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- Introducing Conviviality as a property of Multi …orbilu.uni.lu/bitstream/10993/8877/1/ECAI12-arcoe.pdf· Introducing Conviviality as a property of Multi-Context Systems? ... (MCS)

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Introducing Conviviality as a property of Multi-Context Systems ? Antonis Bikakis 1 and Vasileios Efthymiou 2 and Patrice Caire 2 and Yves Le Traon 2 1 Department of Information Studies, University College London [emailprotected] 2 University of Luxembourg, Interdisciplinary Center for Security, Reliability and Trust (SnT) [emailprotected] Abstract. Multi-Context Systems (MCS) are rule-based representation models for distributed, heterogeneous knowledge sources, called contexts, such as ambi- ent intelligence devices and agents. Contexts interact with each other through the sharing of their local knowledge, or parts thereof, using so-called bridge rules to enable the cooperation among different contexts. On the other hand, the concept of conviviality, introduced as a social science concept for multiagent systems to highlight soft qualitative requirements like user friendliness of systems, was recently proposed to model and measure cooperation among agents in multia- gent systems. In this paper, we introduce conviviality as a property to model and evaluate cooperation in MCS. We ﬁrst introduce a formal model, then we pro- pose conviviality measures, and ﬁnally we suggest an application consisting in a conviviality-driven method for inconsistency resolution. 1 Introduction Multi-Context Systems (MCS) [1–3] are logical formalizations of distributed context theories connected through a set of bridge rules, which enable information ﬂow be- tween contexts. A context can be thought of as a logical theory - a set of axioms and inference rules - that models local knowledge. Intuitively, MCS can be used as a rep- resentation model for any information system that involves distributed, heterogeneous knowledge agents including peer-to-peer systems, distributed ontologies (e.g. Linked Open Data) or Ambient Intelligence systems. In fact, several applications have already been developed on top of MCS or other similar formal models of context including (a) the CYC common sense knowledge base [4], (b) contextualized ontology languages, such as Distributed Description Logics [5] and C-OWL [6], (c) context-based agent ar- chitectures [7, 8], and (d) distributed reasoning algorithms for Mobile Social Networks [9] and Ambient Intelligence systems [10]. While designing a real system based on the MCS model, comparing MCSs in order to select the most appropriate conﬁguration, evaluations and measures are needed. For example, consider a house environment, which consists of various devices, sensors and appliances connected through a wireless network. The role of an Ambient Intelligence system is to transform this network of devices into a smart home environment by en- abling devices to share and reason with their context knowledge. However, techniques ? Thanks to: National Research Fund, Luxembourg (I2R-SER-PFN-11COPA) CoPAInS project

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Introducing Conviviality as a propertyof Multi-Context Systems?

Antonis Bikakis1 and Vasileios Efthymiou2 and Patrice Caire2 and Yves Le Traon2

1 Department of Information Studies, University College [emailprotected]

2 University of Luxembourg, Interdisciplinary Center for Security, Reliability and Trust (SnT)[emailprotected]

Abstract. Multi-Context Systems (MCS) are rule-based representation modelsfor distributed, heterogeneous knowledge sources, called contexts, such as ambi-ent intelligence devices and agents. Contexts interact with each other through thesharing of their local knowledge, or parts thereof, using so-called bridge rules toenable the cooperation among different contexts. On the other hand, the conceptof conviviality, introduced as a social science concept for multiagent systemsto highlight soft qualitative requirements like user friendliness of systems, wasrecently proposed to model and measure cooperation among agents in multia-gent systems. In this paper, we introduce conviviality as a property to model andevaluate cooperation in MCS. We first introduce a formal model, then we pro-pose conviviality measures, and finally we suggest an application consisting in aconviviality-driven method for inconsistency resolution.

1 Introduction

Multi-Context Systems (MCS) [1–3] are logical formalizations of distributed contexttheories connected through a set of bridge rules, which enable information flow be-tween contexts. A context can be thought of as a logical theory - a set of axioms andinference rules - that models local knowledge. Intuitively, MCS can be used as a rep-resentation model for any information system that involves distributed, heterogeneousknowledge agents including peer-to-peer systems, distributed ontologies (e.g. LinkedOpen Data) or Ambient Intelligence systems. In fact, several applications have alreadybeen developed on top of MCS or other similar formal models of context including (a)the CYC common sense knowledge base [4], (b) contextualized ontology languages,such as Distributed Description Logics [5] and C-OWL [6], (c) context-based agent ar-chitectures [7, 8], and (d) distributed reasoning algorithms for Mobile Social Networks[9] and Ambient Intelligence systems [10].

