How strong to believe information ? a trust model in the logical belief function theory

Abstract

To which extent an agent can believe a piece of information it gets? This is the question we address in this paper. More precisely, we provide a model for expressing the relations between the trust an agent puts in the information sources and its beliefs about the information they provide. This model is based on Demolombe's model and extends it by considering imperfect information sources i.e information sources that can report false information. Furthermore, this model is defined in the belief function theory allowing degrees of trust to be modelled in a quantitative way. Not only this model can be used when the agent directly gets information from the source but it can also be used when the agent gets second hand information i.e., when the agent is not directly in contact with the source.

Motivation

When it gets a new piece of information from an information source, a rational agent has to update or revise its belief base if it considers this piece of information sufficiently supported [ 1 ], [20], [12], [ 13 ] . Thus an important question for the agent is to estimate how this new piece of information is supported i.e, how strong it can believe it ?

Obviously, the agent may believe a new piece of information if it trusts the information source for delivering true information [ 7 ], [ 14 ]. For instance, assume that in order to know if it will rain tomorrow, I look at M´et´eoFrance web site and read that indeed it will rain. If I trust M´et´eo-France for delivering correct forecast, then I can believe what M´et´eo-France is reporting i.e., I can believe that it will rain.

But in many applications [19], [16] [17], information sources are not necessarly correct all the time. They may deliver false information, intentionnaly or not. This is why, assuming them to be incorrect must also be useful. Furthermore, it may happen that information of interest is second hand i.e., the agent who collects information is not directly in contact with the information source: it obtains information through an agent which cites the information source. The process can even be longer. This is the case when, for instance, I am informed by my neighbour that he read on M´et´eo-France web site that it is going to rain. In this case, the information my neighbour reports is that he read that according to M´et´eo-France it is going to rain. My neighbour does not tell me that it is going to rain. We insist on the fact that here, the very information which interests me (will it rain ?) is reported via two agents, M´et´eo-France and my neighbour, the second citing the first. Second hand information management is a more delicate issue. Indeed, trusting my neighbour for giving true forecast is not useful here. However, trusting him not to lie and trusting M´et´eo France for giving true forecast will lead me to believe that it is going to rain.

One very interesting contribution in the trust community is Demolombe’s one [ 7 ],[ 8 ],[ 9 ]. The author considers trust as an attitude of an agent who believes that another agent has a given property. As for an agent which provides information (i.e an information source), it can be attributed six properties: it can be sincere (roughly speaking, it believes what it reports), competent (information it believes is true), valid (information it reports is true), cooperative (it reports what it believes), vigilant (it believes what is true) or complete (it reports what is true). In this model, validity is defined as sincerity and competence and completeness is defined as cooperativity and vigilance. This work shows that the trust an agent puts in an information source relatively to some of these properties influences the fact that this agent believes what this source produces or does not produce. This is formally shown by considering a formal logical framework expressed in modal logic. The operators of this modal logic are : Bi (Bip means “agent i believes that p”) and Iij (Iij p means “agent i informs1 agent j that p”). Operators Bi obey KD system and their semantics are defined by serial accessibility relations between words. Operators Iij satisfy Iij p ∧ Iij q → Iij (p ∧ q) and obey rule of equivalence substitutivity [ 2 ]. Their semantics are defined by neighborhood functions. Furthermore, it is assumed that there is no failure in the communication process when an agent informs another one. I.e, Iij p → BiIij p and¬Iij p → Bi¬Iij p are considered as true. The different sorts of trusts considered in this model are2: • T sincerea,b(p) =def Ba(Ibap → Bbp) i.e., a trusts b for p in regard to sincerity iff a believes that if b tells it p then b believes p. • T competenta,b(p) =def Ba(Bbp → p) i.e., a trusts b for p in regard to competence iff a believes that if b believes p then p is true. • T cooperativea,b(p) =def Ba(Bpp → Ibap) i.e., a trusts b for p in regard to cooperativity iff a believes that if b believes p then b tells it p. • T vigilanta,b(p) =def Ba(p → Bbp) i.e., a trusts b for p in regard to vigilance iff a believes that if p is true then b believes it. • T valida,b(p) =def Ba(Ibap → p) i.e., a trusts b for p in regard to validity iff a believes that if b tells it p then p is true. • T completea,b(p) =def Ba(p → Ibap) i.e., a trusts b for p in regard to completeness iff a believes that if p is true then b tells it.

