Orchestration of Domain Specific Test Languages with a Behavior Driven Development approach

Robin Bussenot, Hervé Leblanc and Christian Percebois Institut de Recherche en Informatique de Toulouse 118 Route de Narbonne, 31400 Toulouse, France robin.bussenot@irit.fr, leblanc@irit.fr, percebois@irit.fr Abstract-An airplane is composed by many complexes and embedded systems. During the integration testing phase, the design office produces requirements of the targeted system, and the test center produces concrete test procedures to be executed on a test bench. In this context, integration tests are mostly written in natural language and manually executed step by step by a tester. In order to formalize integration tests procedures dedicated to each system with domain specific languages approved by testers, and in order to automatize integration tests, we have introduced agile practices in the integration testing phase. We have chosen a Behavior Driven Development (BDD) approach to orchestrate Domain Specific Test Languages produced for the ACOVAS FUI project.

I. INTRODUCTION

An airplane is composed by many complexes and embedded systems. Due to the criticality of such systems and constraints of verification and validation, a process of certification is necessary in order to commercialize them. In this context, the system integration testing phase is an important topic of the certification process. As a System Under Test (SUT) is composed by hardware and software, a dedicated test bench is needed to interact with it. We focus our work on integration test procedures.

In most cases, integration tests are written in natural language and are executed on a test bench step by step manually. The lack of formalism of the natural language introduces misinterpretations. The step by step execution can be fastidious, expensive and error prone. Tests written in natural language and executed manually are more focused on how the test has to be performed rather than the elicitation of the feature to be tested. The test intention is lost because it is mixed with implementation details. When test procedures are transformed into some scripting languages (Python, Lua, XML, C . . . ) executed on a test bench, the gap between a test scenario and the corresponding script is too large and does not permit debug. Testers have domain skills on the kind of system to test and not necessary computer knowledge to understand test scripts. The lack of test automation implies the waste of time during test replays. The lack of formalization implies the loss of intention in test procedures and then the lack of reusing for other avionic programs.

Our problematic is to improve the formalization of test procedures by introducing several best practices coming from software engineering and the efficiency of the integration testing activities. We used capabilities of Domain Specific Languages (DSLs) to formalize specific test languages dedicated to each system of an airplane. From these languages, program transformations are designed to generate executable code for some kind of test benches with their script languages. BDD promotes test cases naturally described in languages that focuses on preoccupations of a specific domain. The business value of test cases is improved because tests are centered on their intent and can be shared by all the stakeholders of the system under test development. Our work was supported by a FUI1 project named ACOVAS aiming at introducing agile methods with a new generation of test benches.

After presenting system integration testing goals, basis, and needs (Section II), we introduce Domain Specific Languages and Behavior Driven Development, the two main concepts of our proposition (Section III). Then we present our behavior driven development framework for system integration testing (Section IV). The proof of concept is detailed by the two cases studies we have implemented during the project (Section V). We position our work (Section VI) and we conclude and give some perspectives (Section VII).

II. SYSTEM INTEGRATION TESTING

The process development of an entire avionic system is more complex than a software development process because an avionic system is composed by many systems and each system is composed by hardware and software. This process follows several V -cycles, and the test management process belongs to all activities of the process. Then, we recall the objectives of our proposition from observations of the ACOVAS partners.

A. V -cycles

An avionic system has several specific characteristics: real time, safety critical, embedded, and fault tolerant. The development process of each of them follows a V -model. An embedded system merges hardware and software with their specific V -cycle too. These two V -models are named the W -model. The resulting system of these two processes is integrated itself into a main process that follows a V -model too.

Two kinds of testing activities coexist: classic testing in isolation and In-the-Loop Testing. The latter provides models and simulators to put the SUT in simulated flight conditions. This kind of testing allows integration testing as soon as possible. More precisely, testing is also performed since requirements level thanks to Model-In-the-Loop testing (MiL). The SUT behavior and the external environment are simulated through models to ensure that requirements are correct. In Software-In-the-Loop testing (SiL), the real software is tested with an emulated hardware. In Hardware-In-the-Loop testing (HiL), the real software is integrated into the target hardware. Inthe-Loop testing responds to reactive and real time constraints of embedded systems. This kind of testing needs environment models to simulate collaborations between the SUT and other systems.

B. Test procedures management

Test procedures management has a dedicated process too, as explained by Sommerville [START_REF] Sommerville | Software Engineering[END_REF] for a Software Engineering point of view. In System Engineering, this process depends on companies. In the ACOVAS context, the design office and the test center are the two stakeholders that manage integration test procedures. The design office produces requirements of the target system to develop, while the test center produces concrete test procedures to be executed on a test bench to verify that the system conforms to its requirements.

More precisely, the design office produces a test plan providing the integration strategy and acceptance criteria refined from system requirements.

C. Needs

Actually, many test procedures are written in natural language. We identify two different ways to use test procedures written in natural language:

1) Test procedures are read and interpreted by a tester that directly executes each instruction step by step on a test bench. 2) Test procedures are translated via ad hoc transformations into a dedicated scripting language that allows automatic execution. The manual intervention of a tester for the comprehension or the translation of a test procedure written in natural language is fastidious and error prone. More precisely, the list below enumerates the main preoccupations of the ACOVAS stakeholders:

• improve tests automation,

• reduce the time of tests implementation,

• manage changes in tests configuration,

• identify test solutions from the design phase,

• introduce specific languages for procedures definitions. A framework dedicated to support the design and the implementation of test procedures will help testers to produce formalized tests. This formalization allows to compile a test procedure into some scripting languages and then increases the automation of tests. The formalization has been implemented using a DSL workbench.

Moreover, we would like to give an holistic approach [2] by focusing on the communication means provided by test procedures. Our goal is not to build a complete automatic tool chain but to help testers, developers and designers to achieved their tasks in a collaborative manner. Our solution is a framework supporting a BDD approach.

III. BACKGROUND

A. Domain Specific Languages

Each system of an aircraft requires specific domain skills. The ATA 100 (Air Transport Association of America) [START_REF]S-tech Entreprises[END_REF] standard contains the reference to the numbering system which is a common standard for all commercial aircraft documentation. It contains some chapters dedicated to systems that compose an aircraft (Aircraft general, Airframe systems, Structure, Propeller/Rotor, Power plant). This commonality allows greater ease of learning and understanding for pilots, aircraft maintenance technicians, and engineers alike. Testers must be specialized on each system and it is difficult for them to gain competencies both on the SUT and programming languages. The goal of a tester is to focus on domain specifications and not to design complex test procedures. This point requires the introduction of small computer languages understandable by testers and that produce executable procedures on a test bench.

A Domain Specific Language (DSL) is a computer language that allows to provide a solution for a particular class of problems [START_REF] Fowler | Domain-Specific Languages[END_REF]. We can cite for instance SQL, initially named SEQUEL (Structured English Query Language), which was designed to manipulate and retrieve data stored in relational databases. Other examples are HTML and XML configuration languages. A DSL clearly focuses on a small domain in order to be easier to learn by domain experts than general purpose languages (GPLs).

Producing a DSL brings many advantages [START_REF] Völter | DSL Engineering Designing, Implementing and Using Domain-Specific Languages[END_REF]:

• makes easier expressing domain concerns,

• increases the common knowledge about a domain,

• improves team communication,

• can be used by domain experts that not necessarily have computer programming knowledge,

• can be managed by specific tools as IDE for GPL languages,

• hides GPLs complexity,

• can generate many lines of code in GPL from few lines.

Our specific testing languages focus on the description of test scenarii for ATA 21 and ATA 42. Each DSL addresses a particular domain and are designed for test engineers only.

We named Domain Specific Test Language (DSTL) this new category of DSL [START_REF] Bussenot | A domain specific test language for systems integration[END_REF].

B. Projectional vs textual editors

There are many tools to manage and help the implementation of new languages [START_REF] Fowler | Language workbenches: The killer-app for domain specific languages[END_REF]. The most used are Xtext [START_REF] Itemis | XText[END_REF], Spoofax [START_REF] Visser | Spoofax[END_REF] and Meta-Programming System (MPS) [START_REF] Jetbrains | Meta-programming system[END_REF]. All these tools are doing almost the same thing but the main difference is the way of editing a program. Editors can be projectional as they are in MPS, or textual as usual in Xtext and Spoofax. According to Martin Folwer in [START_REF] Fowler | Domain-Specific Languages[END_REF]: "Projectional editing thus usually displays a wider range of editing environments -including graphical and tabular structures -rather than just a textual form."

With a projectional editor, the end-user is guided by the structure of the language and no syntax errors are allowed. With a textual editor, the end-user has an empty text frame and must know the grammar of the language to produce test scenarii. Syntax errors are allowed and require corrections by the user himself.

From the point of view of a DSTL programmer, generators in MPS come with a list of features that allow to go through Abstract Syntax Trees (AST) [START_REF] Campagne | The MPS Language Workbench[END_REF]. Rules programmers are only focused on the semantic of transformations and do not take care about syntactical problems. Textual editors provide small stub generators and are model-to-text generation oriented. For those reasons, we chose MPS to conduct our experimentation.

C. Behavior Driven Development

BDD [START_REF] North | Introducing BDD[END_REF] has emerged to reduce the gap between unit tests and the elicitation of the specifications of the product's behavior. "BDD leads the development of features by designing functional tests used to validate those features" (Agile Alliance, Glossary BDD).

BDD combines Test Driven Development (TDD) and Domain Driven Design (DDD) principles to encompass the wider picture of agile analysis and automated acceptance testing for software production. BDD argues that the expected behavior of software should be described in testable scenarii related to functional requirements. Those testable scenarii can be seen as isolated tests in Acceptance Test Driven Development (ATDD) with a business view in order to improve communication means.