While designing a real system based on the MCS model, comparing MCSs in orderto select the most appropriate configuration, evaluations and measures are needed. Forexample, consider a house environment, which consists of various devices, sensors andappliances connected through a wireless network. The role of an Ambient Intelligencesystem is to transform this network of devices into a smart home environment by en-abling devices to share and reason with their context knowledge. However, techniques? Thanks to: National Research Fund, Luxembourg (I2R-SER-PFN-11COPA) CoPAInS project

to allow sharing of knowledge among the contexts could still be improved, to makeAmbient Intelligence sustainable in the long term. A MCS may be used as the repre-sentation and reasoning model for such a system. Bridge rules are used to enable thissharing, by allowing each context to access the knowledge acquired by the other con-texts. For example, consider two devices in our smart-house, that share their knowledgeabout the user’s location within the house, to reason and optimize their service to thisuser’s needs and desires. But, how can we then evaluate the ways in which the systemenables this cooperation? How can we characterise a MCS based on the opportunitiesfor information exchange that it provides to its contexts? To answer such questions, weintroduce in MCS the notion of conviviality.

Defined by Illich as “individual freedom realized in personal interdependence” [11],conviviality has been introduced as a social science concept for multiagent systems tohighlight soft qualitative requirements like user friendliness of systems. Multiagent sys-tems technology can be used to realize tools for conviviality when we interpret “free-dom” as choice [12]. Tools for conviviality are concerned in particular with dynamicaspects of conviviality, such as the emergence of conviviality from the sharing of prop-erties or behaviors whereby each member’s perception is that their personal needs aretaken care of [11]. We measure conviviality by counting the possible ways to cooperate,indicating degree of choice or freedom to engage in coalitions. Our coalitional theoryis based on dependence networks [13, 14]; labeled directed graphs where the nodes areagents, and each labeled edge represents that the former agent depends on the latter oneto achieve some goal. Here, we draw a parallel between, on the one hand an agent anda context, and on the other hand between a goal and a bridge rule. Particularly, we use acontext to encode an agent’s knowledge in some logic language, and a bridge rule to de-scribe how an agent achieves its goal, namely to acquire knowledge from other agents,as illustrated in Figure 1. The focus on dependence networks and more specifically ontheir cycles, is a reasonable way of formalizing conviviality as something related to thefreedom of choice of individuals plus the subsidiary relations –interdependence for taskachievement– among fellow members of a social system.

Agent 2Agent 1goal 1

bridge rule 1Context 1

Context 2

Fig. 1. The dependence network parallelism of contexts as agents, and bridge rules as goals. Alabeled arrow, representing a goal, from a to b means that a depends on b to achieve this goal.

In distributed information systems, individual freedom is linked with the choiceto keep personal knowledge and beliefs at the local level, while interdependence isunderstood as reciprocity, i.e. cooperation. Participating entities depend on each otherto achieve the enrichment of their local knowledge.

Considering the potential applications of MCS and the notion of conviviality as de-scribed above, our main research question is the following:

How to introduce the concept of conviviality to Multi-Context Systems?

This main research question breaks into the following questions:

1. How to define and model conviviality for Multi-Context Systems?2. How to measure the conviviality of Multi-Context Systems?3. How to use conviviality as a property of Multi-Context Systems?

Building on the ideas of [15], where we first identified ways in which convivialitytools, and specifically dependence networks and conviviality measures can be used toevaluate cooperation in Contextual Defeasible Logic, we propose:

1. a formal model for representing information dependencies in MCS based on de-pendence networks,

2. conviviality measures for MCS, and3. a potential application of these tools (model and measures) for the problem of in-

consistency resolution in MCS.

So far, most approaches for inconsistency (such as occurrence of α, ¬α) resolutionin MCS have been based on the invalidation or unconditional application of a subset ofthe bridge rules that cause inconsistency [16–19]. They differ in the preference criterionthat is applied for choosing among two or more candidate solutions. Here, we proposeusing the conviviality of the system as a preference criterion. This is based on the ideathat removing (or applying unconditionally) a bridge rule affects the information depen-dency between the connected contexts, and, as a result, the conviviality of the system.We suggest that the optimal solution is the one that minimally affects conviviality.

