Rallou Thomopoulos
email: rallou.thomopoulos@supagro.inra.fr

Madalina Croitoru
email: madalina.croitoru@lirmm.fr

Nouredine Tamani
email: nouredine.tamani@lirmm.fr

Decision Support for Agri-Food Chains: A Reverse Engineering Argumentation-Based Approach

Keywords: decision support, knowledge representation, argumentation, reverse engineering, backward chaining, agrifood chain control, goal, viewpoint

Evaluating food quality is a complex process since it relies on numerous criteria historically grouped into four main types: nutritional, sensorial, practical and hygienic qualities. They may be completed by other emerging preoccupations such as the environmental impact, economic phenomena, etc. However, all these aspects of quality and their various components are not always compatible and their simultaneous improvement is a problem that sometimes has no obvious solution, which corresponds to a real issue for decision making. This paper proposes a decision support method guided by the objectives dened for the end products of an agrifood chain. It is materialized by a backward chaining approach based on argumentation.

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Introduction

In agrifood chains, the products traditionally go through the intermediate stages of processing, storage, transport, packaging and reach the consumer (the demand) from the producer (the supply). More recently, due to an increase in quality constraints, several parties are involved in production process, such as consumers, industrials, health and sanitary authorities, etc.

expressing their requirements on the nal product as dierent point of views which could be conicting. The notion of reverse engineering control, in which the demand (and not the supply) sets the specications of desired products and it is up to the supply to adapt and nd its ways to respond, can be considered in this case.

In this article, we discuss two aspects of this problem. First, we accept the idea that specications cannot be established and several complementary points of view -possibly contradictory -can be expressed (nutritional, environmental, taste, etc.). We then need to assess their compatibility (or incompatibility) and identify solutions satisfying a maximum set of viewpoints. To this end we propose a logical framework based on argumentation and introduce a method of decision making based on backward chaining for the bread industry.

Since a joint argumentation -decision support approach is highly relevant to the food sector (Thomopoulos et al., 2009), the contribution of the paper is twofold. First we present a real use case of an argumentation process in the agrifood domain. Second we introduce the notion of viewpoint / goal in this setting based on the notion of backwards chaining reasoning and show how to use those techniques in a concrete application.

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The main alternative method to deal with the problem is the multicriteria decision approach. However multicriteria decision aims at evaluating several alternative options, whereas argumentation-based decision focuses on whether several options make sense together, which is a dierent perspective, addressed in this paper. Moreover, multicriteria decision is not connected to the backward chaining procedure as the argumentative approach is, by construction of the arguments, as will be explained in Section 5.2.

In Section 2, we introduce the real scenario considered in the application. In Section 3, we motivate our technical and modeling choices. In Section 4, the developed approach is introduced. It relies on an instantiation of a logic based argumentation framework based on a specic fragment of rst order logic. In Section 5, we explain the technical results that ensure the soundness and completeness of our agronomy application method. In Section 6, some evaluation results are presented. Finally, Section 7 concludes the paper.

Scenario

The case of study considered in this paper relates to the debate around the change of ash content in our used for common French bread. Various actors of the agronomy sector are concerned, in particular the Ministry for Health through its recommendations within the framework of the PNNS (National Program for Nutrition and Health), the millers, the bakers, the nutritionists and the consumers.

The PNNS recommends to privilege the whole-grain cereal products and in particular to pass to a common bread of T80 type, i.e made with our containing an ash content (mineral matter rate) of 0.8%, instead of the type

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M A N U S C R I P T T65 (0.65% of mineral matter) currently used. Increasing the ash content comes down to using a more complete our, since mineral matter is concentrated in the peripheral layers of the wheat grain, as well as a good amount of components of nutritional interest (vitamins, bers). However, the peripheral layers of the grain are also exposed to the phytosanitary products, which does not make them advisable from a health point of view, unless one uses organic our.

Other arguments (and of various nature) are in favour or discredit wholegrain bread. From an organoleptic point of view for example, the bread loses out in its being crusty. From a nutritional point of view, the argument according to which the bers are benecial for health is discussed, some bers could irritate the digestive system. From an economic point of view, the bakers fear selling less bread, because whole-grain bread increases satiety which is benecial from a nutritional point of view, for the regulation of the appetite and the ght against food imbalances and pathologies. However whole-grain bread requires also less our and more water for its production, thus reducing the cost. The millers also fear a decrease in the quality of the technical methods used in the our production.

Beyond the polemic on the choice between two alternatives (T65 or T80), one can take the debate further by distinguishing the various points of view concerned, identifying the desirable target characteristics, estimating the means of reaching that point. The contribution of this paper is showing how using argumentation can help towards such practical goals.

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Motivation

In this paper we will elicit the points of view and the desirable target characteristics by the means of interviews with agronomy experts. Once the target characteristics identied, nding the means of reaching them will be done automatically by a combination of reverse engineering and argumentation. The reverse engineering will be used in order to nd the complete set of actions to take towards a given characteristic, for all characteristics.

In certain cases the actions to take will be inconsistent. Argumentation will then be employed in order to identify actions that can be accepted together.