Tests in BDD are structured following a Given-When-Then canvas [START_REF] Solís | A study of the characteristics of behaviour driven development[END_REF], corresponding to the Gherkin language, which is the simplest way of describing a behavior: given a nominal state, when an event changes this nominal state, then the system should produce the expected response. This canvas helps testers to discover and describe a scenario. Scenarii, corresponding to unit tests are used as specifications, and also have in mind to produce automated tests easily understandable by domain experts.

Another advantage of BDD is the communication improvement between stakeholders by using the vocabulary of the business domain and understandable notations by the whole development team. BDD provides a common view of the domain and helps developers and test specialists to understand domain experts needs.

In our approach, a DSTL encapsulates vocabulary and specific actions for testing and managing a kind of SUT as promoted by the BDD approach by using the Gherkin language. In teams adopting a BDD style of development, specialists of the domain write acceptance tests in a formalized language near a natural language. Programmers complete acceptance tests with glue code that provides links with the real software under test. In our case, we have designed a pivot language that a test programmer can use to link acceptance tests to the parameters of the SUT and to the capabilities of the test bench.

IV. A DSTL WORKBENCH

Our first idea is to use DSLs to design a specific test language dedicated for each kind of avionic domain (ATA). All domains involved in the development process of an airplane are covered by a family of such languages that we named DSTLs. Our second idea is to orchestrate three levels of testing languages (languages dedicated to testers, a unique language for test programmers, and scripting languages for test benches) in a BDD approach. The Fig. 1 shows those three levels of abstraction. In this section, we explain how we merge BDD concepts with our DSTLs.

A. The big picture explained

State Chart XML (SCXML) and Python with Data Distribution Services are two examples of executable languages on a test bench. The pivot language has been designed to reduce the semantic gap between languages for testers and languages for test benches. This language is programming oriented and serves to give a readable view of the executable code and to reuse transformation rules to several target languages. Finally, some business specific testing languages are proposed to testers for each kind of system.

Nowadays, the process to design those languages is empirical. To add a new DSTL, programmers have to design projectional editors, semantic constraints and an AST-to-AST transformation into the pivot language. To add a new target language, programmers have to develop an AST-to-text transformation based on transformation patterns used by other transformations proposed by the framework.

B. DSTLs and Gherkin language

DSTLs are testers oriented. Each language is dedicated to a specific domain to take into account testers practices and testing constraints. As example of testing domain we can quote network features, air flow control, flight commands, etc. We describe in the next section two experiments on DSTLs, one for ATA 21 (Air flow control) and one for ATA 42 (IMA). Gherkin language patterns can be used to explain the intention of each unit test. However, for the integration testing phase of reactive systems, tests must follow a flight plan that focus on a unique test objective. The Given-When-Then pattern is transposed into a flight plan pattern.

C. DSTLs and BDD approach

High level test languages are designed to replace test procedures in natural language. Since those languages use a common vocabulary of a specific domain, they can be considered as ubiquitous languages. Those languages are an efficient way of communication between testers, testers and designers, and also testers and test programmers. "BDD offers more precise guidance on organizing the conversation between developers, testers and domain experts" (Agile Alliance, Glossary BDD). Moreover, as BDD extends TDD and ATDD approaches, a test must focus on a single aspect of a program. Tests describe a unique intention to which the code has to conform. During system integration testing phase, test cases must focus on a unique test objective.

D. The pivot language

The semantic gap between high level testing languages and scripting languages for test benches is too wide. On the one hand, high level languages must be closed to natural languages. On the other hand, scripting languages are similar to assembly languages enhanced by the management of interactions with the test bench. The pivot language is a simple imperative language that serves to map high level statements to executable code. It is composed by a set of four atomic statements: set, check during, check until and call externalTool that cover about 80% of basic actions required by tests. The set statement of the pivot language allows to assign a value to an avionic parameter. The goal of the check until statement is to ensure that a state can be reached until an amount of time given by the statement. The check during statement ensures that a state of the SUT is stable during an amount of time given by the statement. This language allows to compose test suites, test cases and unit tests as a xUnit framework. It is open to the addition of new statements and minimizes the effort to translate test procedures into another scripting language. It is test programmer oriented, however it could be used by testers to directly encode their test procedures.

E. Expected benefits

The first benefit is that test procedures can be automated depending on the capabilities of test benches and supported scripting languages at our disposal. Test procedures keep a human readable form nearest of old test procedures written in natural language and executed manually. They are able to explicit a unique intention by test case. As in TDD approaches, tests are isolated from each other and a specific error message can be thrown when a test failed or when a test cannot be run due to a problem with the SUT. The debugging phase during new test campaigns is simplified. The second benefit is the separation between preoccupations of testers (only focusing on expected behavior explained) and programmers (only focusing on glue code to improve the automation of tests) as it is promoted by the BDD approach. The last benefit is the reuse of transformation patterns for the generation of executable code from a unique pivot language.

V. CASES STUDIES

The global specification of an airplane is refined into system domains, systems, sub-systems, and finally equipments. These systems are regrouped in the ATA classification in more than hundred chapters [START_REF]S-tech Entreprises[END_REF].

Two kinds of test procedures are provided by the ACOVAS project:

1) Airflow control test procedures (ATA 21) provided by an equipment manufacturer are textual descriptions in an intermediate level language that will be transformed manually into Excel spreadsheets. These spreadsheets are automatically transformed into an XML dialect. 2) Integrated Modular Avionic (IMA) test procedures (ATA 42) provided by an aircraft integrator are textual descriptions in natural language. Due to the wide expressiveness of natural languages, a similar procedure written by two different testers may have significant differences. Moreover, these procedures will be manually executed by another tester.

A. ATA 21

The ATA 21 focuses on airflow control systems of an airplane. Those systems manage the temperature of each area by controlling the airflow rate of each duct supplying them.

The mind map presented by the Fig. 2 has been retroconceived closely with experts of the ATA 21 from procedures written in pivot language. It shows the main The generator coming with the DSTL translates each statement into the pivot language. Generally, variables manipulated by DSTL statements correspond to a set of real avionic parameters. Statements used to decrease or increase a temperature or an airflow rate are translated into a collection of set statements of the pivot language. Statements to verify a behavior or to wait an event are translated into a single check until statement of the pivot language. The statement to check the temperature of an area is translated into a single check during statement of the pivot language.

We implemented our conceptual framework for an air flow control test procedure. The DSTL procedure contains 48 lines of code, and the generated one into the pivot language contains 68 lines. We generated 2010 lines of SCXML code and 290 lines of Python code.

B. ATA 42

The ATA 42 focuses on Integrated Modular Avionic systems. Those systems are used as data concentrator and provide data to other systems of the airplane.

Procedures dedicated to ATA 42 provided by ACOVAS partners are described thanks to natural language. Procedures are composed by a set of test cases and test cases are completed by sentences divided in five kinds (Step, Check, Trace, Log and Reminder). We have studied 10 procedures containing around 3700 tagged sentences. We have used Natural Language Processing tools [START_REF] Manning | The stanford corenlp natural language processing toolkit[END_REF] to provide statistics about verbs the most used for each kind of sentence. The table I shows the number of sentences and the rate of each kind of sentence.

We have collaborated with experts of ATA 42 to produce the mind map of all needs for this domain formalization. We have designed seventeen kinds of Step statements, one for Log statements, one for Check statements and five for Trace statements. The Fig. 3 presents the mind map for the Step statements that formalize interactions with the SUT. Testers have formalized their procedures thanks to our DSTL workbench successfully. To improve the formalization, some semantic contraints must be added to complete projectional editors. To assume test automation, a generator must be provided for Step, Trace and Log statements. To provide an oracle, a new formalization and a new generator must be provided for Check statements.

VI. RELATED WORK

We considered some contributions related to the transposition of best practices coming from software engineering into avionic systems testing. We identify three scientific axes:

1) Test language ( [15]), 2) Test automation in an avionic context ( [START_REF] Ott | System Testing in the Avionics Domain[END_REF], [START_REF] Guduvan | A Meta-Model for Tests of Avionics Embedded Systems[END_REF]), 3) Agility in an avionic context ( [START_REF] Salo | Agile methods in european embedded software development organisations: a survey on the actual use and usefulness of extreme programming and scrum[END_REF], [START_REF] Manhart | Breaking the Ice for Agile Development of Embedded Software: An Industry Experience Report[END_REF], [START_REF] Smith | E-TDD -embedded test driven development a tool for hardware-software co-design projects[END_REF], [START_REF] Karlesky | Effective test driven development for embedded software[END_REF], [START_REF] Wils | Agility in the avionics software world[END_REF]).

A. Test language

The European Telecommunications Standards Institute (ETSI) has standardized a general purpose language named Test Description Language (TDL) [START_REF] Etsi | The Test Description Language (TDL); Part 1 : Abstract Syntax and Associated Semantics[END_REF] dedicated to the specification of test descriptions and the presentation of test execution results. This language is between test purposes described in natural language and the necessary complexity to implement tests in Test Control Notation version 3 (TTCN-3) for example.

Unfortunately, a tester without computer programming skills cannot understand TDL test descriptions. None of the partners of the ACOVAS project have adopted this standard which comes from the telecommunications industry.

B. Test automation in an avionic context

Ott [START_REF] Ott | System Testing in the Avionics Domain[END_REF] focuses on new testing activities arising from IMA architectures. These activities concern automated bare IMA modules and a network of configured IMA modules. The automation of these activities is based on generic test templates. In the first case, a user must instantiate data configurations for a bare IMA module. In the second case, data configurations are automatically generated from the ICD (Interface Control Document) of a network of IMA modules. In this work, the effort of automation uniquely concerns unit tests of IMA modules and networks of IMA modules. A more general framework is proposed by Guduvan et al. [START_REF] Guduvan | A Meta-Model for Tests of Avionics Embedded Systems[END_REF] who use a model-driven development approach to generate test cases for all kinds of avionic tests. The structure of test procedures is defined thanks to a dedicated meta-model. Test cases are components of a test suite or a test group. They must be decomposed into behavioral sequences of statements, and each statement corresponds to a specific test order. As in our work, the test procedure structure is domain independent, while specific statements are related to a specific domain.