The rest of the paper is structured as follows: Section 2 presents formal definitionsfor MCS, as these were originally proposed in [3]. Section 3 proposes a model andmeasures for conviviality in MCS. Section 4 describes a potential use of conviviality asa property of MCS for the problem of inconsistency resolution. Last section summarizesand presents directions for future work in the field.

2 Multi-Context Systems - Formal Definitions

For the needs of this paper we will use the definition of heterogeneous nonmonotonicMCS given in [3], according to which a MCS is a set of contexts, each composed of aknowledge base with an underlying logic, and a set of bridge rules which control theinformation flow between contexts. A logic L = (KBL, BSL, ACCL) consists of thefollowing components:

– KBL is the set of well-formed knowledge bases of L. We assume each element ofKBL is a set of “formulas”.

– BSL is the set of possible belief sets, where the elements of a belief set is a set of“formulas”.

– ACCL: KBL → 2BSL is a function describing the semantics of the logic by as-signing to each knowledge base a set of acceptable belief sets.

A bridge rule can add information to a context, depending on the belief sets whichare accepted at other contexts. Let L = (L1, . . ., Ln) be a sequence of logics. An Lk-bridge rule r over L is of the form

r = (k : s)← (c1 : p1), . . . , (cj : pj),not(cj+1 : pj+1), . . . ,not(cm : pm). (1)

where 1 ≤ ci ≤ n, pi is an element of some belief set of Lci , k refers to the contextreceiving information s. We denote by hb(r) the belief formula s in the head of r.

An MCS M = (C1, . . . , Cn) is a collection of contexts Ci = (Li, kbi, bri), 1 ≤ci ≤ n, where Li = (KBi, BSi, ACCi) is a logic, kbi ∈ KBi a knowledge base, and bria set of Li-bridge rules over (L1, . . ., Ln). For each H ⊆ hb(r)|r ∈ bri it holds thatkbi ∪H ∈ KBLi

, i.e., bridge rule heads are compatible with knowledge bases.A belief state of an MCS M = (C1, . . . , Cn) is a sequence S = (S1, . . . , Sn)

such that Si ∈ BSi. A bridge rule of form (1) is applicable in a belief state S iff for1 ≤ i ≤ j: pi ∈ Sci and for j < l ≤ m: pl /∈ Scl . By brM =

⋃ni=1 bri we denote the

set of all bridge rules of M .The above definitions are exemplified below. It is not in the scope of this paper to

illustrate the many different logics that can be used in MCS. Furthermore, for the sakeof clarity, our example is built on propositional logics only.

Example 1. Consider an MCS M , through which the software agents of three researchstudents exchange information and classify research articles that they access in onlinedatabases. M contains contexts C1−C3, each of which encodes the knowledge of eachof the three agents. The knowledge bases for the three contexts are:

kb1 = sensors, corba, centralizedComputing ↔ ¬distributedComputingkb2 = profAkb3 = ubiquitousComputing ⊆ ambientComputing

C1 collects information about the keywords of the articles and encodes this in-formation in propositional logic. In this case, the article under examination is aboutsensors and corba (Common Object Request Broker Architecture). C1 also possessesthe knowledge that centralized computing and distributed computing are two comple-mentary concepts. C2 uses propositional logic to encode additional information aboutarticles, including the names of their authors; in this case profA is the author of thearticle under examination. Finally, C3 is an ontology of computing-related concepts,according to which ubiquitousComputing is a type of ambientComputing.

The bridge rules that the three agents use to exchange information and collectivelydecide about the classification of the article are as follows:

r1 = (1 : centralizedComputing)← (2 : middleware)r2 = (1 : distributedComputing)← (3 : ambientComputing)r3 = (2 : middleware)← (1 : corba)r4 = (3 : ubiquitousComputing)← (1 : sensors), (2 : profB)

Rule r1 links the concept ofmiddleware used by C2 to the concept of centralized-Computing of C1. r2 expresses that ambientComputing (a term used by C3) im-plies distributedComputing (a term used by C1). r3 expresses that corba is a typeof middleware, while r4 expresses the belief of the third agent (C3) that articlesthat are written by profB and that contain sensors among their keywords are aboutubiquitousComputing.