Reverse Engineering

While reverse engineering has been widely employed in other Computer Science domains such as multi agent systems or requirements engineering (e.g. Brunelière et al. ( 2014)), it is quite a novel methodology when applied in agronomy. In agrifood chains, the products traditionally go through the intermediate stages of processing, storage, transport, packaging and reach the consumer (the demand) from the producer (the supply). It is only recently, due to an increase in quality constraints, that the notion of reverse engineering control has emerged (Perrot et al., 2011). In this case the demand (and not the supply) sets the specications of desired products and it is up to the supply to adapt and nd its ways to respond. In what follows, starting from the desired target criteria for the nal product, the methods allowing one to identify ways to achieve these criteria (by intervention on the various stages of the supply chain) are named reverse engineering.

Reverse engineering is known to be challenging from a methodological

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viewpoint. This is due to two main aspects. First, the diculty of dening the specications for the expected nished product. The desired quality criteria are multiple, questionable, and not necessarily compatible. The second diculty lies in the fact that the impact of dierent steps of food processing and their order is not completely known. Some steps are more studied than others, several successive steps can have opposite eects (or unknown eects), the target criteria may be outside of the characteristics of products.

Second, reconciling dierent viewpoints involved in the food sector still raises unaddressed questions. The problem does not simply consist in addressing a multi-criteria optimisation problem (Bouyssou et al., 2009): the domain experts would need to be able to justify why a certain decision (or set of possible decisions) is taken.

Argumentation

Argumentation is a reasoning model based on the construction and the evaluation of interacting arguments. It has been applied to nonmonotonic reasoning, decision making, or for modeling dierent types of dialogues including negotiation. Most of the models developed for these applications are grounded on the abstract argumentation framework proposed by [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF]. This framework consists of a set of arguments and a binary relation on that set, expressing conicts among arguments. An argument gives a reason for believing a claim, for doing an action.

Argumentation theory in general [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF][START_REF] Besnard | Elements of Argumentation[END_REF]Rahwan and Simari, 2009) is actively pursued in the literature. Some approaches combine argumentation and multi criteria decision making (Amgoud and Prade, 2009).

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Value based Argumentation Frameworks (Bench-Capon, 2003a) have been proposed, where the strength of an argument corresponds to the values it promotes. What we call viewpoint later on in this paper would then correspond to the notion of audience in such setting. Although intuitive, this approach is not adapted in the case of the considered application. Here a value can be split into several audiences: there could be contradictory goals even from the same viewpoint. The notion of viewpoint and goals introduced in this setting also remind those proposed by (Assaghir et al., 2011).

Logic-based Argumentation

In this paper we present a methodology combining reverse engineering and logical based argumentation for selecting the actions to take towards the agronomy application at hand. The logical instantiation language is a subset of rst order logic denoted in this paper SRC equivalent to Datalog+-(Calì et al., 2010), Conceptual Graphs or Description Logics (more precisely the EL fragment [START_REF] Baader | Pushing the el envelope[END_REF] and DL-Lite families (Calvanese et al., 2007)). All above mentioned languages are logically equivalent in terms of representation or reasoning power. The reason why this application is using SRC is the graph based representation proper to SRC (and not to the other languages). This graph based representation (implemented in the Cogui tool [START_REF] Chein | Graph-based Knowledge Representation and ReasoningComputational Foundations of Conceptual Graphs[END_REF][START_REF] Chein | Visual reasoning with graphbased mechanisms: the good, the better and the best[END_REF]) makes the language suitable for interacting with non computing experts [START_REF] Chein | Visual reasoning with graphbased mechanisms: the good, the better and the best[END_REF].

Here we use the instantiation of [START_REF] Croitoru | What can argumentation do for inconsistent ontology query answering?[END_REF] for dening what an argument and an attack are. While other approaches such as [START_REF] García | Defeasible logic programming: An argumentative approach[END_REF], [START_REF] Besnard | Practical rst-order argumentation[END_REF], [START_REF] Muller | An argumentation-based approach for decision making[END_REF] etc. address rst order logic based argumentation, the work of (Croitoru and Vesic, 2013) uses the same SRC syntax and graph reasoning foundations. In Figure 1 the visual interface of Cogui is depicted: knowledge is represented as graph which is enriched dynamically by rule application. More on the visual appeal of Cogui for knowledge representation and reasoning can be found in [START_REF] Chein | Visual reasoning with graphbased mechanisms: the good, the better and the best[END_REF].

Approach

As mentioned above, in this paper we use an instantiation of logic based argumentation based on a specic fragment of rst order logic. This subset is equivalent to Datalog+-(Calì et al., 2010), Conceptual Graphs or Description Logics (the EL fragment [START_REF] Baader | Pushing the el envelope[END_REF] and the DL-Lite families (Calvanese et al., 2007)). The reason for which our application required this

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specic logic fragment is related to the information capitalisation needs of the food sector. The long term aim is to enrich ontologies and data sources based on these ontologies and join the Open Data movement. This entails that the language used by the food applications needs to be compatible with the Semantic Web equivalent languages as mentioned before.

The choice of the SRC syntax and graph reasoning mechanism is justied by the visual appeal of this language for non computing experts.

