3.16) does hardly depend on ω.

Once this choice of big annulus is made, observe that if we constrain the spins in these two intervals to be either + or -, within these two intervals the measures on the unfixed spins are close to those of the Dyson-type model in a positive, c.q. negative, magnetic field. As those measures are unique Gibbs measures, no influence from the boundary can be transmitted. Indeed, in contrast to the case of the purely alternating configuration, in the case when we condition on all primed spins to be + (resp. -) in these large annuli, there is no phase transition and the system of unprimed spins has a unique Gibbs measure. It is a Dyson model, again at a heightened temperature, but now in a homogeneous external field, with positive (resp. negative) magnetisation +M 0 (β, α) > 0 (resp. -M 0 (β, α) < 0), stochastically larger (resp. smaller) than the zero-field + ( resp. -)-measure.

In the --case, in the annulus the magnetisation of the -even-distance-Dyson-Ising model is essentially that of the model with a negative homogeneous external field -h everywhere, which at low enough temperature and for L large enough is close to (and in fact smaller than) the magnetisation of the Dyson-Ising model under the zero-field --measure, i.e to -M 0 (β, α) < 0. Thus the inner interval where the constraint is alternating feels a --like condition from outside its boundary. On the other hand, the magnetisation with the constraint ω + will be close to or bigger than +M 0 (β, α) > 0 so that a non-zero difference is created at low enough temperature. One needs again to adjust the sizes of L and N to be sure that boundary effects from outside the annulus are negligible in the inner interval.

Thus, for a given δ > 0, e.g. δ = 1 2 M 0 (β, α), for arbitrary L one can find N (L) large enough, such that the expectation of the spin at the origin differs by more than δ. One can therefore feel the influence from the decimated spins in the far-away annulus, however large the central interval of decimated alternating spins is chosen. Thus, indeed it holds that M + -M -> δ, uniformly in L.

In our choice of decimated lattice we made use of the fact that the constrained system, due to cancellations, again formed a zero-field Dyson-like model. This does not work for decimations to more dilute lattices, but although the original proofs of Dyson [START_REF] Dyson | Existence of a Phase Transition in a One-Dimensional Ising Ferromagnet[END_REF] and of Fröhlich and Spencer [START_REF] Fröhlich | The Phase Transition in the One-Dimensional Ising Model with 1/r 2 interaction energy[END_REF], or the Reflection Positivity proof of [START_REF] Fröhlich | Phase Transitions and Reflection Positivity. I. General Theory and Long Range Lattice Models[END_REF] do no longer apply to such periodic-field cases, the contour-like arguments of [START_REF] Cassandro | Geometry of Contours and Peierls Estimates in d = 1 Ising Models with Long Range Interactions[END_REF] and [START_REF] Johansson | Condensation of a One-Dimensional Lattice Gas[END_REF] could presumably still be modified to include such cases. Compare also [START_REF] Kerimov | A block effect of external field in the one-dimensional ferromagnetic Ising model with long-range interaction[END_REF].

The analysis of [START_REF] Cassandro | Phase Transition in the 1D Random Field Ising Model with Long Range Interaction[END_REF] which proves existence of a phase transition for Dyson models in random magnetic fields for a certain interval of α-values should imply that in that case there are many more, random, configurations which all are points of discontinuity. We note that choosing independent spins as a constraint provides a random field which is correlated. However, these correlations decay enough that this need actually not spoil the argument. Similarly, one should be able to prove that decimation of Dyson models in a weak external field will result in a non-Gibbsian measure. An interesting question would be to perform the analysis of [START_REF] Maes | Percolation, Path Large deviations and Weak Gibbsianity[END_REF]) or [START_REF] Ny | Almost Gibbsianness and Parsimonious Description of the Decimated 2d-Ising model[END_REF] to get a.s. configuration-dependent correlation decays.

On the other side of the Gibbs-non-Gibbs analysis, when the range of the interaction is It turns out that for a one-dimensional long-range model in a decaying field, depending on the relation between α and γ, there can be either one or two extremal Gibbs measures. We can prove the following theorem. we have M 0 (β, α, γ) > 0, i.e. µ + = µ -.