Equilibrium semantics selects certain belief states of an MCS M = (C1, . . . , Cn)as acceptable. Intuitively, an equilibrium is a belief state S = (S1, . . . , Sn) where eachcontext Ci respects all bridge rules applicable in S and accepts Si. Formally, S is anequilibrium of M , iff for 1 ≤ i ≤ n,

Si ∈ ACCi(kbi ∪ hb(r)|r ∈ bri applicable in S).

Example 2. In the example given above, S = (S1, S2, S3) is the only equilibrium of thesystem:

S = (sensors, corba, centralizedComputing, profA,middleware, ∅).

S3 is an empty set, since kb3, the knowledge base of context C3, is an empty set, br3 =r4, namely the set of bridge rules for context C3 only consists of bridge rule r4, andr4 is not applicable in S, because profB /∈ S2.

3 Modelling and measuring conviviality in MCS

We mentioned in the introduction that dependence networks have been proposed asa model for representing social dependencies among the agents of a multiagent sys-tem. They have also been used as the underlying model for formalizing and measuringconviviality in such systems. In this section, we describe how dependence networkscan be used to model the information dependencies among the contexts of a MCS andhow conviviality measures can then be applied to measure conviviality in MCS. Ourapproach is based on the following ideas: (a) cooperation in MCS can be understoodas information sharing among the contexts; (b) it is enabled by the bridge rules of thesystem; (c) therefore, bridge rules actually represent information dependencies amongthe contexts. Intuitively, that means conviviality will be captured through the differentbridge rules that link the contexts.

3.1 Dependence Networks Model for MCS

According to [20], conviviality may be modeled by the reciprocity-based coalitionsthat can be formed. Some coalitions, however, provide more opportunities for their

participants to cooperate with each other than others, being thereby more convivial. Torepresent the interdependencies among agents in the coalitions, [20] use dependencenetworks.

In this subsection, we first present Definition 1 from [20], which abstracts fromtasks and plans. Then, building on [20]’s definition, we introduce our definition for adependence network corresponding to a MCS.

A dependence network is defined by [20] as follows:

Definition 1 (Dependence networks). A dependence network (DN) is a tuple〈A,G, dep,≥〉 where: A is a set of agents, G is a set of goals, dep : A× A → 2G is afunction that relates with each pair of agents, the sets of goals on which the first agentdepends on the second, and ≥: A → 2G × 2G is for each agent a total pre-order onsets of goals occurring in its dependencies: G1 >(a) G2.

To capture the notions of contexts and bridge rules, we now introduce our definition,Definition 2, for a dependence network corresponding to a MCS, as follows:

Definition 2 (Dependence networks for MCS). A dependence network correspondingto a MCS M , denoted as DN(M), is a tuple 〈C,R, dep,≥〉 where: C is the set ofcontexts inM ,R is the set of bridge rules inM , dep : C×C → 2R is a function that isconstructed as follows: for each bridge rule r (in the form of (1)) inR add the followingdependencies: dep(k, ci) = r where k is the context appearing in the head of r andci stands for each distinct context appearing in the body of r, and ≥: C → 2R × 2R

is for each context a total pre-order on sets of bridge rules that the context appears intheir heads.

In other words, a bridge rule r creates one dependency between context k, whichappears in the head of r, and each of contexts ci that appear in the body of r. Theintuition behind this is that k depends on the information it receives from each of thecontexts ci to achieve its goal, which is to apply r in order to infer s. It follows fromDefinition 2 that we can have two or more dependencies labeled by the same rule. Theapplication of this rule relies upon all dependencies labeled with this rule. An alternativeway to label dependencies would be to use the heads of the rules that these dependenciesare derived from, instead of the rules themselves. This is based on the intuition that,when using a rule, a context has actually the goal to derive the conclusion that labelsthe head of the rule. In that case, however, a new definition of dependence networksmay be needed to support both conjunctions and disjunctions of dependencies.

We should also note here that the total preorder that each context defines on the setsof bridge rules may reflect the local preferences of a context, e.g. in the way that theseare defined and used in Contextual Defeasible Logic [18, 10]. For sake of simplicity, wedo not use this feature in the conviviality model that we describe below. However, it isamong our plans to integrate it in future extensions of this work.

To graphically represent dependence networks, we use nodes for contexts and la-beled arrows for dependencies among the contexts that the arrows connect. An arrowfrom context a to context b, labeled as r, means that a depends on b to apply bridge ruler.

In our running example, the dependence network that corresponds to MCSM is theone depicted in Figure 2.