In a nutshell our methodology is as follows. The set of goals, viewpoints as well as the knowledge associated with the goals / viewpoints is elicited either by the means of interviews with the domain experts or manually from dierent scientic papers. This step of the application is the most time consuming but the most important. If the knowledge elicited is not complete, sound or precise the outcome of the system is compromised. Then, based on the knowledge elicited from the knowledge experts and the goals of the experts, we enrich the knowledge bases using reverse engineering (implemented using backwards chaining algorithms). Putting together the enriched knowledge bases obtained by backwards chaining from the dierent goals will lead to inconsistencies. The argumentation process is used at this step and the extensions yield by the applications computed. Based on the extensions and the associated viewpoints we can use voting functions to determine the application choice of viewpoints.

Use Case Real Data

Expressing the target characteristics or goals according to various points of view consists of identifying the facets involved in the construction of product quality: points of view, topics of concern such as nutrition, envi-

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ronment, technology, etc. In addition, such viewpoints have to be addressed according to their various components (bers, minerals, vitamins, etc). Desirable directions need to be laid down, and in a rst step we consider them independent one from another.

The considered sources of information include, from most formal to less formal: (1) peer reviewed scientic papers;

(2) technical reports or information posted on websites;

(3) conferences and scientic meetings around research projects; (4) expert knowledge obtained through interviews. The scientic articles we have analysed with the supervision of experts in agrifood include: (Bourre et al., 2008;Slavin and Green, 2007;[START_REF] Dubuisson-Quellier | De la routine à la délibération. les arbitrages des consommateurs en situation d'achat[END_REF][START_REF] Ginon | Eect of bre information on consumer willingness to pay for french baguettes[END_REF][START_REF] Layat | Place du pain dans l'équilibre alimentaire[END_REF]). (Bourre et al., 2008) compares the dierent types of our from a nutritional point of view. (Slavin and Green, 2007) explores the link between ber and satiety. [START_REF] Dubuisson-Quellier | De la routine à la délibération. les arbitrages des consommateurs en situation d'achat[END_REF][START_REF] Ginon | Eect of bre information on consumer willingness to pay for french baguettes[END_REF] deal with consumer behaviour and willingness to pay.

They focus on French baguette when information concerning the level of bers is provided, and they base their results on statistical studies of consumer panels. [START_REF] Layat | Place du pain dans l'équilibre alimentaire[END_REF] provides a summary of the nutritional aspects of consumption of bread and the link with technological aspects.

We also reviewed technical reports available on ocial websites on health policy: the public PNNS (National Program for Nutrition and Health, www.

mangerbouger.fr/pnns) (PNNS (documents statutaires), 2010), the European project Healthgrain (looking at improving nutrition and health through grains) [START_REF] Dean | Report on consumer expectations of health benets of modied cereal products[END_REF]HEALTHGRAIN, 2009), as well as projects and symposia on sanitary measures regarding the nutritional, technological and organoleptic properties of breads [START_REF] Dinabio | Proceedings of dinabio développement et innovation en agriculture biologique[END_REF]CADINNO, 2008; AQUA- NUP, 2009;FCN, 2009). Finally, several interviews were conducted to collect domain expert knowledge, in particular for technology specialists in our laboratory.

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A summary of the results obtained in the baking industry is synthesised in Figure 2 regarding nutritional and organoleptic aspects. Figure 2(a) shows the main identied goals to reach for a nutritionally optimised bread (for instance, containing a high level of soluble bers, vitamins and minerals, low salt, etc.), whereas Figure 2(b) sums up the main goals to achieve for an enjoyable bread regarding sensorial concerns (for example, crusty, etc.).

Technical Soundness

In this section we explain the technical results that ensure the soundness and completeness of our agronomy application method. The section is composed of three parts. A rst subsection explains the logical subset of rst order logic language employed in the paper. The second subsection shows how to construct arguments and attacks in order to obtain extensions when a knowledge base expressed under this language is inconsistent. Last, the third section shows how we used reverse engineering to complete the knowledge base with all possible actions and how argumentation can be used in order to select consistent subsets of knowledge which support given actions.

The Logical Language

In the following, we give the general setting knowledge representation language used throughout the paper.

A knowledge base is a 3-tuple K = (F, R, N ) composed of three nite sets of formulae: a set F of facts, a set R of rules and a set N of constraints. Facts Syntax. Let C be a set of constants and P = P 1 ∪ P 2 . . . ∪ P n a set of predicates of the corresponding arity i = 1, . . . , n. Let V be a countably innite set of variables. We dene the set of terms by T = V ∪ C. As usual, given i ∈ {1 . . . n}, p ∈ P i and t 1 , . . . , t i ∈ T we call p(t 1 , . . . , t i ) an atom. A fact is the existential closure of an atom or an existential closure of a conjunction of atoms. (Note that there is no negation or disjunction in the facts and that we consider a generalised notion of facts that can contain several atoms.)

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• Bread, Cereal, LowSalt, ContaminantFree are examples of unary predicates (arity 1) and IsIngredientOf is a binary predicate (arity 2).

• Wheat, oats, rye, barley are constant examples.

• Cereal (wheat) is an atom.