These notions can then formally be used to derive the beliefs of an agent who receives a piece of information. For instance, T validi,j (p) → (Ijip → Bip) is a theorem. It shows that if agent i trusts agent j for information p in regard to validity, then if j tells it p then i believes p. An instance of this theorem is: T validi,MF (rain) → (IMiF rain → Birain) which means that if I trust M´et´eo-France for being valid for forecast, then if M´et´eoFrance reports that it will rain then I believe that it will rain. Notice that we also have the following theorem: T completei,MF (rain) → (¬IMiF rain → Bi¬rain). This means that if I trust M´et´eo-France for being complete for forecast, then if M´et´eo-France does not report that it will rain then I believe that it will not rain.

This model can be used to make more complex inferences, in particular for the problem of trust propagation when agents inform each other about their own trusts [ 9 ]. For instance, suppose that due to a gaz failure at home, my husband calls a plumber. Suppose that my husband tells me that the plumber told him that our gaz installation must be renewed. He adds that he trusts this person because he owns the Qualigaz agreement. Then, if I trust my husband to tell the truth, I can conclude that our gaz installation must be renewed.

Furthermore, agents may be uncertain about the trust they put in information sources. In [ 8 ], Demolombe presented a qualitative framework to model this uncertainty and defines a modal logic for graded beliefs . Graded beliefs are modelled by operators Big so that Bigp means that the strength level of agent i about p is g. However, 1We will sometimes say “tells” or “reports” or “produces” 2We adopt here the definitions given in [ 9 ] which defines trust with the belief operator Bi and not the definitions of [ 7 ] which defines trust with a “strong belief” Ki operator the author did not show the application of this framework to second hand information management. In [ 6 ], Pichon et al address very close questions by using the belief function theory [18] which is a framework for modelling uncertainty in a quantitative way. They propose a mechanism for computing the plausibility of a piece of information which is emitted by an agent given our uncertain belief about its reliability. For the authors, the reliability of an agent is defined by its relevance and its truthfulness so that (1) information provided by a non-relevant information source is ignored i.e a non-relevant source brings no information; (2) we can believe the piece of information provided by a relevant and truthful source; (3) we can believe the negation of the piece of information provided by a relevant but non-truthful source. It must be noticed that “being relevant and truthful” is quite close to “being valid” as introduced by Demolombe. However, this work did not address the case of second hand information either.

In the same time, we suggested to extend Demolombe’s model by considering the negative counterparts of the previous six properties [ 3 ], [ 4 ]. In particular, we considered negative counterparts of validity and completeness, which led us to consider misinformer agents and falsifier agents. Roughly speaking, when a misinformer reports a piece of information, then we can conclude that it is false. When a piece of information is false, a misinformer reports it is true. In Demolombe’s terms, this could be defined by: • T misinf ormera,b(p) =def Ba(Ibap → ¬p) i.e., a trusts b for being misinformer relatively to p iff a believes that if b tells it p then p is false. Notice that “being misinformer” is quite close to “being relevant and non truthful” as introduced by Dubois et al. • T f alsif iera,b(p) =def Ba(¬p → Ibap) i.e., a trusts b for being falsifier relatively to p iff a believes that if p is false then b tells it is true.

In the present paper, we go on our research by proposing a model which (i) allows an agent to express to which extent it trusts an information source for being valid, complete, misinformer or falsifier; (ii) which can also be used to manage second hand information. This model is defined in a framework recently defined, the logical belief function theory [ 5 ]. This framework extends the belief function theory and allows the user to assign degrees of beliefs to propositional formulas and to express integrity constraints.

This paper is organized as follows. Section 2 describes a model which allows an agent to express uncertainty about the trust it puts in sources. Section 3 shows how to use this model to evaluate information. We first study the case when the agent is in direct contact with the source and then the case when it is not. Examples will illustrate the model. Finally section 4 is devoted to a discussion. 2

A trust model in the logical belief function theory

The belief function theory [18] is a framework which offers several interesting tools to manage uncertain beliefs such as mass functions and belief functions which are used to quantify the extent to which one can believe a piece of information and rules of combination which are used to combine uncertain beliefs in many different way [ 10 ]. Ignorance can explicitely be quantified in this framework and consequently this model does satisfy the rule of additivity, i.e., the degree of belief in a proposition plus the degree of belief in its negation can be strictly less than 1.