C. Agility in an avionic context

The usefulness of Scrum and XP was studied by Salo et al. [START_REF] Salo | Agile methods in european embedded software development organisations: a survey on the actual use and usefulness of extreme programming and scrum[END_REF] in many embedded software development projects from European organizations. This study reveals that these organizations seem to be able to apply agile methods in their projects. A process named Safe Scrum was proposed by Stålhane et al. [START_REF] Stålhane | The application of safe scrum to IEC 61508 certifiable software[END_REF] to introduce agile principles in an embedded software certification context. A test first approach was proposed by Manhart et al. [START_REF] Manhart | Breaking the Ice for Agile Development of Embedded Software: An Industry Experience Report[END_REF] for high-speed software engineering for embedded software. This work focuses on unit testing and mixes agile practices with conventional process activities. All these works are dedicated to software embedded development only.

A DSL is very closed to an ubiquitous language used by the BDD approach [START_REF] Solís | A study of the characteristics of behaviour driven development[END_REF]. The structure of an ubiquitous language comes from the business model and contains terms which will be used to define the behavior of a system. The main idea is that customers and developers share the same language without ambiguity.

VII. CONCLUSION

Integration test procedures are conceived to verify several behaviors expected of a system responding to several stimuli during a flight plan. Usually, integration test procedures are described by a specific domain vocabulary. These procedures are good candidates to adopt a behavior driven approach. We have formalized and automatized test procedures for ATA 21 and two demonstrations have been made on two different test benches with StateChart XML and Python scripting languages.

We will design a new DSTL dedicated to another ATA to propose a global process to guide the design of new DSTLs. These future works will be supported by the ESTET (Early Systems TEsTing) project leaded by the DGAC 2 .

Fig. 1 .

 1 Fig. 1. The big picture Fig big pic

Fig. 2 .

 2 Fig. 2. Mind map of ATA 21

Fig. 3 .

 3 Fig. 3. Mind map for Step statements Fig p fo

  Test strategy includes test means needed to perform test cases. Test procedures are refined from test cases descriptions according to test means. Generally, test procedures are stored in textual documents.

	A Lab Test Request (LTR) is
	derived from a test plan. It is focused on test objectives and
	is related to a specific version of the SUT. A test objective
	corresponds to one expected feature of the SUT. It is described
	by a single sentence completed by a flight scenario which is
	independent of test means and an environment model from
	which results of a test can be exploited. The test center
	produces a test strategy that matches with test objectives of the
	LTR.

TABLE I NUMBER

 I OF SENTENCES SORTED BY KIND

		Step	Check Log Trace Reminder
	Number 1274	1047	594	520	269
	%	34,4	28,3	16	14	7,2

Fonds Unique Interministériel (FUI) is a French program dedicated to support applied research, to help the development of new products and services susceptible to be marketed in short or middle term.

ACKNOWLEDGMENT

We would like to thank the ACOVAS (outil Agile pour la COnception et la VAlidation Système) partners for their collaboration and their involvement during the project.


Guifen Chen

Wang Ke

Ma Li

Research on Intelligent decision of pulmonary tuberculosis disease based on Data Mining

Keywords: pulmonary tuberculosis disease, rough set, decision tree, intelligent diagnosis

Aiming at the problem that the low diagnostic efficiency and low accuracy of the single data mining method for Diagnosis of pulmonary tuberculosis, In this study, the electronic records of 1203 cases of tuberculosis patients in Changping District City, Beijing City of Beijng and Beijing Institute of tuberculosis control and tuberculosis control were build, Tuberculosis disease diagnosis model is built by application of rough set and decision tree method, On the basis of this, the diagnosis system of pulmonary tuberculosis was constructed. In this study, The combining method of rough set and decision tree was approached to attribute reduction, the model reduced redundant 57attributes and remained 22 attributes, and articled 7 the decision rules. The model accuracy is 89.46%. Compared with the non reduction method, the decision rule was reduced by 128%, and the accuracy of the model remained unchanged. The research results showed that the algorithm can reduce the time and space complexity of the algorithm while ensuring the accuracy of the model, so as to improve the efficiency of the mining, and provide some references for clinical diagnosis.

Introduction

The digitization of medical equipment and instruments makes the information capacity of the hospital database expand, including a large number of clinical information about the patient's medical history, diagnosis, examination and treatment [1] . How through the efficient, intelligent algorithm of massive pulmonary tuberculosis disease diagnosis and treatment data data mining. According to the hidden relationship between treatment outcome and medical records data, looking for a reliable and feasible method for diagnosis and treatment of that early and effective for medical personnel to provide for auxiliary diagnosis and treatment, has important clinical significance [START_REF] Xin | A case control study on the risk factors of pulmonary tuberculosis[END_REF][START_REF] Qi | Decision tree model for the classification and prediction of tuberculosis treatment program [J][END_REF][START_REF] Zheng | The epidemiological characteristics and trends of the time of onset of pulmonary tuberculosis in China from 2005 to 2011[END_REF] .

Pulmonary tuberculosis is a chronic respiratory disease which is seriously harmful to human health. At present ,the diagnosis of typical tuberculosis is mainly through the observation of the clinical manifestations, sputum sputum, chest imaging, bronchoscopy judgments [START_REF] Dachuan | Research progress in the diagnosis of pulmonary tuberculosis [J][END_REF] .How to treatment of TB suspicious symptoms and suspected pulmonary tuberculosis patients with reasonable examination and early diagnosis and treatment, Reduce the further spread of Mycobacterium tuberculosis [START_REF] Wu Tengyan | Study on the application effect of tuberculosis control mode in Guangxi[END_REF] ; How to get rid of the single index, establish the patient's multi modal clinical information, and dig out the clinical index which is closely related to the pathology, to realize the clinical identification of pulmonary tuberculosis is one of the important clinical needs in the Department of respiration [START_REF] Lu | Vega electronic medical records system and the data of diagnosis and treatment of chronic respiratory diseases [D]. mining research of[END_REF] . This paper attempts to improve the existing mining algorithms according to the data features of pulmonary tuberculosis, by the combination of rough set and decision tree method. Attribute reduction is carried out, and the decision tree rule set is extracted, Mining the data of medical records with implicit diagnosis rules, to obtain new knowledge discovery, to provide reference for clinical treatment of tuberculosis patients. The method of mining the medical record data with implicit diagnosis rules, obtain new knowledge discovery, to provide reference for clinical treatment of tuberculosis patients.

Guifen Chen (1956-),College of Information Technology, Jilin Agricultural University,ChangChun,China 2 Data Mining Foundation

Rough Set Theory

Rough set theory is the core of knowledge and approximate set,and deals with imprecise, uncertain and incomplete data. In the problem solving process, it is not required to provide any prior information of the problem data set, so it can deal with the uncertainty problem more objectively [START_REF] Guifen | Clustering, rough set and combination algorithm of decision tree in soil fertility evaluation[END_REF] .

Rough set theory can describe knowledge with four yuan ordered groups, namely: k = (U, a, V, d). Where u is the universe; A is all attributes; V= ∪a∈ A Va, Va is the attribute range; d: U× A→ V is a function of information, dx: A→V, x∈ U,says the object x in K complete information, in which dx (a)= d(x , a). For such information systems, each attribute subset defines an equivalence relation on the domain, namely B A, defined RB: xRB y dx (b) = dy (b) , b∈ B [9] .

Note by the equivalence relation attribute set B  A T, is derived for the RB. a∈ A T , if RAT = RA T\ {a}, said the a attribute is redundant; If there is no redundant attribute in the system, then the AT is independent; IfRB = RAT and B is not redundant subset of attributes, then B AT is called A T reduction, denoted as red (AT),

The intersection of all reduction of A T is called the A T kernel, denoted as core (AT) [10] .

Decision Tree Method

The decision tree method represents as a decision tree on the basis of the function that will be learned from a set of training data by a large amount of data to be classified according to a certain target, from which to find useful, potential information, commonly used in the classification of the prediction algorithm [START_REF] Xinghua | Research on fuzzy decision tree algorithm based on axiomatic fuzzy set[END_REF] . Decision tree method has the characteristics of high speed, high precision, simple model and so on.

It has a wide application in data mining [12 , 13] . The decision tree is constructed including two steps: generating decision trees and decision tree pruning.

The decision tree is generated from a root node, which is to construct a tree from top to bottom by constantly dividing the sample into subsets [START_REF] Rui | ID3 decision tree algorithm analysis and improvement of[END_REF] . The test value for each attribute is represented as a non leaf node on the tree. Each result is represented as a branch of the tree, and the final classification class is the leaf node of the tree. In the decision tree structure, the information gain is used as the criterion for dividing the nodes [START_REF] Wenbin | Network anomaly detection and filtering based on decision tree classification[END_REF] . Because of noise and outliers, so the decision tree can cause abnormal branching, so need pruning strategy [START_REF] Jian | Decision Tree Mining Journal of research and application of technology in the surgical diagnosis[END_REF] . In the decision tree pruning, usually choose leaf nodes to replace one or more sub tree, then select the class with the highest probability for the node of the category, can also replace the subtree with the branches.

Data Preprocessing

Because medical records are manually entered into the text or database by doctors or non medical professionals, there will be a large number of records, records are not uniform, record errors and noise, and therefore need to pre process the data of the original medical records [START_REF] Main Fu | The design and implementation of electronic medical record system[END_REF] . Data preprocessing mainly includes case data acquisition, attribute selection, discretization of continuous attributes, noise and missing data in data processing, case selection and so on [18]. In order to further data mining, it is necessary to encode the values and fields in the information data table. The coding of the tuberculosis disease is shown in Table 1, and the pre processed data is shown in table 1. (2)the results

Attribute Reduction Based on Rough Set

There are many attributes in the medical record data, and there is often a certain degree of dependence between each attribute, which can not be simply deleted.