• ∃ x (Bread(x) ∧ IsIngredientOf(wheat, x)) is a fact. Due to lack of space we do not show the full semantic denitions of facts (or rules and constraints in the following section). For a complete semantic depiction of this language please check [START_REF] Chein | Graph-based Knowledge Representation and ReasoningComputational Foundations of Conceptual Graphs[END_REF][START_REF] Chein | Visual reasoning with graphbased mechanisms: the good, the better and the best[END_REF][START_REF] Croitoru | What can argumentation do for inconsistent ontology query answering?[END_REF]. It is well known that F ′ |= F (read the fact F ′ entails the fact F ) if and only if there is a homomorphism from F to F ′ [START_REF] Chein | Graph-based Knowledge Representation and ReasoningComputational Foundations of Conceptual Graphs[END_REF].

Rules. A rule R is a formula of the form

∀x 1 , . . . , ∀x n ∀y 1 , . . . , ∀y m (H(x 1 , . . . , x n , y 1 , . . . , y m ) → ∃z 1 , ...∃z k C(y 1 , . . . , y m , z 1 , ...z k ))
where H, the hypothesis, and C, the conclusion, are atoms or conjunctions of atoms, n, m, k ∈ {0, 1, . . .}, x 1 , . . . , x n are the variables appearing in H,

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M A N U S C R I P T y 1 , . . . , y m are the variables appearing in both H and C and z 1 , . . . , z k the new variables introduced in the conclusion. An example of a rule is the following:

∀ x (Bread(x) ∧ PesticideFree(x) ∧ MycotoxinFree(x)
→ ContaminantFree(x)).

In the following we will consider rules without new existential variables in the conclusion.

Reasoning consists of applying rules on the set F and thus inferring new knowledge. A rule R = (H, C) is applicable to set F if and only if there exists F ′ ⊆ F such that there is a homomorphism σ from the hypothesis of R to the conjunction of elements of F ′ . A rule R = (H, C) is inversely applicable to a fact F if there is a homomorphism π from C to F . In this case, the inverse application of R to F according to π produces a new fact

F ′ such that R(F ′ ) = F .
We then say that the new fact is an immediate inverse derivation of F by R, abusively denoted R -1 (F ).

Note that this technique is commonly used, for example, for backward chaining query answering [START_REF] Baget | Rules dependencies in backward chaining of conceptual graphs rules[END_REF][START_REF] Konig | A sound and complete backward chaining algorithm for existential rules[END_REF] where a query is rewritten according to the rules. The same mechanism is also discussed by abductive reasoning algorithms [START_REF] Klarman | Abox abduction in the description logic alc[END_REF] where minimal sets of facts (in the set inclusion sense) are added to the knowledge base in order to be able to deduct a query.

Let F = Bread(bleuette) ∧ PesticideFree(bleuette) ∧ MycotoxinFree(bleu- ette) and R the rule ∀ x (Bread(x) ∧ PesticideFree(x) ∧ MycotoxinFree(x) → ContaminantFree(x)).
R is applicable to F and produces by derivation the following fact: Bread

(bleuette) ∧ PesticideFree(bleuette) ∧ MycotoxinFree(bleuette) ∧ Contaminant- A C C E P T E D M A N U S C R I P T

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Free(bleuette).

Let F = Bread(bleuette) ∧ ContaminantFree(bleuette) and R the rule ∀ x (Bread(x) ∧ PesticideFree(x) ∧ MycotoxinFree(x) → ContaminantFree(x)).

R inversely applicable to F and produces by inverse derivation the fact:

F ′ = Bread(bleuette) ∧ PesticideFree(bleuette) ∧ MycotoxinFree(bleuette).
Let F be a subset of F and let R be a set of rules. A set F n is called an R-derivation of F if there is a sequence of sets (called a derivation sequence

) (F 0 , F 1 , . . . , F n ) such that F 0 ⊆ F , F 0 is R-consistent, for every i ∈ {1, . . . , n -1}, it holds that F i is an immediate derivation of F i-1 .
Given a set {F 0 , . . . , F k } ⊆ F and a set of rules R, the closure of {F 0 , . . . , F k } w.r.t. R, denoted Cl R ({F 0 , . . . , F k }), is dened as the smallest set (with respect to ⊆) which contains {F 0 , . . . , F k }, and is closed for R-derivation (that is, for every R-derivation F n of {F 0 , . . . , F k }, we have Constraints. A constraint is a formula ∀x 1 . . . ∀x n (H(x 1 , . . . , x n ) → ⊥), where H is an atom or a conjunction of atoms and n ∈ {0, 1, 2, . . .}.

F n ⊆ Cl R ({F 0 , . . . , F k })).
Equivalently, a constraint can be written as ¬(∃x 1 , ..., ∃x n H(x 1 , ...x n )). As an example of a constraint, consider N = ¬(∃ x (Growth(x) ∧ Decrease(x))).