Sketch of proof:

The main idea of the proof is to extend the analysis in [START_REF] Cassandro | Geometry of Contours and Peierls Estimates in d = 1 Ising Models with Long Range Interactions[END_REF][START_REF] Cassandro | Phase Transition in the 1D Random Field Ising Model with Long Range Interaction[END_REF], combined with [START_REF] Littin | Quasiadditive estimates on the Hamiltonian for the One-dimensional Long-Range Ising Model[END_REF]. Consider a finite-volume Gibbs measure on an interval, say Λ = [-N, N ] and fix +-boundary conditions. Each spin configuration σ can be uniquely mapped into a triangle configuration T = (T 1 , ..., T n ) where endpoints of the triangles are defined by interface points dividing plus from minus spins. Contours Γ are collections of triangles T i such that they are in some sense well separated from each other and subadditive, so that we obtain a lower bound for the energy of given triangle configuration T . Phase coexistence will follow from the well-known Peierls argument for d > 1, i.e. fromthe estimate that for for β sufficiently large

µ + Λ [σ o = -1] ≤ µ + Λ [{o ∈ Γ}] ≤ 1 Z + Λ Γ∋o Γ compatible e -βH(T ) < 1 2 .
The main difficulty then is to obtain a good energetic lower bound for the Hamiltonian including the effect of the external field.

Physically, an argument explaining the statement of the theorem goes as follows: There is a competition between the effect of the pair interaction and that of the external field. Having minus boundary conditions means that inserting a large interval [-L, L] of plus spins will cost an energy of order |i|<L |j|>L |i -j| -α = O(L 2-α ).

However, the gain in energy due to the spins following the external magnetic field is of order |i|<L |i| -γ = O(L 1-γ ).

Thus, (somewhat similar to an Imry-Ma argument), we see that for γ > α -1 we should expect that the field is too weak to overcome the boundary conditions and the plus and minus measures are different: µ + = µ -.

When the opposite case pertains, that is γ < α -1, there should be a unique Gibbs measure, with a magnetisation in the direction of the field, whatever the boundary conditions employed. We are in the process of rigorising this picture.

In fact, the analogous prediction in the 2-dimensional short-range model has been fully proved by [START_REF] Bissacot | Phase transition in ferromagnetic Ising models with spatially dependent magnetic fields[END_REF][START_REF] Cioletti | Graphical representations for Ising and Potts models in general external fields[END_REF], also giving that the critical value for γ equals 1, where is possible to prove the phase transition even in the critical case, assuming that h is small enough. Here we have the same situation, we can extend the theorem above for the case when γ = max{α-1, α * -1} if we take h small enough.

The restriction on γ involving α * seems due to technical reasons, since we use arguments developed in [START_REF] Cassandro | Geometry of Contours and Peierls Estimates in d = 1 Ising Models with Long Range Interactions[END_REF]. Also, again based on their paper, for technically reasons we require the nearest-neighbor term to be strong enough. However, from the physical argument sketched above, we expect that these limitations should not be required and the argument should work, just assuming the inequality between γ and α .

Theorem 2 (

 2 [START_REF] Bissacot | Phase transition of long-range Dyson models in inhomogeneous fields[END_REF]) Let α ∈ (1, 2] and γ > max{α -1, α * -1} be the exponents of the Dyson model w.r.t. an interaction Φ D given by (4.17). Then, there exists β D c,h > 0 s.t. for all β > β D c,h

We denote µ[f ] for the expectation Eµ[f ] under a measure µ.

expressing that ∀Λ ∈ S, ∀A ∈ FΛ, ρ(A) > 0 implies that γΛ(A|ω) > 0 for any ω ∈ Ω.

in the sense that for all bounded increasing functions f , and Λ ∈ S, the function γ D Λ f is increasing.

Acknowledgments: We gratefully dedicate this paper to Chuck Newman on the occasion of his 70th birthday. Especially AvE has over many years been inspired by and profited from his scientific contributions, his always friendly stimulating questions and discussions, and his encouraging attitude, whether it was about Dyson models, non-Gibbsian measures, spin glasses or any other topic from statistical mechanics. We thank J. Littin for making [44] available to us and P. Picco for very helpful discussions. RB is supported by CAPES, FAPESP grant 2011/16265-8 and CNPq grants 453985/2016-5 and 312112/2015-7. EOE is supported by FAPESP Grants 14/10637-9 and 15/14434-8. We also thank the universities of Delft and Groningen and STAR-NWO for making various research visits possible.