Reduction under the premise without loss of information, can simply indicate the decision system,and the decision attributes set dependencies of condition attribute set to remove unnecessary attributes from the condition attributes , and simplify the condition attribute, and improves the efficiency of data mining [19 , 20] . In this paper, the remaining attributes are 22, and the redundant attributes are reduced by 54, and the attributes of the reduced attributes are shown in Table 2.

Decision Tree Model

The C5.0 algorithm is used to construct decision tree, importance degree sequence is as follows according to the information gain to establish attribute: before treatment of In this study, the combination of rough set and decision tree method and a single decision tree method are compared in terms of rule number, accuracy, and confidence interval and modeling time. The comparison results are shown in Table 4. From the table, we can see that a single decision tree method is extensive in classification, there are still some redundant attributes ,that resulting in the construction of a large decision tree size, more rules are extracted, and mining efficiency is not high. In this study, we use existing pulmonary tuberculosis disease archives data, by combining rough set and decision tree optimization algorithm for screening of 22 attribute variables to establish TB prediction and classification model and delete redundant attributes, simplifying the decision model, the mining efficiency is improved.

Redundant attributes are removed, the decision model is simplified, and the mining efficiency is improved. (3) Data layer: this layer mainly use database virtualization technology, middleware technology, a unified scheduling and allocation of medical information resources in each node of the database resources. Data layer, including all kinds of electronic information resources, the layer is the core task of data synchronization, data management, data reduplication, data backup.

results and discussion

Medicine development has been the experience of medical and medical experiments to evidence as the basis of evidence-based medicine; this produces the large amount of medical data and objectivity. Clinicians should be combined with history, clinical symptoms, and gradually learn to use large samples to establish the model of data mining, and analyze the condition, and make reasonable treatment and predicting disease development [START_REF] Hehua | Evidence based medicine in intensive care informatization development[END_REF] .Computer aided medical data mining implements the redundancy elimination of medical data, standardized storage and data seamless integration and sharing, knowledge extraction automation and visual expression and other functions [START_REF] Yao Yuan | The design and application of the chronic disease diagnosis and treatment information system of military hospital [D]. Chinese people's Liberation Army Military Medical Science Academy of the PLA[END_REF] . Decision tree method of data mining in the diagnosis of pulmonary tuberculosis has been applied, such as Zhangqi "decision tree model for classification of tuberculosis treatment regimens and pre sentenced", which illustrate that the data mining method is suitable for pulmonary tuberculosis disease diagnosis and classification problems,

1 ( 1 )( 2 )( 3 )Fig. 1

 11231 Fig. 1 part of the original data3.2 Tuberculosis Diagnosis Based on Data Mining

  acid fast bacilli in sputum culture have to do as a result of, rapid culture of Mycobacterium tuberculosis in sputum results and treatment of chest DR is the, tuberculosis antibody, sputum Mycobacterium tuberculosis X-pert detection, tuberculin test, gamma -IFN release test /T-SPOT. (lung tissue biopsy / pleura in water). This and clinical diagnosis of pulmonary tuberculosis patients diagnosis and treatment standard is basically the same. The results are shown in Figure 2

Fig. 2 1 and 7 3. 3

 2173 Fig.2 attribute importance ranking According to C5.0 decision tree construction, in order to avoid the effect of sampling errors of a single partition of the results, improve the accuracy of the model, the decision tree model uses of ten fold cross validation, model accuracy is 83.46 percent, support data number is 1004. and the 7 decision rules are dig out. In accordance with the support, confidence in descending order as shown in Table3.

Fig 3 ( 2 )

 32 Fig 3 the system architecture of the diagnosis of pulmonary tuberculosis

  but single application of decision tree constructing diagnosis model, there are a large number of research variables included, which may cause problems in the test of efficacy decreased.This study uses existing pulmonary tuberculosis disease archives data, by combining rough set and decision tree optimization algorithm for screening of 22 attribute variables establish TB prediction and classification model, this model deletes redundant attributes, improves the efficiency of mining, and provide a reference for clinical diagnosis. In addition, the research ideas and methods of this study can also apply to the choice of regimen in the treatment of other chronic diseases, such as hypertension, diabetes.Practice has proved that the optimization algorithm based on the combination of rough set and decision tree can effectively deal with the data of uncertainty reasoning, is a powerful tool for data mining, and in the future of medical data mining and utilization has broad application prospects.

Table 1

 1 part of Pulmonary tuberculosis disease attribute code table

	Attribute	Data discretization and	Attribute	Data discretization
		coding		and coding
	sex	(1) male (2) female	Cough and	(1) no (2)yes
			expectoration
	Grouping	(1) the treatment group (2)	Cough,	(1) no (2)yes
		in the first group	expectoration less
			than 2 weeks

Table 2

 2

		attributes after reduction
	General information of	Contact history, diabetes, tuberculosis pleurisy
	patients (3 items)
	Main symptoms before	Cough, sputum or hemoptysis / 2 weeks, bloody sputum, chest
	treatment (5 items)	pain, physical examination found, without any other symptoms.
	Check project development	Erythrocyte sedimentation rate, C-reactive protein value, before
	before treatment (14 items)	treatment of acid fast bacilli in sputum culture have to do as a result
		of, sputum Mycobacterium tuberculosis rapid culture results has
		been done, tuberculin test, tuberculosis antibody has something to
		do with the results, interferon gamma release test /T-SPOT, sputum

Table 4

 4 Comparison of model results

	Model	rules number accuracy rate	confidence interval	modeling time
	Combinatorial	7	83.46%	60%-99%	0.02
	optimization
	method
	Single decision	16	82.18%	63% -98%	0.25
	tree method
	4

Establishment of the Decision System

	According to the function and application requirement of the diagnosis of the disease,
	this paper constructs the system architecture of the tuberculosis disease diagnosis
	system, which includes three levels: data layer, service layer, application layer, as
	shown in figure 3, In System construction, Java is used as the development language,
	B/S mode is used as development model, the application of Spring3 +Hibernate4
	+Struts2 model is used to develop. Development platform is : the operating system is
	Windows 7 Professional Edition, server is Tomcat_7.0.14, the data base tool is
	MySQL Server 5.0, using JDBC is use to link database, LINQ to SQL is used to
	realize data access layer.

Acknowledgments.

Funding for this research was provided by The national Spark plan "based on the IOT of maize precision technology integration and demonstration" (2015GA66004)


Sabine Cassat
email: sabine.cassat@irit.fr

Marcos Serrano
email: marcos.serrano@irit.fr

Emmanuel Dubois
email: emmanuel.dubois@irit.fr

Pourang Irani
email: pourang.irani@cs.manitoba.ca

A Novel Interaction Paradigm For Exploring Spatio-Temporal Data

Keywords: Spatio-temporal dataset, data visualization, tangible interaction. ACM Classification H.5.m. Information interfaces and presentation (e.g., HCI): Miscellaneous

Complex spatio-temporal data is difficult to visualize and even further to interact with, especially by several users at the same time. However, visualization and exploration of such data are essential for experts to understand complex data environments, such as for mitigating the adverse effects of disease spread. This paper presents an alternative approach to that of current spatiotemporal data visualizations to access, interpret, and manipulate spatio-temporal datasets as a single user or as a team. Our approach uses tangible and visual tools such as mini-robots, tabletop displays and augmented reality tools, to facilitate the data exploration and interpretation. We also introduce a simple use case that illustrates one of the possible utilization of the system. While tangibles have been introduced to represent information, we are investigating manners in which we can depict even more complex datasets. Our system will provide a novel approach to manipulate 3D and 4D datasets that classic tools such as a 2D mouse or a tactile screen would not allow.

Introduction

Spatio-temporal datasets are generated daily, to collect disease, migration or any other movement data. Their visualization usually includes animated visualizations or representations of space-time cubes [START_REF] Kapler | GeoTime Information Visualization[END_REF], as illustrated in Figure 1. When facing such rich and complex data environments, the exploration and interpretation of data is crucial for acting on the systems, in order to, for example, limit the spread of a disease or identify traffic patterns. To facilitate this process, novel interactive tools are thus required.

Three key elements emerge from spatio-temporal datasets [START_REF] Amini | The Impact of Interactivity on Comprehending 2D and 3D Visualizations of Movement Data[END_REF]. The first represents the object that is often the key entity. The second is the location of the movement they represent. Finally, the last key element is the time element that defines how fast the object moves or appears at a given location. For such threedimensional datasets, finding a coherent and efficient interaction support to perform a set of operations to better explore these data or to manipulate them is a major challenge [START_REF] Bach | A Review of Temporal Data Visualizations Based on Space-Time Cube Operations[END_REF]. Solutions explored so far include the mouse, tangible user interfaces and mid-air interaction [START_REF] Cordeil | Design space for spatio-data coordination: Tangible interaction devices for immersive information visualisation[END_REF]. Using a 2D device, such as the mouse, for a 3D task greatly limits the efficacy of the interaction. Although tangible interfaces add physicality, it does not intrinsically provide feedback and raises issues in terms of the number of physical bricks that can be used simultaneously. Finally, mid-air interaction supports direct data manipulation, but may be limiting in terms of gesture detection and 3D perception (related to depth detection). These solutions, taken separately, allow limited manipulation of spatio-temporal datasets, but combined with other novel interaction technologies, can open radically new vistas for data exploration. This article presents a different approach to spatiotemporal dataset exploration by combining the inherent utility of tangible objects [5][9] to represent entities in a set of spatio-temporal data with the opportunity to enrich the information they represent in space and time by using embedded visualizations [START_REF] Willett | Embedded Data Representations[END_REF]. By doing so, we are developing an agenda that will push the boundaries of what is possible with tangible user interfaces for data exploration and interaction.

Approach

As is common, most approaches in human-machine interfaces are iterative and build on several designimplementation-evaluation steps, allowing a continuous refinement of the initial project conception, while eliminating technical assumptions and validating empirically the various concepts involved.