Given a knowledge base K = (F, R, N ), a set {F 1 , . . . , F k } ⊆ F is said to be inconsistent if and only if there exists a constraint N ∈ N such that {F 1 , . . . , F k } |= H N , where H N denotes the existential closure of the hypoth-

esis of N . A set is consistent if and only if it is not inconsistent. A set {F 1 , . . . , F k } ⊆ F is R-inconsistent if and only if there exists a constraint A C C E P T E D M A N U S C R I P T N ∈ N such that Cl R ({F 1 , . . . , F k }) |= H N
, where H N denotes the existential closure of the hypothesis of N .

Let K = (F, R, N ) where:

• F contains the following facts:

-F 1 = Bread(bleuette) ∧ ContaminantFree(bleuette)

-F 2 = ∃ e ExtractionRate(e,bleuette)

-F 3 = ∃ f (FiberContent(f,bleuette) ∧ High(f))
• R consists of the following rules:

-

R 1 = ∀ x,y (Bread(x) ∧ ExtractionRate(y,x) ∧ PesticideFree(x) → Decrease(y)) -R 2 = ∀ x,y,z (Bread(x) ∧ ExtractionRate(y,x) ∧ FiberContent(z,x) ∧ High(z) → Growth(y)) -R 3 = ∀ x (Bread(x) ∧ ContaminantFree(x)
→ PesticideFree(x) ∧ MycotoxinFree(x))

• N contains the following negative constraint:

-N = ¬(∃ x (Growth(x) ∧ Decrease(x)))

K is inconsistent since (F, R) |= N . Indeed, F 1 and R 3 allow to deduce PesticideFree(bleuette). Combined to F 2 and R 1 we obtain Decrease(e). F 3 and R 2 deduce Growth(e), violating the negative constraint N .

Given a knowledge base, one can ask a conjunctive query in order to know whether something holds or not. Without loss of generality we consider

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boolean conjunctive queries (which are facts). As an example of a query, take

∃x 1 cat(x 1
). The answer to query α is positive if and only if F, R |= α.

Answering Q, traditionally, has two dierent algorithmic approaches: either forward chaining or backwards chaining. The two approaches come to either ( 1) nding an answer of Q in the R-derivations of the facts in the knowledge base or (2) computing the inverse R-derivations of the query and nding if there is a match in the facts. We will focus on the latter approach in the following.

Arguments and Attacks

This section shows that it is possible to dene an instantiation of Dung's abstract argumentation theory [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF] that can be used to reason with an inconsistent ontological KB.

We rst dene the notion of an argument. For a set of formulae G = {G 1 , . . . , G n }, notation G is used as an abbreviation for G 1 ∧ . . . ∧ G n .

Denition 1. Given a knowledge base K = (F, R, N ), an argument a is a tuple a = (F 0 , F 1 , . . . , F n ) where:

• (F 0 , . . . , F n-1 ) is a derivation sequence with respect to K

• F n is an atom, a conjunction of atoms, the existential closure of an atom or the existential closure of a conjunction of atoms such that

F n-1 |= F n .
This denition, following the denition of [START_REF] Croitoru | What can argumentation do for inconsistent ontology query answering?[END_REF]) is a straightforward way to dene an argument, since an argument corresponds to a derivation.

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To simplify the notation, from now on, we suppose that we are given a xed knowledge base K = (F, R, N ) and do not explicitly mention F, R nor N if not necessary. Let a = (F 0 , ..., F n ) be an argument. Then, we denote Supp(a) = F 0 and Conc(a) = F n .

Arguments may attack each other, which is captured by a binary attack relation Att ⊆ Arg(F) × Arg(F).

Denition 2. Let K = (F, R, N ) be a knowledge base and let a and b be two arguments. The argument a attacks argument b, denoted (a, b) ∈ Att, if and only if there exists ϕ ∈ Supp(b) such that the set {Conc(a), ϕ} is

R-inconsistent.
This attack relation is not symmetric. To see why, consider the following example.

Let F = {p(m), q(m), r(m)}, R = ∅, N = {∀x 1 (p(x 1 ) ∧ q(x 1 ) ∧ r(x 1 ) → ⊥)}. Let a = ({p(m), q(m)}, p(m) ∧ q(m)), b = ({r(m)}, r(m)).
We have (a, b) ∈ Att and (b, a) / ∈ Att. This will ensure that the naive extension is dierent, at least in theory, from the preferred, stable, etc. semantics. However, in our application they all entail the same information as shown later on. Denition 3. Given a knowledge base K = (F, R, N ), the corresponding argumentation framework AF K is a pair (A = Arg(F), Att) where Arg(F) is the set of all arguments that can be constructed from F and Att is the corresponding attack relation as specied in Denition 2.

Let E ⊆ A and a ∈ A. We say that E is conict free i there exists no arguments a, b ∈ E such that (a, b) ∈ Att. E defends a i for every argument b ∈ A, if we have (b, a) ∈ Att then there exists c ∈ E such that (c, b) ∈ Att.

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E is admissible i it is conict free and defends all its arguments. E is a complete extension i E is an admissible set which contains all the arguments it defends. E is a preferred extension i it is maximal (with respect to set inclusion) admissible set. E is a stable extension i it is conict-free and for all a ∈ A \ E, there exists an argument b ∈ E such that (b, a) ∈ Att. E is a grounded extension i E is a minimal (for set inclusion) complete extension.