In this paper, we will use the logical belief function theory [ 5 ] which extends the belief function theory by allowing the user to express its beliefs in propositional logic and to consider integrity constraints. It has been proved that this formalism is not more expressive than the belief function theory. However, it allows to express uncertain beliefs directly and in a more compact way. Furthermore, it enlightens the role of integrity constraints. This framework is summarized below. 2.1

The logical belief function theory

Let Θ be a finite propositional language and σ be a satisfiable formula of Θ.

We consider the equivalence relation denoted ↔σ which is defined by: A ↔σ B iff σ |= A ↔ B. Thus, two formulas are in relation by ↔σ if and only if they are equivalent when σ is true. By convention, we will say that all the formulas of a given equivalent class of ↔σ are identical. With this convention, we can consider that the set of formulas is finite. It is denoted F ORM σ. Finally, in the following, true denotes any tautology and f alse denotes any inconsistant formula.

In the logical belief function theory, the user can assign masses to propositional formulas. This is shown by the following definition. and

Like the belief function theory, the logical belief function theory offers several interesting functions for decision. In particular, the logical belief function and the logical plausibility functions are defined by:

Definition 2 Given a logical mass function m, the logical belief function which is associated to m is defined by:

Bel : F ORM σ → [ 0, 1 ] such that: m(f alse) = 0 ϕ∈F ORMσ

Definition 3 Given a logical mass function, the logical plausibility function which is associated to m is defined

by: P l : F ORM σ → [ 0, 1 ] such that

P l(ϕ) = 1 − Bel(¬ϕ) It can be noticed that

P l(ϕ) =

m(ψ) ψ∈F ORMσ (σ∧ψ∧ϕ) is satisfiable

Like the belief function theory, the logical belief function theory offers several rules for combining logical mass functions. Let us only detail the logical DS rule defined by: Definition 4 Let m1 and m2 be two logical mass functions. The logical DS rule defines the logical mass function m1 ⊕ m2 : F ORM σ → [ 0, 1 ] by:

m1 ⊕ m2(f alse) = 0 and for any ϕ such that ϕ ↔σ f alse: m1 ⊕ m2(ϕ) = if

σ (ϕ1∧ϕ2)↔ϕ σ∧ϕ1∧ϕ2 is satisfiable m1(ϕ1).m2(ϕ2)

m1(ϕ1).m2(ϕ2) σ∧ϕ1∧ϕ2 is satisfiable m1(ϕ1).m2(ϕ2) = 0

It can easily be shown that this combination rule is the reformulation, into the logical belief function theory, of the Demspter’s rule of combination. 2.2

A trust model

The trust model we define here will be used to express the degrees at which an agent i thinks that another agent j is valid (resp is a misinformer, or complete, or is a falsifier) when it reports information ϕ. The model does not specify how these degrees are defined. For instance, they can be given by agent i on the basis on its past experience with j. More precisely, if i has already get information from j, it can evaluate how many times it believes it and how many times it believes the opposite. We can also imagine that if i does not know j, these degrees are provided by a reputation fusion mechanism or some other trust assessment model [15].

We consider a propositional language Θ with two kinds of letters. The “information letters” p, q..... will be used to model the information which is reported by the agents; the “reporting letters” of the form Ij ϕ will be used to represent “agent j reports information ϕ”, for any propositional formula ϕ made of information letters. For instance Iap is a letter which represents “agent a reports information p”, Ib(p ∧ q) is a letter which represents “agent b reports information p ∧ q”. Definition 5 Consider two agents i and j and a piece of information ϕ i.e., a formula made of information letters. Let vj ∈ [ 0, 1 ] and mj ∈ [ 0, 1 ] two real numbers 3 such that 0 ≤ vj + mj ≤ 1. vj is the degree to which i trusts j for being valid relatively to ϕ and mj is the degree to which i trusts j for being a misinformer relatively to ϕ (written V M (i, j, ϕ, vj , mj )) iff i’s beliefs can be modelled by the mass assignment mV M(i,j,ϕ,vj,mj) defined by :

mV M(i,j,ϕ,vj,mj)(Ij ϕ → ϕ) = vj mV M(i,j,ϕ,vj,mj)(Ij ϕ → ¬ϕ) = mj mV M(i,j,ϕ,vj,mj)(true) = 1 − (vj + mj )