First, we propose to represent objects using physical entities such as Zooid mini-robots [START_REF] Goc | Zooids: Building Blocks for Swarm User Interfaces[END_REF] or Ozobots [START_REF] Ozobot | Robots to code, create, and connect with[END_REF], or even the Cellulo robots [START_REF] Özgür | Haptic-Enabled Handheld Mobile Robots: Design and Analysis[END_REF] that can provide additional haptic feedback. They are dynamic, can move along predefined trajectories and represent the tangible element of our dataset exploration. Each robot representing an object can be touched, held and probed to facilitate quick data exploration. Because of their autonomous behavior, they are able to adapt to physical interaction from the user or to the reception of new digital inputs that offer additional information.

Secondly, we propose to represent the spatial element through a tabletop display at first, and finally with visual displays superimposed on our tangible bots (via augmented reality). The mini-robots will be able to move freely on the tabletop display indicating their spatial extent and locations, as illustrated in Figure 2. Tabletop displays are an intermediate physical support to minirobot movement, but they are also interactive. The operators can request more information directly on the table, as if they interacted with their own laptop. Finally, the temporal dimension is represented by the movement of the mini-robots in place.

An often overlooked aspect of visual analysis is the need for a personal "scratch pad" [START_REF] Mcgrath | Branch-explore-merge: Facilitating Real-time Revision Control in Collaborative Visual Exploration[END_REF]. In addition to the spatio-temporal information provided by the mini-bots moving on the interactive table, we propose to add personalized visual information associated with each of the bots to add or merge the spatio-temporal data communicated by the tabletop displays and the movement of mini-robots. This will then greatly facilitate the visualization of data and their variation based on physical interactions or the reception of new data.

Use case

An ideal case of using such a platform could be the following scenario. A group of scientists have assembled to mitigate what is considered to be one of the deadliest and fasting spread of an epidemic. They need to contain the disease. Assembled around a digital table in a control room, they have objects they can hold, and that move around the table, each representing an entity they suspect as being a carrier of the disease. As the pieces move autonomously or as a scientist picks a piece and moves it to a location, the underlying spatio-temporal data gathered from surveillance cameras or sensors, shows the reciprocal movement of all the other agents or objects. This quickly gives them clues about the displacement of such entities, their locations, the speed at which they travel, and their future movement direction. Through multiple interactions, the scientists quickly ascertain which agents are moving the most, the ones that cover vast distances as well as those that have the most interactions with other agents. They quickly make a decision about which individuals to quarantine and the disease is soon under control.

Given the widespread movement of individuals, the multitude of daily interactions and the complexities inherent in these, we cannot resort to mouse and computer solutions, any longer. These do not allow for the exploration of data groups, they limit the visualization for the information to a display and offer minimal interactivity. Instead, we propose to develop a platform composed of tangible moving objects, with representations of their interactions on the digital tabletop, such that group can interact and dissect with vast repositories of data. Such a development will be matched with equal effort to advance our knowledge for how tangibles and augmented displays can be used for displaying, interacting and analyzing complex data.

Conclusion

A system that simultaneously employs tangible and visual interaction to access, interpret, and manipulate spatio-temporal datasets can offer novel interaction methods. Mobile mini-bots on a tabletop display, that an analyst can physically manipulate (grasp, hold, displace) allows direct interaction with complex spatio-temporal data, a service that other classical tools cannot provide. Furthermore, personalized data visualization displayed above each robot will greatly reduce the workload and will allow each user to consult the adapted data to its profile. As such, the system will provide a better understanding of complex data for each operator, but also for the team working in collaboration. In addition, the interactive table will enhance teamwork while allowing physical support for the movements of mini- bots and tracking their trajectory. We are embarking on a project to implement this method for complex data visualization and manipulation by focusing on the consistency between the behavior of the mini-robot swarm, the tabletop displays and the additional visual information displayed above the physical entities using augmented reality. This will provide us an environment to examine a multitude of hypotheses.

Figure 1 :

 1 Figure 1: Different examples of space-time visualizations [2].

Figure 2 :

 2 Figure 2: Use of tangible robots to visualize simple data [5].

Acknowledgements

This work is supported by a CNRS PICS grant and NSERC.


Claudette Cayrol
email: ccayrol@irit.fr

Marie-Christine Lagasquie-Schiex

Keywords: argumentation, higher-order attacks, logical theory

We propose a logical encoding of extended abstract argumentation frameworks, that is frameworks with higherorder attacks (i.e. attacks whose targets are other attacks). Our purpose is to separate the logical expression of the meaning of an attack (simple or higher-order) from the logical expression of acceptability semantics. We consider semantics which specify the conditions under which the arguments (resp. the attacks) are considered as accepted, directly on the extended framework, without translating the original framework into a Dung's argumentation framework. We characterize the output of a given framework in logical terms (namely as particular models of a logical theory). Our proposal applies to the particular case of Dung's frameworks, enabling to recover standard extensions.

I. INTRODUCTION

Formal argumentation has become an essential paradigm in Artificial Intelligence, e.g. for reasoning from incomplete and/or contradictory information or for modeling the interactions between agents [START_REF] Rahwan | Argumentation in Artificial Intelligence[END_REF]. Formal abstract frameworks have greatly eased the modeling and study of argumentation. The original Dung's argumentation framework (AF) [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF] consists of a collection of arguments interacting with each other through a relation reflecting conflicts between them, called attack, and enables to determine acceptable sets of arguments called extensions.

AF have been extended along different lines, e.g. by enriching them with positive interactions between arguments (usually expressed by a support relation), or higher-order attacks (i.e. attacks whose targets are other attacks). The idea of encompassing attacks to attacks in abstract argumentation frameworks has been first considered in [START_REF] Barringer | Temporal dynamics of support and attack networks : From argumentation to zoology[END_REF] in the context of an extended framework handling argument strengths and their propagation. Then, higher-order attacks have been considered for representing preferences between arguments (second-order attacks in [START_REF] Modgil | Reasoning about preferences in argumentation frameworks[END_REF]), or for modeling situations where an attack might be defeated by an argument, without contesting the acceptability of the source of the attack [START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF]. Attacks to attacks and supports have been first considered in [START_REF] Gabbay | Fibring argumentation frames[END_REF] with higher level networks, then in [START_REF] Villata | Modelling defeasible and prioritized support in bipolar argumentation[END_REF]; and more generally, [START_REF] Cohen | An approach to abstract argumentation with recursive attack and support[END_REF] proposes an Attack-Support Argumentation Framework which allows for nested attacks and supports, i.e. attacks and supports whose targets can be other attacks or supports, at any level. Here is an example of higher-order attack in the legal field.

Example 1: The lawyer says that the defendant did not have intention to kill the victim (Argument b). The prosecutor says that the defendant threw a sharp knife towards the victim (Argument a). So, there is an attack from a to b. And the intention to kill should be inferred. Then the lawyer says that the defendant was in a habit of throwing the knife at his wife's foot once drunk. This latter argument (Argument c) is better considered attacking the attack from a to b, than Argument a itself. Now the prosecutor's argumentation seems no longer sufficient for proving the intention to kill. A natural idea that has proven useful to define semantics for these frameworks, known as "flattening technique", consists in turning the original extended framework into an AF by introducing meta-arguments and a new simple (first-order) attack relation involving these meta-arguments [START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF], [START_REF] Cohen | An approach to abstract argumentation with recursive attack and support[END_REF], [START_REF] Cayrol | Towards a new framework for recursive interactions in abstract bipolar argumentation[END_REF]. More recently, alternative acceptability semantics have been defined in a direct way for argumentation frameworks with higher-order attacks [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF] or for higher-order attacks and supports [START_REF] Cohen | On the acceptability semantics of argumentation frameworks with recursive attack and support[END_REF]. The idea is to specify the conditions under which the arguments (resp. the interactions) are considered as accepted directly on the extended framework, without translating the original framework into an AF.

In this paper, we propose a logical encoding of argumentation frameworks with higher-order attacks. Our purpose is [START_REF] Rahwan | Argumentation in Artificial Intelligence[END_REF] to characterize in a logical way the meaning of an attack (simple or higher-order) (2) to encode the acceptance conditions for arguments and attacks proposed in [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF] and (3) to characterize the outputs of the framework in logical terms, thus enabling to use logical tools for computational issues.

The connection between abstract argumentation and logics goes back to the seminal work of Dung, where a translation from an AF to a logic program was given. This line of research has been pursued with other kinds of translation e.g. in [START_REF] Caminada | On the equivalence between logic programming semantics and argumentation semantics[END_REF]. Other works have encoded acceptance conditions as logical formulae of a first-order theory e.g. [START_REF] Besnard | Encoding argument graphs in logic[END_REF], or defined a logical language for expressing the dynamics of a framework e.g. [START_REF] Dupin De Saint-Cyr | Argumentation update in YALLA (Yet Another Logic Language for Argumentation)[END_REF]. To the best of our knowledge, all these works consider only attacks between two arguments. In [START_REF] Gabbay | The attack as strong negation, part I[END_REF] a representation of a standard abstract argumentation network is provided by adding Logical Encoding of Argumentation Frameworks with Higher-order Attacks strong negation to classical logic. Then, networks with higher level attacks can be represented via a translation of the original network into a standard one by the addition of nodes.

The paper is organized as follows: the necessary background is given in Sect. II; the logical encoding is presented in Sect. III and IV; some related works are discussed in Sect. V and Sect. VI concludes the paper. Due to lack of space, the proofs are omitted but are available in a technical report [START_REF] Cayrol | Acceptability semantics in recursive argumentation frameworks: Logical encoding and computation[END_REF].