For an argumentation framework AS = (A, Att) we denote by Ext x (AS)

(or by Ext x (A, Att)) the set of its extensions with respect to semantics x.

We use the abbreviations c, p, s, and g for respectively complete, preferred, stable and grounded semantics.

An argument is sceptically accepted if it is in all extensions, credulously accepted if it is in at least one extension and rejected if it is not in any extension.

Based on this denition of arguments and attacks in [START_REF] Croitoru | What can argumentation do for inconsistent ontology query answering?[END_REF] was also shown that the rationality postulates of (Caminada and Amgoud, 2007) are respected. This instantiation respects the direct, indirect consistency as well as the closure.

Formalising the use case

In this subsection we formalise the notions presented in section 4.

Let K = (F, R, N ) be a consistent knowledge base. This is the knowledge base that all actors share and agree upon. In this paper we assume that the rules and negative constraints are common to everybody.

The goals of the dierent actors can be seen as a set of existentially closed conjuncts. We denote them by G 1 , G 2 , ..., G n .

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Let G i be a goal and K the knowledge base. K is consistent and K does not entail G i . We compute the inverse R-derivations of G i (where R is the set of rules of the knowledge base). We add all of the R -1 (G i ) to the facts. We thus obtain a new knowledge base K i which diers from K solely by its facts set (which now also includes

R -1 (G i )): K = (F ∪ R -1 (G i ), R, N ) . We also impose that K i is consistent. Given G = {G 1 , G 2 , .
.., G n }, the goals correspond to a set of viewpoints V (there exists a function κ : G → 2 V ). This function can assign a goal to one or more viewpoints and each viewpoint can be associated with one or more goals. Given a goal G i , the (set of) viewpoint(s) associated with this goal is denoted by κ(G i ). Similarly, given a viewpoint v i , the set of goals associated with it is denoted by κ -1 (v i ).

Example 1. Let the set of viewpoints V = {nutrition, sanitary, organolep-tic} and G consisting of the following goals:

G 1 = ∃ x (Bread(x) ∧ LowSalt(x)), G 2 = ∃ x (Bread(x) ∧ ContaminantFree(x)), G 3 = ∃ x (Bread(x) ∧ Crusty(x)), G 4 = ∃ x (Bread(x) ∧ TraceElementRich(x)). We have κ(G 1 ) = κ(G 4 ) = nutrition, κ(G 2 ) = sanitary and κ(G 3 ) = organoleptic. Conversely κ -1 (nutrition) = {G 1 , G 4 }, κ -1 (sanitary) = {G 2 } and κ -1 (organoleptic) = {G 3 }.
The rules will correspond to the set of sucient conditions needed for the goal G i . In the context of our practical application this is illustrated in Figure 3 (with respect to nutrition goals).

Example 2. To reach the goal G 1 = ∃ x (Bread(x) ∧ LowSalt(x)), the Let us now consider the set of goals G = {G 1 , G 2 , ..., G n } and the initial knowledge base K = (F, R, N ). As described above we compute the n knowledge bases, corresponding to each goal:

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K i = (F ∪ R -1 (G i ), R, N ) for each i = 1, ..., n.
We consider the union of all these knowledge bases:

K agg = (F i=1,...,n R -1 (G i ), R, N ) Example 3. Let K = (F, R, N ) where : • F = {F 1 } = {CurrentExtractionRate(T65)}
• R contains the following rules:

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT -R 1 = ∀ x,y (Bread(x) ∧ ExtractionRate(y,x) ∧ Decrease(y) → Digestible(x)) -R 2 = ∀ x,z (Bread(x) ∧ SaltAdjunction(z,x) ∧ Decrease(z) → LowSalt(x)) -R 3 = ∀ x,y (Bread(x) ∧ ExtractionRate(y,x) ∧ Growth(y) → TraceElementRich(x)) -R 4 = ∀ x,y (Bread(x) ∧ ExtractionRate(y,x) ∧ Decrease(y)
→ PesticideFree(x))

• N contains the following negative constraint:

-

N = ¬(∃ x (Growth(x) ∧ Decrease(x)))
Let the goal set G as follows:

•

G 1 = ∃ p (Bread(p) ∧ Digestible(p)), where κ(G 1 ) = nutrition • G 2 = ∃ p (Bread(p) ∧ LowSalt(p)), where κ(G 2 ) = nutrition • G 3 = ∃ p (Bread(p) ∧ TraceElementRich(p)), where κ(G 3 ) = nutrition • G 4 = ∃ p (Bread(p) ∧ PesticideFree(p)),
where κ(G 4 ) = sanitary.