lower, i.e. for α > 2, or the temperature is too high, uniqueness holds, for all possible constraints and the transformed measures should be Gibbsian. Some standard high-temperature results apply, which were already discussed in [START_REF] Van Enter | Regularity Properties and Pathologies of Position-Space R.G. Transformations: Scope and Limitations of Gibbsian Theory[END_REF]. About these shorter-range models, (i.e. long-range models with faster polynomial decay), Redig and Wang [START_REF] Redig | Transformations of One-Dimensional Gibbs Measures with Infinite Range Interaction[END_REF] have proved that Gibbsianness was conserved, providing in some cases (α > 3) a decay of correlation for the transformed potential. In our longer-range models, for intermediate temperatures (below the transition temperature but above the transition temperature of the alternating-configuration-constrained model) decimating, both +-and --measures, should imply Gibbsianness, essentially due to the arguments as proposed for short-range models in [START_REF] Haller | Absence of renormalization pathologies near the critical temperature. Two examples[END_REF].

Dyson models in decaying fields

In this section we consider one-dimensional Dyson models in a decaying field with decay parameter γ. The corresponding interaction Φ D A (ω) is defined by

for some J, h > 0. The question raised in [START_REF] Bissacot | Phase transition of long-range Dyson models in inhomogeneous fields[END_REF] is whether it is possible to extend results from e.g. [START_REF] Cassandro | Geometry of Contours and Peierls Estimates in d = 1 Ising Models with Long Range Interactions[END_REF][START_REF] Cassandro | Phase Transition in the 1D Random Field Ising Model with Long Range Interaction[END_REF], and in particular to investigate whether and under which conditions the existence of two distinct phases prevails in the presence of an external field. Let us mention that one-dimensional Dyson models in a field were considered before, for example in [START_REF] Kerimov | A block effect of external field in the one-dimensional ferromagnetic Ising model with long-range interaction[END_REF], where uniqueness was proven for fields which are either strong enough (Φ D {i} = h i ω i , where there exists h 0 > 0 such that |h i | > h 0 ) or periodic in large enough blocks.

The main tool we will use are the one-dimensional contours of [START_REF] Cassandro | Geometry of Contours and Peierls Estimates in d = 1 Ising Models with Long Range Interactions[END_REF]. Recall that in [START_REF] Cassandro | Geometry of Contours and Peierls Estimates in d = 1 Ising Models with Long Range Interactions[END_REF] the authors prove that for α ∈ (α * , 2] where α * := 3 -log (3) log( 2) and h = 0 there exists β D c,0 > 0 such that for all β > β D c,0

there is spontaneous magnetization yielding non-uniqueness of the Gibbs measures, µ + = µ -. This result was generalized to all values of α ∈ (1, 2] in [START_REF] Littin | Quasiadditive estimates on the Hamiltonian for the One-dimensional Long-Range Ising Model[END_REF].

Phase coexistence in a positive external field is an unusual phenomenon, since typically Gibbs measures for models in a field are unique. It was previously observed in nearest neighbour pair potentials with polynomially decaying fields in d ≥ 2, see [START_REF] Bissacot | Phase transition in ferromagnetic Ising models with non-uniform external fields[END_REF][START_REF] Bissacot | Phase transition in ferromagnetic Ising models with spatially dependent magnetic fields[END_REF][START_REF] Cioletti | Graphical representations for Ising and Potts models in general external fields[END_REF] or for sufficiently fast decaying (but not necessarily summable) fields on trees [START_REF] Bissacot | Stability of the phase transition of criticalfield Ising model on Cayley trees under inhomogeneous external fields[END_REF]. In [START_REF] Bissacot | Phase transition in ferromagnetic Ising models with spatially dependent magnetic fields[END_REF] it is proven that in nearest-neighbour models for γ > 1 and low enough temperatures, there are multiple Gibbs states, whereas for γ < 1 there is a unique one.

Pirogov-Sinai is a robust and often applicable version of the Peierls contour argument, applicable in d ≥ 2, which is the most generally applicable approach in higher dimensions. In [START_REF] Cassandro | Geometry of Contours and Peierls Estimates in d = 1 Ising Models with Long Range Interactions[END_REF], inspired by and extending results of the seminal paper of Fröhlich and Spencer [START_REF] Fröhlich | The Phase Transition in the One-Dimensional Ising Model with 1/r 2 interaction energy[END_REF], the authors presented a contour argument which works even for long-range models in one dimension, in particular, it worked for one-dimensional Dyson models with α * < α ≤ 2. The techniques used in [START_REF] Cassandro | Geometry of Contours and Peierls Estimates in d = 1 Ising Models with Long Range Interactions[END_REF] rely on developing a graphical representation of spin configurations in terms of triangular contours.