II. BACKGROUND

A. The Standard Abstract Framework

The standard case handles only one kind of interaction: attacks between arguments. A graphical representation can be used for an AF: an attack (a, c) ∈ R is represented by two nodes a, c and an edge from a to c:

a c
We recall the definitions1 of some well-known extensionbased semantics. Such a semantics specifies the requirements that a set of arguments should satisfy. The basic requirements are the following ones:

An extension can "stand together". This corresponds to the conflict-freeness principle. An extension can "stand on its own", namely is able to counter all the attacks it receives. This corresponds to the defence principle. Reinstatement is a kind of dual principle. An attacked argument which is defended by an extension is reinstated by the extension and should belong to it. The characteristic function of AF is defined by: F(S) = {a ∈ A such that a is acceptable wrt S}. S is admissible iff S is conflict-free and S ⊆ F(S). S is a complete extension of AF iff it is conflict-free and a fixed point of F. S is the grounded extension of AF iff it is the minimal (wrt ⊆) fixed point2 of F. S is a preferred extension of AF iff it is a maximal (wrt ⊆) complete extension. S is a stable extension of AF iff it is conflict-free and for each a ∈ S, there is b ∈ S with(b, a) ∈ R.

Note that the complete (resp. grounded, preferred, stable) semantics satisfies the conflict-freeness, defence and reinstatement principles.

B. A Framework with Higher-Order Attacks

We consider a framework that allows representing both simple and higher-order attacks, i.e. attacks from an argument to either another argument or another attack. Such a framework has been usually called "recursive argumentation framework" in literature. So we keep this latter expression, even it is not completely satisfactory.

Definition 3: [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF] A recursive argumentation framework (RAF) is a tuple A, R, s, t where A is a finite and non-empty set of arguments, R is a finite set disjunct from A representing attack names, s is a function from R to A mapping each interaction to its source, and t is a function from R to (A∪R) mapping each interaction to its target.

Note that an AF can be viewed as a particular RAF with t being a mapping from R to A.

A RAF can also be graphically represented: an attack named α (with s(α) = a and t(α) = c ∈ A) being the target of an attack β with s(β) = b is represented by:

a α c b β
(arguments are in a circle and attack names are in a square)

Acceptability semantics for argumentation frameworks with higher-order attacks have been defined in a direct way in [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF]. The idea is to specify the conditions under which the arguments are considered as accepted directly on the extended framework, without translating the original framework into an AF. Moreover, due to the defeasible nature of attacks (attacks may be affected by other attacks), conditions under which the attacks are accepted must also be specified. Indeed, some attacks may not be "valid", in the sense that they cannot defeat the argument or attack they are targeting. So, acceptability conditions for arguments should be given with respect to valid attacks and conversely attacks should be declared valid with respect to other arguments or attacks. For instance, the fact that two arguments may be conflicting depends on the validity of the attack between them. Hence, the notion of extension (set of arguments) is replaced by a pair of a set of arguments and a set of attacks, called a "structure".

Definition 4:

[10] Consider RAF = A, R, s, t . A structure of RAF is a pair (S, Γ) with S ⊆ A and Γ ⊆ R.
Intuitively, given a structure U = (S, Γ), S contains the arguments that are accepted "owing to" U and Γ contains the attacks which are valid "owing to" U (the meaning of "owing to" depending on the considered semantics).

In the following, we recall the acceptability conditions for structures, and the definitions of the semantics that are given in [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF]. The key notion is the fact that a set of arguments (resp. attacks) can be "defeated" (resp. "inhibited") wrt a given structure.

Definition 5:

[10] Consider RAF = A, R, s, t . Given U = (S, Γ) a structure of RAF . Let a ∈ A and α ∈ R.
a is defeated wrt U iff there is β ∈ Γ with s(β) ∈ S and t(β) = a, α is inhibited wrt U iff there is β ∈ Γ with s(β) ∈ S and t(β) = α. Def (U ) (resp. Inh(U )) denotes the set of arguments (resp. attacks) that are defeated (resp. inhibited) wrt U .

1) Conflict-free structures:

The minimal requirement for a structure (S, Γ) is that two arguments of S cannot be related by an attack of the structure, and similarly there cannot be an attack grounded in S and whose target is an element of Γ. Formally:

Definition 6: [10] Consider RAF = A, R, s, t . A struc- ture U = (S, Γ) of RAF is conflict-free iff S ∩ Def (U ) = ∅ and Γ ∩ Inh(U ) = ∅.
2) Admissible structures: Acceptability (for an argument or an attack) is also relative to a structure.

Definition 7:

[10] Consider RAF = A, R, s, t . Given a structure U = (S, Γ) of RAF . Let a ∈ A and α ∈ R. a (resp. α) is acceptable wrt U iff for each β ∈ R with t(β) = a (resp. t(β) = α), either β ∈ Inh(U ) or s(β) ∈ Def (U ).
U is admissible iff it is conflict-free and for each x ∈ (S ∪ Γ), x is acceptable wrt U . Acc(U ) denotes the set containing all acceptable arguments and attacks wrt U .

Remark: 3) Complete, stable, preferred and grounded structures: For any pair of structures U = (S, Γ) and

Let α ∈ R with t(α) = b ∈ A (resp. t(α) = β ∈ R).
U ′ = (S ′ , Γ ′ ), U ⊆ U ′ means that (S ∪ Γ) ⊆ (S ′ ∪ Γ ′ ). The structure U is ⊆-maximal iff every structure U ′ that satisfies U ⊆ U ′ also satisfies U ′ ⊆ U . Similarly, U is ⊆-minimal iff every structure U ′ that satisfies U ′ ⊆ U also satisfies U ⊆ U ′ . Definition 8: [10] Consider RAF = A, R, s, t . A structure U = (S, Γ) of RAF is: complete iff it is conflict-free and Acc(U ) = S ∪ Γ. stable iff it is conflict-free and satisfies A \ S ⊆ Def (U ) and R \ Γ ⊆ Inh(U ).
preferred iff it is a ⊆-maximal admissible structure. grounded iff it is the ⊆-minimal conflict-free structure U = (S, Γ) satisfying Acc(U ) ⊆ S ∪ Γ. It has been proved in [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF] that usual properties of Dung's extensions also hold for structures:

A complete structure contains all the unattacked arguments and all the unattacked attacks. Every complete structure is admissible, every preferred structure is also complete and every stable structure is also preferred.

The grounded structure is the ⊆-minimal complete structure. It is unique.

Example 2: Consider the RAF depicted by the following figure:

a α b d δ β c
There is only one complete (resp. preferred, stable, grounded structure): ({a, c, d}, {α, δ}).

4) D-structures:

The notion of structure has been strengthened in order to obtain a conservative generalization of Dung's frameworks for the conflict-free, admissible, complete, stable and preferred semantics. It is worth to note that in an AF, each attack is considered as valid, in the sense that it may affect its target. The next definition strengthens the notion of structure by adding a condition on attacks that will force every acceptable attack to be valid.

Definition 9: [10] Given RAF = A, R, s, t . 1) A d-structure on RAF is a structure U = (S, Γ) such that (Acc(U ) ∩ R) ⊆ Γ.
2) A conflict-free (resp. admissible, complete, preferred, stable) d-structure is a conflict-free (resp. admissible, complete, preferred, stable) structure which is also a dstructure.

It follows from Def. 8 that every complete (resp. stable, preferred) structure of a RAF is a d-structure of this RAF. However it is not the case for admissible and conflict-free structures. The conservative generalization proved in [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF] relies upon a correspondence between a Dung's framework (and its extensions) and a "nonrecursive" RAF (and its d-structures), where a nonrecursive RAF is a RAF in which no attack targets another attack.

III. LOGICAL DESCRIPTION OF A RAF

We propose a logical description of a RAF, that allows an explicit representation of arguments, attacks and their properties (accepted argument, attacked argument, valid attack, . . . ). We have been inspired by works in bioinformatics (see [START_REF] Demolombe | Molecular Interaction Automated Maps (regular paper)[END_REF], [START_REF] Alliot | Reasoning on molecular interaction maps[END_REF]), where metabolic networks are used to describe the chemical reactions of cells; these reactions can be negative (inhibition of a protein) or positive (production of a new protein) and they can depend on other proteins or other reactions. A translation from metabolic networks to classical logic has been proposed in [START_REF] Alliot | Reasoning on molecular interaction maps[END_REF], which allows for the use of automated deduction methods for reasoning on these networks.

Given RAF a recursive argumentation framework, Σ(RAF ) will denote the set of first-order logic formulae describing RAF .

A. Vocabulary

The following unary predicate symbols are used: Acc, N Acc, V al, Attack, Arg and the following unary functions symbols : T , S, with the following meaning:

Acc(x) (resp. N Acc(x)) means "x is accepted" (resp. "x cannot be accepted"), when x denotes an argument V al(α) means "α is valid" when α denotes an attack Attack(x) means "x is an attack" Arg(x) means "x is an argument" T (x) (resp. S(x)) denotes the target (resp. source) of x, when x denotes an attack The binary equality predicate is also used. Note that the quantifiers ∃ and ∀ range over some domain D. To restrict them to subsets of D, bounded quantifiers will be used:

∀x ∈ E (P (x)) means ∀x (x ∈ E → P (x)) or equivalently ∀x(E(x) → P (x)).

So we will use:

∀x ∈ Attack (Φ(x)) (resp. ∃x ∈ Attack (Φ(x)))
and ∀x ∈ Arg (Φ(x)) (resp. ∃x ∈ Arg (Φ(x))). Note that the meaning of N Acc(x) is not "x is not accepted" but rather "x cannot be accepted" (for instance because x is the target of a valid attack whose source is accepted). Hence, N Acc(x) is not logically equivalent to ¬Acc(x). However, the logical theory will enable to deduce ¬Acc(x) from N Acc(x), as shown below.

B. Logical theory

Two kinds of formulae describe RAF : the formulae describing the general behaviour of an attack, possibly recursive, in an argumentation framework, i.e. how an attack interacts with arguments and other attacks related to it. and the formulae encoding the specificities of the current framework.

The meaning of an attack is described under the form of constraints on its source (an argument) and its target (an argument or an attack). Moreover, as attacks may be attacked by other attacks, some attacks may not be valid.

If an attack from an argument to an attack is valid, then if its source is accepted, its target is not valid.