C1

C2 C3

r3

r1

r4

r4

r2

Fig. 2. The dependence network DN(M) of MCS M of the running example. Nodes representcontexts and arrows represent dependencies. An arrow from context a to context b, labeled as r,means that a depends on b to apply bridge rule r.

In this graph, each node corresponds to one of the contexts in M . Dependenciesare derived from the four bridge rules of M . For example, there are two dependencieslabeled by r4: each of them connects C3, which appears in the head of r4, to one ofthe contexts C1 and C2, which appear in the body of r4. This actually means that toapply rule r4 in order to prove that the paper under examination is about ubiquitouscomputing, C4 depends on information about the keywords of the paper that it importsfrom C1 and information about the authors of the paper that it imports from C2.

3.2 Conviviality Measures

Conviviality measures have been introduced to compare the conviviality of multi agentsystems [20], for example before and after, making a change such as adding a newnorm, or policy. Furthermore, to evaluate conviviality in a more precise way, [20] in-troduce formal conviviality measures for dependence networks using coalitional gametheoretic framework. Based on Illich’s definition of conviviality as “individual freedomrealized in personal interdependency”, the notions of interdependency and choice, if weinterpret freedom as choice, are stressed. Such measures provide insights into the typeof properties that may be measured in convivial systems and thus reveal the quality ofthe system.

The conviviality measures presented in this work reflect the following Hypotheses:

H1 the cycles identified in a dependence network are considered as coalitions. Thesecoalitions are used to evaluate conviviality in the network. Cycles are the smallestgraph topology expressing interdependence, thereby conviviality, and are thereforeconsidered atomic relations of interdependence. When referring to cycles, we areimplicitly signifying simple cycles, i.e., where all nodes are distinct [21]; we alsodiscard self-loops. When referring to conviviality, we always refer to potential in-teraction not actual interaction.

H2 conviviality in a dependence network is evaluated in a bounded domain, i.e., overa [min,max] interval. This allows the comparison of different systems in terms ofconviviality.

H3 there is more conviviality in larger coalitions than in smaller ones.H4 the more coalitions in the dependence network, the higher the conviviality measure

(ceteris paribus).

Hypothesis H1 is consistent with Definition 2, according to which each bridge ruleis mapped to a set of potential dependencies between MCS contexts. The intuition forHypothesis H3, is that a greater number of collaborating contexts in a MCS offers agreater source of knowledge. This means that each context participating in a largecoalition has more available data, than the data it would have in a smaller coalition.Hypothesis H4 is motivated by the fact that a large number of coalitions indicates moreinteractions among contexts, which is positive in term of conviviality.

Our top goal is to maximize conviviality in the MCS. Some coalitions provide moreopportunities for their participating contexts to cooperate than others, being therebymore convivial. Our two sub-goals (or Requirements) are thus:

R1 maximize the size of the contexts’ coalitions, i.e. to maximize the number of con-texts involved in the coalitions,

R2 maximize the number of these coalitions.

Following the definition of the conviviality of a dependence network [20], we definethe conviviality of a dependence network of a MCS M as

Conv(DN(M)) =

∑ci,cj∈C,i6=j

coal(ci, cj)

Ω, (2)

Ω = |C|(|C| − 1)×Θ, (3)

Θ =

L=|C|∑L=2

P (|C| − 2, L− 2)× |R|L, (4)

where |C| is the number of contexts in M , |R| is the number of bridge rules in M ,L is the cycle length, P is the permutation defined in combinatorics, coal(ci, cj) forany distinct ci, cj ∈ C is the number of cycles that contain both ci and cj in DN(M)and Ω denotes the maximal number of pairs of contexts in cycles (which produces thenormalization mentioned in Hypothesis H2).

This way, the conviviality measurement of a dependence network, which is a ra-tional number in [0,1], can be used to compare different dependence networks, with0 being the conviviality of an acyclic dependence network and 1 the conviviality of afully-connected dependence network.

Example 3. Following Equation 2 and the dependence network of M , which is graph-ically represented in Figure 2, we calculate the conviviality of DN(M) of our runningexample, as:

Conv(DN(M)) =10

Ω= 0.208,

where Ω = 480.

The result of Example 3 is just a way of comparing the conviviality of differentsystems. By itself it cannot be used to classify the conviviality of a MCS.