Then:

• K 1 = (F 1 , R, N ) where F 1 = F ∪ R -1 (G 1
) contains the following facts:

-

F 1 = CurrentExtractionRate(T65) -F 2 = Bread(p) ∧ ExtractionRate(τ ,p) ∧ Decrease(τ ) • K 2 = (F 2 , R, N ) where F 2 = F ∪ R -1 (G 2
) contains the following facts:

-

F 1 = CurrentExtractionRate(T65) -F 3 = Bread(p) ∧ SaltAdjunction(s,p) ∧ Decrease(s) • K 3 = (F 3 , R, N ) where F 3 = F ∪ R -1 (G 3 ) contains the following facts: A C C E P T E D M A N U S C R I P T -F 1 = CurrentExtractionRate(T65) -F 4 = Bread(p) ∧ ExtractionRate(τ,p) ∧ Growth(τ) • K 4 = (F 4 , R, N ) where F 4 = F ∪ R -1 (G 4
) contains the following facts:

-

F 1 = CurrentExtractionRate(T65) -F 2 = Bread(p) ∧ ExtractionRate(τ,p) ∧ Decrease(τ) Finally K agg = (F i=1,...,n R -1 (G i ), R, N )
where

F i=1,...,n R -1 (G i ) = {F 1 , F 2 , F 3 , F 4 }.
As observed in the previous example, it may happen that K agg is inconsistent (and it does so even for goals belonging to the same viewpoint). We then use argumentation, which, by the means of extensions will isolate subsets of facts we can accept together (called extensions). Furthermore, the extensions will allow us to see which are the viewpoints associated to each maximal consistent subset of knowledge (by the means of the function κ). A choice procedure then has to be used (see example below).

The argument framework we can construct from the above knowledge base is (A, Att) where A contains the following:

• a = ({F 2 }, F 2 , R 1 (F 2 )) where R 1 (F 2 ) = Bread(p) ∧ ExtractionRate(τ,p) ∧ Decrease(τ) ∧ Digestible(p). • b = ({F 4 }, F 4 , R 3 (F 4 )) where R 3 (F 4 ) = Bread(p) ∧ ExtractionRate(τ,p) ∧ Growth(τ) ∧ TraceElementRich(p). • c = ({F 2 }, F 2 , R 4 (F 2 )) where R 4 (F 2 ) = Bread(p) ∧ ExtractionRate(τ,p) ∧ Decrease(τ) ∧ PesticideFree(p). • d = ({F 3 }, F 3 , R 2 (F 3 )) where R 2 (F 3 ) = Bread(p) ∧ SaltAdjunction(s,

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ACCEPTED MANUSCRIPT

In this argumentation system dened we now obtain:

• Ext stable (A, Att) = Ext semi-stable (A, Att) = Ext pref ered (A, Att) = {{a, c, d}, {b, d}}.

Starting from the extensions Ext x (A, Att), the proposed decision support system functions as follows: for every extension ε ∈ Ext x (A, Att) :

• Consider the facts occurring in the arguments of ε ;

• Identify the knowledge bases K i where these facts occur;

• Obtain the goals G i which are satised by the extension;

• Using the κ function to obtain the viewpoints corresponding to these goals;

• Show domain experts the set of goals, and compatible viewpoints corresponding to the given extension.

This method allows us to obtain a set of options equal to the cardinality of Ext x (A, Att). For taking a nal decision several possibilities can be considered and presented to the experts:

• Maximise the number of goals satised;

• Maximise the number of viewpoints satised;

• Use preference relations of experts on goals and / or viewpoints.

In the previous example (please recall that the goals G 1 and G 2 are associated with the nutritional viewpoint while G 4 is associated with the sanitary viewpoint) we have:

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• The rst extension {a, c, d} is based on the facts F 2 and F 3 obtained from K 1 , K 2 and K 4 that satisfy the goals G 1 , G 2 and G 4 .

• The second extension {b, d} is based on F 3 and F 4 obtained from K 2 and K 3 satisfying G 2 and G 3 both associated with the nutritional viewpoint.

One rst possibility (corresponding to the extension {a, c, d}) consists of accomplishing F 2 and F 3 and allows to satisfy the biggest number of goals and viewpoints.

The second possibility (corresponding to the extension {b, d}) consists of accomplishing F 3 and F 4 . It would satisfy two goals and one viewpoint. It could be considered though if the goal G 3 (not satised by the rst option) is preferred to the others.

Evaluation

The evaluation of the implemented system was done via a series of interviews with domain experts. The above knowledge and reasoning procedures were implemented using the Cogui knowledge representation tool [START_REF] Chein | Visual reasoning with graphbased mechanisms: the good, the better and the best[END_REF], with an extension of 2000 lines of supplemental code. Three experts have validated our approach: two researchers in food science and cereal technologies of the French national institute of agronomic research, specialists respectively of the grain-to-our transformation process and of the breadmaking process, and one industrial expert -the Director of the French National Institute of Bread and Pastry.

The rst meeting dealt with the delimitation of the project objectives and addressed fundamental questions such as: Is it possible to uniquely dene a

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M A N U S C R I P T good bread? Which scenarii of good bread should be considered? How could they be dened from a nutritional, sanitary, sensorial and economic point of view? Which are the main known ways to achieve them?

Then a series of individual interviews constituted the elicitation phase.

Each expert gave more arguments which were complementing one each other.

In parallel, the writing of specications for the demonstrator and the denition of the knowledge base structure were conducted.

In the following plenary meeting the real potential of the approach was shown. The experts were formulating goals and viewpoints they were interested in and the Cogui system together with the argumentation extension was yielding the associated possible propositions. Figure 6 shows a screenshot of the demonstrator answers for a two-goal query: a nutritional goal (high ber content) and an organoleptic goal (crusty bread). Two sets of compatible actions are proposed, some choices (such as increasing or decreasing the extraction rate) being incompatible for both goals, and thus separated in the two alternative sets.