If an attack between two arguments is valid and if its source is accepted, then its target cannot be accepted. In that case, the target is not accepted. Using the vocabulary defined above, these constraints can be expressed by the following formulae:

(1)

∀x ∈ Attack (∀y ∈ Attack ( (V al(y) ∧ (T (y) = x) ∧ Acc(S(y))) → ¬V al(x) )) (2) ∀x ∈ Arg (∀y ∈ Attack ( (V al(y) ∧ (T (y) = x) ∧ Acc(S(y))) → N Acc(x) )) (3) ∀x ∈ Arg (N Acc(x) → ¬Acc(x))
Two other formulae limit the domain to arguments and attacks.

(4) ∀x (Attack(x) → ¬Arg(x)) [START_REF] Baroni | AFRA: Argumentation framework with recursive attacks[END_REF] ∀x (Arg(x) ∨ Attack(x)) Note that we assume that the argumentation framework is finite, with A = {a 1 , . . . a n } and R = {α 1 , . . . , α m }. Then, the logical encoding of specificities of the RAF leads to the following set of formulas:

(6)

(S(α) = a) ∧ (T (α) = b) for all α ∈ R with s(α) = a and t(α) = b (7) ∀x (Arg(x) ↔ (x = a 1 ) ∨ . . . ∨ (x = a n )) (8 ∀x (Attack(x) ↔ (x = α 1 ) ∨ . . . ∨ (x = α m )) (9)
a i = a j for all a i , a j ∈ A with i = j (10) α i = α j for all α i , α j ∈ R with i = j In the following, we will write s α (resp. t α ) in place of S(α) (resp. T (α)) for simplicity.

The logical theory Σ(RAF ) corresponding to RAF consists of the above 10 formulae. It is obviously consistent.

Example 2 (cont'd): Using the equality axioms, a simplified version of Σ(RAF ) can be obtained (in particular tautologies are omitted): 3

Σ(RAF ) = {(V al(β) ∧ Acc(c)) → ¬V al(α) (from (1)), (V al(δ) ∧ Acc(d)) → ¬V al(β) (from (1)), (V al(α) ∧ Acc(a)) → N Acc(b) (from (2)), N Acc(b) → ¬Acc(b) (from (3)), N Acc(a) → ¬Acc(a) (from (3)), N Acc(c) → ¬Acc(c) (from (3)), N Acc(d) → ¬Acc(d) (from (3))}
In the particular case of a nonrecursive RAF, formula (1) is a tautology. However, formula (2) cannot be simplified as it cannot be deduced that the attacks are valid. Indeed, the logical theory Σ(RAF ) only captures the description of RAF and is not concerned with the semantics of the framework (the logical description of the semantics is handled in the next section).

Example 3: Consider the nonrecursive RAF depicted by the following figure: However, it cannot be deduced that Acc(b) → N Acc(a). Indeed, the predicate N Acc allows for the representation of the direction of an attack between two arguments and avoids the contraposition of the attack.

IV. LOGICAL FORMALIZATION OF SEMANTICS

A. Logical Encoding of Semantics

In presence of higher-order attacks, the conflict-freeness, defence and reinstatement principles must take into account the fact that attacks might be not valid. Moreover, for each of these principles, two versions will be given, one for arguments and another one for attacks. Then, for each principle, we give a logical expression, thus leading to add formulas to the base Σ(RAF ) and producing new bases.

1) Conflict-freeness:

The conflict-freeness principle is formulated as follows (for arguments and for attacks):

If there is a valid attack between two arguments, they cannot be jointly accepted.

If there is an attack from an accepted argument to an attack, these attacks cannot be both valid. Note that these properties are already expressed in Σ(RAF ) by the formulae ( 1), ( 2), (3).

2) Defence: The idea is to claim that an argument a is defended by a set of arguments S if S weakens each attack α to a, either by attacking the source of α, or by attacking α itself. Moreover the defence should be obtained with valid attacks. So, the defence principle is formulated as follows (for arguments and for attacks):

An attacked argument may be accepted only if for each attack to it, either the source or the attack itself is in turn attacked by a valid attack from an accepted argument. An attack may be valid only if for each attack to it, either the source or the attack itself is in turn attacked by a valid attack from an accepted argument. These properties are expressed by the following formulae: [START_REF] Cohen | On the acceptability semantics of argumentation frameworks with recursive attack and support[END_REF]

∀α ∈ Attack ( Acc(t α ) → (∃β ∈ Attack (t β ∈ {s α , α} 4 ∧ V al(β) ∧ Acc(s β ))) ) (12) ∀α ∈ Attack (∀δ ∈ Attack ( ((δ = t α ) ∧ V al(δ)) → (∃β ∈ Attack β ∈ {s α , α} ∧ V al(β) ∧ Acc(s β )))
)) These formulae are added to the base Σ(RAF ), thus producing the base Σ d (RAF ).

3) Reinstatement: Based on the previous notion of defence, the reinstatement principle is formulated as follows (for arguments and for attacks):

An argument must be accepted provided that, for each attack to it, the source or the attack itself is in turn attacked by a valid attack from an accepted argument. An attack may be valid provided that for each attack to it, either the source or the attack itself is in turn attacked by a valid attack from an accepted argument. These properties are expressed by the following formulae: [START_REF] Besnard | Encoding argument graphs in logic[END_REF] 4 Strictly speaking, should be written as follows :

∀c ∈ Arg ( (∀α ∈ Attack (t α = c → (∃β ∈ Attack (t β ∈ {s α , α} ∧ V al(β) ∧ Acc(s β ))))) → Acc(c) ) (14) ∀δ ∈ Attack ( (∀α ∈ Attack (t α = δ → (∃β ∈ Attack
t β = sα ∨ t β = α. (t β ∈ {s α , α} ∧ V al(β) ∧ Acc(s β ))))) → V al(δ) )
These formulae are added to the base Σ(RAF ), thus producing the base Σ r (RAF ).

4) Stability:

The stability requirement can be formulated as follows (one for arguments and one for attacks):

If an argument is not accepted, it must be attacked by a valid attack from an accepted argument. If an attack is not valid, it must be attacked by a valid attack from an accepted argument. These properties are expressed by the following formulae: [START_REF] Gabbay | The attack as strong negation, part I[END_REF]

∀c ∈ Arg ( ¬Acc(c) → (∃β ∈ Attack ((t β = c) ∧ V al(β) ∧ Acc(s β ))) ) (16) ∀α ∈ Attack ( ¬V al(α) → (∃β ∈ Attack ((t β = α) ∧ V al(β) ∧ Acc(s β )
) ) These formulae are added to the base Σ(RAF ), thus producing the base Σ s (RAF ). Example 2 (cont'd): Σ d (RAF ) is obtained from Σ(RAF ) by adding the formulas:

Acc(b) → (V al(β) ∧ Acc(c)), ¬V al(β) and V al(α) → (V al(δ) ∧ Acc(d)). Σ r (RAF ) is obtained from Σ(RAF )

B. Characterizing Semantics of a RAF

We propose characterizations of the structures under different semantics in terms of models of the bases Σ(RAF ), Σ d (RAF ), Σ r (RAF ), Σ s (RAF ).

Let RAF = A, R, s, t . Given I an interpretation of Σ(RAF ), we define:

S I = {x ∈ A|I(Acc(x)) = true} Γ I = {x ∈ R|I(V al(x)) = true} Moreover, let I be a model of Σ(RAF ): I is a ⊆-maximal model of Σ(RAF ) iff there is no model I ′ of Σ(RAF ) with (S I ∪ Γ I ) ⊂ (S I ′ ∪ Γ I ′ ). I is a ⊆-minimal model of Σ(RAF ) iff there is no model I ′ of Σ(RAF ) with (S I ′ ∪ Γ I ′ ) ⊂ (S I ∪ Γ I ).
We have the following characterizations: Proposition 1: Let RAF = A, R, s, t . Let U = (S, Γ) a structure on RAF .

1) U is conflict-free iff there exists I model of Σ(RAF ) with S I = S and Γ I = Γ. D-structures can also be characterized. Let us recall that dstructures are particular structures in which acceptable attacks are forced to be valid. So, we consider the base Σ(RAF ) augmented with the formula that expresses the reinstatement principle for attacks, that is formula [START_REF] Dupin De Saint-Cyr | Argumentation update in YALLA (Yet Another Logic Language for Argumentation)[END_REF].

As said before, complete structures are d-structures. So we just have to complete Prop.1 with the characterizations of conflict-free and admissible d-structures.

C. Case of AF

As said before, an AF can be viewed as a particular RAF. So we can consider the associated logical theory, which we denote by Σ(AF ) for simplicity. Moreover, in the particular case of an AF, the semantics recalled in SectionII-A assume that each attack is valid. As a consequence, the logical theory Σ(AF ) can be replaced by a logically equivalent theory built as follows: For each (a, b) ∈ R, the attack from a to b is described by the formulae Acc(a) → N Acc(b) and N Acc(b) → ¬Acc(b).

Then, the standard defence, reinstatement and stability principles are encoded with simplified versions of formulae ( 11), ( 13) and ( 15) (as attacks are never attacked, formulae ( 12), ( 14) and ( 16) would be tautologies). Let AF = A, R . For x ∈ A, let R -(x) denote the set of its attackers. For each principle, a set of formulas is provided, one for each argument:

Defence: For each x ∈ A, Acc(x) → (∧ y∈R -(x) (∨ z∈R -(y) Acc(z))) Reinstatement: For each x ∈ A, (∧ y∈R -(x) (∨ z∈R -(y) Acc(z))) → Acc(x) Stability: For each x ∈ A, ¬Acc(x) → (∨ y∈R -(x) Acc(y))
We denote by Σ d (AF ) (resp. Σ r (AF ), Σ s (AF )) the logical theories obtained by adding all the formulas encoding defence (resp. reinstatement, stability) to Σ(AF ). Given I be an interpretation of Σ(AF ), we still denote by S I the set {x ∈ A|I(Acc(x)) = true}. If I is a model of Σ(AF ), I is said to be a ⊆-maximal (resp. minimal) model of Σ(AF ) iff there is no model I ′ of Σ(AF ) such that S I ⊂ S I ′ (resp. S I ′ ⊂ S I ).