According to this definition, if i believes at degree vj that j is valid for ϕ and believes at degree mj that j is a misinformer then its belief degree in the fact “if j reports ϕ then ϕ is true” is vj ; its belief degree in the fact “if j reports ϕ then ϕ is false” is mj ; and its total ignorance degree is 1 − (vj + mj ). The following particular cases are worth pointing out: • (vj = 1) and (mj = 0) In this case, mV M(i,j,ϕ,1,0)(Ij ϕ → ϕ) = 1. I.e, i is certain that if j tells ϕ then ϕ is true, i.e, i is certain that j is valid for ϕ. I.e i totally trusts j for being valid relatively to ϕ. • (vj = 0) and (mj = 1) In this case, mV M(i,j,ϕ,0,1)(Ij ϕ → ¬ϕ) = 1. I.e. i is certain that if j reports ϕ then ϕ is false, i.e, i is certain that j is a misinformer for ϕ. I.e i totally trusts j for being misinformer relatively to ϕ.

Definition 6 Consider two agents i and j and a piece of information ϕ. Let cj ∈ [ 0, 1 ] and fj ∈ [ 0, 1 ] two real numbers4 such that 0 ≤ cj + fj ≤ 1. cj is the degree to which i trust j for being complete relatively to ϕ and fj is the degree to which i trusts j for being a falsifier relatively to ϕ (written CF (i, j, ϕ, cj , fj )) iff i’s beliefs can be modelled by the mass assignment mCF (i,j,ϕ,cj,fj) defined by: mCF (i,j,ϕ,cj,fj)(ϕ → Ij ϕ) = cj mCF (i,j,ϕ,cj,fj)(¬ϕ → Ij ϕ) = fj mCF (i,j,ϕ,cj,fj)(true) = 1 − (cj + fj )

According to this definition, if i believes at degree cj that j is complete for ϕ and believes at degree fj that j is a falsifier then its belief degree in the fact “if ϕ is true then j reports ϕ” is cj ; its belief degree in the fact “if ϕ is false then j reports ϕ” is fj ; and its total ignorance degree is 1 − (cj + fj ). The following particular cases are worth pointing out: • (cj = 1) and (fj = 0) In this case, mCF (i,j,ϕ,1,0)(ϕ → Ij ϕ) = 1. I.e, i is certain that if ϕ is true then j reports ϕ i.e, i is certain that j is complete for ϕ. I.e i totally trusts j for being complete relatively to ϕ. • (cj = 0) and (fj = 1) In this case, mCF (i,j,ϕ,0,1)(¬ϕ → Ij ϕ) = 1. I.e. i is certain that j is a falsifier for ϕ.

I.e i totally trusts j for being falsifier relatively to ϕ. 2.3

A quick comparision with graded trust model

According to Demolombe, [ 8 ], the notion of graded trust envolves two components respectively called graded beliefs and graded regularities. Consequently, graded trust are modelled by formulas of the form: Big(φ →h ψ) where Big is a modal operator so that Biga expresses that the strength of agent i’s beliefs in a is g and →h is a conditional so that (φ →h ψ) expresses that φ entails ψ at degree h.

The trust model we defined in the previous section does not permit to quantify the strength of the implication. So Demolombe’s graded trust model is richer. For instance it can represent the fact that agent i is totally certain that j is highly valid for proposition p. The previous model cannot.

As for graded beliefs, the two models provide different ways to model them since they do not impose the same axioms on graded beliefs. In particular, in the logical belief function theory, we have: if Beli(ϕ1) = g1 and Beli(ϕ2) = g2 then Beli(ϕ1 ∧ ϕ2) ≤ min(g1, g2) and Beli(ϕ1 ∨ ϕ2) ≥ max(g1, g2). These two assertions are less restrictive that axioms (U 2) and (U 3) of [ 8 ] which impose that Beli(ϕ1 ∧ ϕ2) = min(g1, g2) and Beli(ϕ1 ∨ ϕ2) = max(g1, g2).

3These degrees should be indexed by i and by ϕ but this is omitted for readibility 4Again these degrees should be indexed by i and ϕ

Applying this trust model to evaluate information

The question which is addressed in this section is:

To which extent an agent can believe information he gets?

We will successively examine the case when the agent directly gets information from the information source, then the case when there is an intermediary agent between the agent and the information source.

Definition 7 mV MCF denotes the mass assignment obtained by combining the two previous mass assignments. I.e,

mV MCF = mV M(i,j,ϕ,vj,mj) ⊕ mCF (i,j,ϕ,cj,fj).