Four scenarii were more specically evaluated. These scenarii concern four kinds of consumers: obeses (ber preference), people with iron deciency (micronutrient preference), people with cardivascular disease (decreased salt preference) and vegetarians (limited phytic acid), which produces dierent sets of goals. For each scenario, the system proposes several outputed rec- Two interests of the approach were more particularly highlighted. They concern cognitive considerations. Firstly, experts were conscious that the elicitation procedure was done according to their thought processes, that is, in a forward way which is more natural and intuitive. The system was thus able to restitute the knowledge in a dierent manner than the experts usually do. Secondly, from a problem that could initially seem simple, the experts realized that it covered a huge complexity that a human mind could hardly address alone. The tool is currently available to them under restricted access.

The knowledge modeling task can be a very time-consuming step. As presented in Section 4.1, several sources of information were used, from peer reviewed scientic papers and technical reports, to conference meetings and expert interviews. On the one hand, expert interviews appeared to be the For instance, the critical reading a scientic paper of the domain may require a one-day period on its own, for a discerning reader. However they allow one to to grasp the ins and outs of the question.

During the evaluation step, the experts raised the question of the importance attached to the dierent pieces of knowledge modeled in the system.

Moreover, in some cases experts may hesitate on the relevance of some facts or rules. A possibility would thus be to adopt a preference-based argumentation system, as proposed in several works such as (Amgoud and Cayrol, 2002;Bench-Capon, 2003b;[START_REF] Kaci | Preference-based argumentation: Arguments supporting multiple values[END_REF]Amgoud et al., 2000;Bourguet et al., 2013a), able to take into account dierent levels of importance among arguments.

Conclusion

Even if argumentation based decision making methods applied to the food industry were also proposed by [START_REF] Bourguet | Contribution aux methodes d'argumentation pour la prise de decision. application a l'arbitrage au sein de la liere cerealiere[END_REF]Bourguet et al., 2013b), this paper addresses a key point in the context of current techniques used by the food sector and namely addressing reverse engineering. Also, in this approach, an argument is used here as a method computing compatible objectives in the sector. This case study represents an original application and

Figure 1 :

 1 Figure 1: The Cogui visual graph based interface

  Figure 2: Nutritional (a) and organoleptic (b) goals

  Finally, we say that a set F and a set of rules R entail a fact G (and we write F, R |= G) i the closure of the facts by all the rules entails F (i.e. if Cl R (F) |= G).

Figure 3 :

 3 Figure 3: Ways to reach nutritional goals

  p) ∧ Decrease(s) ∧ LowSalt(p) and Att = {(a, b), (b, a), (b, c), (c, b)}.

Figure 4 :

 4 Figure 4: Demonstrator screenshot showing two sets of possible actions

  in terms of time. A one-day period allows both elicitating knowledge through an interview and formalizing it in the software system which constitutes the longest part of the work. However, this relatively short time hides a strong prerequisite: having already a clear view of the case study, a synopsis of the questions to ask the expert and an implemented knowledge model. On the other hand, websites, technical reports and scientic articles are more costly to analyze.

  in the agronomy eld by providing an argumentation based decision-support system for the various food sectors. It requires nevertheless the very expensive task of knowledge modeling. Such task, in its current state cannot be automated. It strongly depends on the quality of expert opinion and elicitation (exhaustiveness, certainty, etc). The current trend for decision-making tools includes more and more methods of argumentation as means of including experts in the task of modeling and the decision-making processes. Another element to take into account, not discussed in this paper, is the diculty of technologically (from an agronomy viewpoint) putting in place the facts of each option. Modeling this aspect in the formalism is still to be studied. Amgoud, L., Cayrol, C., 2002. A reasoning model based on the production of acceptable arguments. Annals of Mathematics and Articial Intelligence 34, 197216. Amgoud, L., Parsons, S., Perrussel, L., 2000. An argumentation framework based on contextual preferences. In: Proceedings of the International Conference on Formal and Applied and Practical Reasoning. pp. 5967. Amgoud, L., Prade, H., 2009. Using arguments for making and explaining decisions. Articial Intelligence 173 (3-4), 413436. AQUANUP, 2009. http://www.inra.fr/inra_cepia/vous_recherchez/ des_projets/france/aquanup.Assaghir, Z., Napoli, A., Kaytoue, M., Dubois, D., Prade, H., 2011. Numeri-

applied to food quality in a public health policy. Expert Systems with Applications 40 (11), 45394546. Bourguet, J.-R., Thomopoulos, R., Mugnier, M.-L., Abécassis, J., 2013b. An articial intelligence-based approach to deal with argumentation applied to food quality in a public health policy.

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Highlights

We present a real use case study of an argumentation process in the agrifood domain, in the wheat to bread chain. We introduce the notions of viewpoint and goal in a backward chaining reasoning procedure.

We provide a logical framework in first order logic. Specific recommended actions are computed depending on the considered concerns.

We present the case study evaluation process.