Then, we get the following characterizations: V. RELATED WORKS From the seminal work of Dung [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF], several works have proposed to connect abstract argumentation with logic programming (see [START_REF] Caminada | On the equivalence between logic programming semantics and argumentation semantics[END_REF] for recent work and more references). The issue is to find an appropriate encoding of an AF into a logic program P , so that applying logic programming semantics to P enables to capture argumentation semantics of the original AF. Dung [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF] has proposed an encoding allowing the capture of (only) grounded and stable semantics. In [START_REF] Caminada | On the equivalence between logic programming semantics and argumentation semantics[END_REF], the encoding allows for the characterization of the standard argumentation semantics (grounded, stable, preferred and complete semantics) through 3-valued models of a logic program.

Proposition 3: Let AF = A, R . Let S ⊆ A. 1) S is conflict-free in A, R iff
In the particular case of an AF, the logical representation of [START_REF] Gabbay | The attack as strong negation, part I[END_REF] using classical propositional logic augmented with strong negation is very close to our proposal. However, taking into account higher-order attacks requires a modification of the original framework with the addition of nodes and joint attacks.

The issue of logical encoding of abstract argumentation has recently been addressed for different other purposes independently of the notion of logic programming. In [START_REF] Besnard | Encoding argument graphs in logic[END_REF], [START_REF] Besnard | SESAME -A System for Specifying Semantics in Abstract Argumentation (regular paper)[END_REF] acceptance conditions and standard semantics are encoded by first-order logical formulae (given a semantics σ and a set S of arguments, a formula is provided which is satisfiable iff S is a σ-extension). However, the argumentation framework itself is not represented. A similar issue is addressed in [START_REF] Grossi | On the logic of argumentation theory[END_REF] with a modal logic, considering that the accessibility relation is the inverse of the attack relation; the same kind of work is presented in [START_REF] Arieli | A QBF-based formalization of abstract argumentation semantics[END_REF] using signed theories and QBF formulae; [START_REF] Wallner | Advanced SAT techniques for abstract argumentation[END_REF] presents algorithms using particular logical notions (minimal correction sets, backbone) in order to compute some semantics (semi-stable and eager); [START_REF] Cerutti | Computing preferred extensions in abstract argumentation: A SAT-based approach[END_REF] translates complete labellings into logical formulae in order to compute preferred extensions with SAT solvers; [START_REF] Bogaerts | Declarative solver development: Case studies[END_REF] proposes a metalevel analysis of the computation problems related to given semantics in order to automatically generate solvers adapted to these problems.

In the more general abstract dialectical framework [START_REF] Brewka | Abstract dialectical frameworks[END_REF], each argument is associated with a propositional formula which represents the acceptance conditions of the argument. This logical translation enables to capture easily the stable semantics. However, recursive interactions are not taken into account.

[14] proposes a complete framework for handling the dynamics on an AF. A first-order logical language is presented, enabling to describe the structure of an AF, to express incomplete knowledge on an AF and to encode change operations on an AF.

Moreover in the context of the First International Competition on Computational Models of Argumentation (ICCMA), different solvers have been proposed and tested (e.g. [START_REF] Beierle | A software system using a SAT solver for reasoning under complete, stable, preferred, and grounded argumentation semantics[END_REF], or [START_REF] Lagniez | Coquiaas: A constraint-based quick abstract argumentation solver[END_REF]). However, in all these works, neither the attack relation itself is logically encoded, nor the recursive aspects are taken into account.

VI. CONCLUSION

In this work, we have proposed a logical encoding of argumentation frameworks with higher-order attacks. Our proposal enables to separate the logical expression of the meaning of an attack (simple or higher-order) and the logical expression of acceptability semantics. These semantics (introduced in [START_REF] Cayrol | Valid attacks in argumentation frameworks with recursive attacks[END_REF]) specify the conditions under which the arguments (resp. the attacks) are considered as accepted, directly on the extended framework, without translating the original framework into an AF.

Then, we are able to characterize the output of a given argumentation framework (under the form of structures) in logical terms (namely as particular models of a logical theory). That opens a way for computational issues by using logical tools. As a preliminary work in that direction, a software has been developed [START_REF] Cayrol | The Grafix website[END_REF] that enables to represent a RAF, to express the associated logical theories Σ(RAF ), Σ d (RAF ), . . ., and to compute the structures under different semantics.

Another feature of our work is its conservative generalization of AF, when d-structures are considered.

Future works will include the study of a logical encoding of frameworks with higher-order attacks and higher-order supports. The difficulty is that there exist different interpretations of the notion of support in argumentation frameworks. In a first step, we plan to consider a recent framework that allows handling both higher-order attacks and higher-order evidential supports, with structure-based semantics [START_REF] Cayrol | Argumentation frameworks with recursive attacks and evidencebased support[END_REF].

Definition 1 :

 1 [2] A Dung's argumentation framework (AF) is a tuple A, R , where A is a finite and non-empty set of arguments and R ⊆ A × A is a binary attack relation on the arguments, with (a, b) ∈ R indicates that a attacks b.

Definition 2 :

 2 [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF] Let AF = A, R and S ⊆ A. S is conflict-free iff (a, b) ∈ R for all a, b ∈ S. a ∈ A is acceptable wrt S (or equivalently S defends a) iff for each b ∈ A with (b, a) ∈ R, there is c ∈ S with (c, b) ∈ R.

  If α and s(α) are unattacked, there is no admissible structure U = (S, Γ) such that b ∈ S (resp. β ∈ Γ).

  ) enables to deduce the formula (V al(α) ∧ Acc(a)) → N Acc(b). Note that if V al(α) is assumed, we obtain Acc(a) → N Acc(b) and from N Acc(b) → ¬Acc(b) it follows that Acc(a) → ¬Acc(b) and so Acc(b) → ¬Acc(a).

  by adding the formulas: Acc(a), Acc(c), Acc(d), (V al(β) ∧ Acc(c)) → Acc(b), V al(δ) and (V al(δ) ∧ Acc(d)) → V al(α).

2 ) 5 Example 2 (

 252 U is admissible iff there exists I model of Σ d (RAF ) with S = S I and Γ I = Γ. 3) U is complete iff there exists I model of Σ d (RAF ) ∪ Σ r (RAF ) with S = S I and Γ I = Γ. 4) U is a stable structure iff there exists I model of Σ s (RAF ) with S I = S and Γ I = Γ. 5) U is a preferred structure iff there exists I ⊆-maximal model of Σ d (RAF ) with S I = S and Γ I = Γ. 6) U is the grounded structure iff S = S I and Γ I = Γ where I is a ⊆-minimal model of Σ r (RAF ). cont'd): There is only one complete structure: ({a, c, d}, {α, δ}). Indeed, every model I of Σ d (RAF ) ∪ Σ r (RAF ) is such that S I = {a, c, d} and Γ I = {α, δ}, in other every model of Σ d (RAF ) ∪ Σ r (RAF ) satisfies Acc(a), Acc(c), Acc(d), V al(δ), V al(α) and falsifies Acc(b), V al(β). An example of admissible (but not complete) structure is ({a, d}, {δ}). Indeed, there is a model I of Σ d (RAF ) with S I = {a, d} and Γ I = {δ}.

Proposition 2 :

 2 Let RAF = A, R, s, t . Let U = (S, Γ) a structure on RAF . 1) U is a conflict-free d-structure iff there exists I model of Σ(RAF ) ∪ {(14)} with S I = S and Γ I = Γ. 2) U is an admissible d-structure iff there exists I model of Σ d (RAF ) ∪ {(14)} with S = S I and Γ I = Γ. Example 2 (cont'd): From (14), the following formulae are obtained: V al(δ) and (V al(δ) ∧ Acc(d)) → V al(α). An example of admissible (but not complete) d-structure is ({a, d}, {α, δ}). Indeed, there is a model I of Σ d (RAF ) ∪ {(14)} with S I = {a, d} and Γ I = {α, δ}. Note that ({a, d}, {δ}) is an admissible structure but not an admissible d-structure.

Example 4 :

 4 there exists I model of Σ(AF ) with S I = S. 2) S is admissible in A, R iff there exists I model of Σ d (AF ) with S = S I . 3) S is a complete extension of A, R iff there exists I model of Σ d (AF ) ∪ Σ r (AF ) with S = S I . 4) S is a stable extension of A, R iff there exists I model of Σ s (AF ) with S I = S. 5) U is a preferred extension of A, R iff there exists I ⊆-maximal model of Σ d (AF ) with S I = S. 6) S is the grounded extension iff S = S I where I is a ⊆-minimal model of Σ r (AF ). Consider the AF represented by: a b c It can be encoded by the following simplified bases: Σ(AF ) = {Acc(a) → N Acc(b), N Acc(b) → ¬Acc(b), Acc(b) → N Acc(c), N Acc(c) → ¬Acc(c), N Acc(a) → ¬Acc(a)} and Σ d (AF ) = Σ(AF ) ∪ {¬Acc(b), Acc(c) → Acc(a)}. Every ⊆-maximal model of Σ d (AF ) satisfies Acc(a), Acc(c) and falsifies Acc(b). That corresponds to the unique preferred extension {a, c}.

Where "iff" (resp. "wrt") stands for "if and only if" (resp. "with respect to").

It can be proved that the minimal fixed point of F is conflict-free.

It also holds that U is the grounded structure iff S = S I and Γ I = Γ where I is a ⊆-minimal model of Σ d (RAF ) ∪ Σr(RAF ). Considering Σ d (RAF ) ∪ Σr(RAF ) instead of Σr(RAF ) might be useful from a computational point of view, when searching for minimal models.