4 Use of conviviality as a property of MCS: InconsistencyResolution

As we previously argued, conviviality is a property that characterizes the cooperative-ness of a MCS, namely the alternative ways in which the contexts of a MCS can shareinformation in order to derive new knowledge. By evaluating conviviality, the systemmay propose the different ways in which it can be increased, e.g. by suggesting newconnections (bridge rules) between the system contexts.

Consider, for example, a MCS, in which a context does not import any informationfrom other contexts. Recommending other contexts that this context could import infor-mation from would be a way to increase the conviviality of the system, which would inturn lead to enriching the local knowledge of the context but also the knowledge of thewhole system.

4.1 Problem Description

Another way of using conviviality as a property of MCS, which we describe in moredetail in this section, is for the problem of inconsistency resolution. In an MCS, even ifcontexts are locally consistent, their bridge rules may render the whole system incon-sistent. This is formally described in [3] as a lack of an equilibrium. All techniques thathave been proposed so far for inconsistency resolution are based on the same intuition:a subset of the bridge rules that cause inconsistency must be invalidated and anothersubset must be unconditionally applied, so that the entire system becomes consistentagain. For nonmonotonic MCS, this has been formally defined in [16] as diagnosis:

”Given an MCS M, a diagnosis of M is a pair (D1, D2), D1, D2 ⊆ brM , s.t.M [brM\D1 ∪ heads(D2)] 6|= ⊥”. D±(M) is the set of all such diagnoses, whileM [brM\D1 ∪ heads(D2)] is the MCS obtained from M by removing the rules in D1

and adding the heads of the rules in D2.In other words, if we deactivate the rules in D1 and apply the rules in D2 in uncon-

ditional form, M will become consistent. As it is obvious, in a MCS it is possible thatthere is more than one diagnosis that can be applied to restore consistency.

Example 4. In our running example, consider the case that profB is also identified byC2 as one of the authors of the paper under examination. In this case kb2 would alsocontain profB:

kb2 = profA, profB

This addition would result in an inconsistency in kb1, caused by the activation of rulesr4 and r2. Specifically, rule r4 would become applicable, ubiquitousComputing andambientComputing would become true in C3, r2 would then become applicable too,and distributedComputing would become true in C1 causing an inconsistency withcentralizedComputing, which has also been evaluated as true. To resolve this con-flict, one of the four bridge rules r1-r4 must be invalidated. Using the definition ofdiagnosis that we presented above, this is formally described as:

D±(M) = (r1, ∅), (r2, ∅), (r3, ∅), (r4, ∅).

Various criteria have been proposed for choosing a diagnosis including:

– the number of bridge rules contained in the diagnosis - specifically in [16] subset-minimal diagnoses are preferred,

– local preferences on diagnoses proposed in [19], and– local preferences on contexts and provenance information, which have been pro-

posed for Contextual Defeasible Logic [18, 10].

4.2 Proposed Solution

Our approach is to use the conviviality of the resulted system as a criterion for choosinga diagnosis. This actually means that for each candidate solution (diagnosis), we mea-sure the conviviality of the system that is derived after applying the diagnosis, and wechoose the diagnosis that minimally decreases the conviviality of the system. The intu-ition behind this approach is that the system should remain as cooperative as possible,and this is achieved by enabling the maximum number of agents to both contribute toand benefit from this cooperation.

Diagnoses contain two types of changes that one can apply in the bridge rules: inval-idation (removal) of a rule; and applying a bridge rule unconditionally, which actuallymeans removing the body of the rule. These changes affect the dependencies of the sys-tem as follows: When invalidating or adding unconditionally rule r (as this is definedin (1)) in a MCS M , all the dependencies that are labeled by r are removed from thedependence network of M .

Assuming that DN(M,Di) is the dependence network that corresponds to MCSM after applying diagnosis D1, the optimal diagnosis is the one that maximizes theconviviality of DN(M,Di):

Dopt = Di : Conv(DN(M,Di)) = max

Example 5. In the running example, there are four diagnoses that we can choose from:D1-D4. Each of them requires invalidating one of the four bridge rules r1 to r4, re-spectively. Figures 3 to 6 depict the four dependence networks DN(M,Di), which arederived after applying Di. Dashed arrows in Figures 3-6 represent the dependenciesthat are dropped in each DN(M,Di), by applying diagnosis Di.

Following Equation 2 and the four dependence networks, which are graphicallyrepresented in Figures 3-6, the conviviality of each DN(M,Di) is:

Conv(DN(M,D1)) =8

Ω= 0.037 and

Conv(DN(M,D2)) = Conv(DN(M,D3)) = Conv(DN(M,D4)) =2

Ω= 0.009,

with Ω = 216.

Since the number of goals |G| is now 3, instead of 4, Ω has a different value thanin DN(M). By applying D1 (Figure 3), only one cycle (C1, C2) is removed from theinitial dependence network DN(M), illustrated in Figure 2. However, by applying any

C1

C2 C3

r3

r1

r4

r4

r2

Fig. 3. DN(M,D1)

C1

C2 C3

r3

r1

r4

r4

r2

Fig. 4. DN(M,D2)

C1

C2 C3

r3

r1

r4

r4

r2

Fig. 5. DN(M,D3)

C1

C2 C3

r3

r1

r4

r4

r2

Fig. 6. DN(M,D4)

of the remaining diagnoses D2-D4, two cycles are removed from DN(M). Specifi-cally, by applying D2 (Figure 4), we remove the cycles (C1, C3) and (C1, C3, C2). Byapplying D3 (Figure 5), we remove the cycles (C1, C2) and (C1, C3, C2). Finally, byapplying D4 (Figure 6), we remove the cycles (C1, C3) and (C1, C3, C2).

Therefore the optimal diagnosis is D1. By applying D1 the system will have thefollowing equilibrium S′:

S′ = (sensors, corba, distributedComputing, profA, profB,middleware,ubiquitousComputing, ambientComputing)

This approach can also be combined with any of the approaches that have beenproposed so far for inconsistency resolution. For example, one may choose to apply theconviviality-based approach only to those diagnoses that comply with some constraintsrepresenting user-defined criteria, as suggested in [19]. It can also be combined withpreferences on diagnoses proposed by [19] or preferences on contexts suggested by[18, 10]. A study of such combined approaches will be part of our future work in thefield.

5 Conclusion

Today, with the rise of systems in which knowledge is distributed in a network of in-terconnected heterogeneous and evolving knowledge resources, such as Semantic Web,Linked Open Data, and Ambient Intelligence, research in contextual knowledge rep-resentation and reasoning has become particularly relevant. Multi-Context Systems

(MCS) are logical formalizations of distributed context theories connected through aset of bridge rules, which enable information flow between contexts. The individualentities, that such systems consist of, cooperate by sharing information through theirbridge rules. By reasoning on the information they import, they are able to derive newknowledge. Evaluating the ways in which the system enables cooperations, and char-acterizing a MCS based on the opportunities for information exchange that it providesto its contexts are. therefore, key issues. The social science concept of conviviality hasrecently been proposed to model and measure the potential cooperation among agentsin multiagent systems. Furthermore, formal conviviality measures for dependence net-works using a coalitional game theoretic framework, have been introduced. Roughly,more opportunities to work with other agents increase the conviviality of the system.

This paper is a step toward extending the concept of conviviality, modeled withdependence networks, to Multi-Context Systems. First, we describe how convivialitycan be used to model cooperation in MCS. Based on the intuition that contexts dependon the information they receive from other contexts to achieve their goals, i.e., applyspecific bridge rules to infer particular information, we define dependence networks forMCS. Furthermore, the aim is for MCSs to be as cooperative as possible, and for con-texts to have as many choices as possible to cooperate with other contexts. This resultsin MCS being as convivial as possible. In order to evaluate the conviviality of a MCS,we apply pairwise conviviality measures and allow for comparisons among MCS. Fi-nally we propose a potential use of conviviality as a property of MCS for the problemof inconsistency resolution. Indeed, without considering contextual information, rea-soning can easily encounter inconsistency problems, for example, when consideringknowledge in the wrong context. Our approach in this case is based on the idea that theoptimal solution is the one that minimally decreases the conviviality of the system.

In further research, we contemplate the need to study alternative ways in which aMCS can be modeled as a dependence network, for example by labeling dependen-cies with the heads of the rules that they are derived from. We also plan to study therelation between the preference order on goals, which is included in the definition ofdependence networks, and preferences on rules, contexts or diagnoses that the systemcontexts may have. Furthermore, we plan to combine the conviviality-based approachfor inconsistency resolution with the preference-based approaches proposed by [19]and [18, 10]. Finally, we want to study how the concept of conviviality and the tools forconviviality can be used in other distributed knowledge models, such as Linked OpenData, Distributed Description Logics [5], E-connections [22] and managed MCS [17].

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