This assignment represents i’s degrees of trust in the fact that j is valid, complete, misinformer or falsifier relatively to information ϕ. One can check that mV MCF is defined by: mV MCF (ϕ ↔ Ij ϕ) = vj .cj

mV MCF (ϕ) = vj .fj mV MCF (Ij ϕ → ϕ) = vj .(1 − cj − fj )

mV MCF (¬ϕ) = mj .cj mV MCF (¬ϕ ↔ Ij ϕ) = mj .fj mV MCF (Ij ϕ → ¬ϕ) = mj .(1 − cj − fj ) mV MCF (ϕ → Ij ϕ) = (1 − vj − mj ).cj mV MCF (¬ϕ → Ij ϕ) = (1 − vj − mj ).fj mV MCF (true) = (1 − vj − mj ).(1 − cj − fj ) Definition 8 mψ is the mass assignments defined by: mψ(ψ) = 1.

In particular, if ψ is Ij ϕ, then mIjϕ represents the fact that agent i is certain that j has reported ϕ. If ψ is ¬Ij ϕ, then m¬Ijϕ represents the fact that agent i is certain that j did not report ϕ. 3.1.1

First case.

Here, we assume that agent i get information from the information source j which reports ϕ. In this case, i’s beliefs are modelled by the following mass assignment mi:

One can check that mi is defined by: mi = mV MCF ⊕ mIjϕ

mi(Ij ϕ ∧ ϕ) = vj mi(Ij ϕ ∧ ¬ϕ) = mj mi(Ij ) = (1 − vj − mj )

Beli(ϕ) = vj , Beli(¬ϕ) = mj Theorem 1 Let Beli be the belief function associated with assignment mi. Then:

Consequently, when i knows that j reported ϕ and when V M (i, j, ϕ, vj , mj ) and CF (i, j, , ϕ, cj , fj ), then i believes ϕ more than ¬ϕ if and only if vj > mj i.e, its belief degree in j’s being valid is greater that its belief degree in j’s being a misinformer. These results are not surprising.

Example 1

In this example, we consider an agent (denoted a) which gets information provided by M´et´eo-France web site (denoted M F ). Θ is the propositional language whose “information letters” are: rain (tomorrow will be rainy), cold (tomorrow will be cold), dry (tomorrow will be dry), warm (tomorrow will be warm), and whose “reporting letters” are: IMF rain (M´et´eo-France forecats rain for tomorrow), IMF ¬rain (M´et´eo-France forecats no rain for tomorrow) IMF cold (M´et´eo-France forecats cold for tomorrow), etc.

σ is here the formula σ = ¬(rain ∧ dry) ∧ ¬(cold ∧ warm)

1. Suppose that M´et´eo-France web site indicates that tomorrow will be rainy. Suppose that, given a’s numerous

previous access to this web site, a believes at degree 0.8 that M´et´eo-France web site is valid for rain forecast and a believes at degree 0.1 that it misinforms for rain forecast i.e., V M (a, M F, rain, 0.8, 0.1). Then, by theorem 1, we have: Bela(rain) = 0.8 and Bela(¬rain) = 0.1, i..e, a believes at degree 0.8 that tomorrow will be rainy and a believes at degree 0.1 that tomorrow will be dry. Notice that we also have Bela(¬dry) = 0.8 and Bela(dry) = 0.1.

2. Now suppose that M´et´eo-France web site forecats rain and cold for tomorrow. Suppose that a trusts this web

site at degree 0.8 for being valid and a trusts it at degree 0.1 for being misinformer, i.e V M (a, M F, (rain ∧ cold), 0.8, 0.1). Then, by theorem 1, we have: • Bela(rain ∧ cold) = Bela(rain) = Bela(cold) = 0.8. • Bela(¬rain ∨ ¬cold) = Bela(dry ∨ warm) = 0.1, • Bela(¬rain) = Bela(¬cold) = Bela(dry) = Bela(warm) = 0.

I.e., a’s degree of belief in the fact that tomorrow will be rainy and cold (resp tomorrow will be rainy,

tomorrow will be cold) is 0.8. a’s degree of belief in the fact that tomorrow will be dry or warm is 0.1. But a’s degree of belief in that the fact that tomorrow will be dry (resp will be warm) is 0.

One can notice however that we also have P la(dry) = P la(warm) = 0.2 i.e., a’s plausibility degree in that the fact that tomorrow will dry (resp will be warm) is 0.2.

As a consequence of theorem 1 we have: • If V M (i, j, ϕ, 1, 0) then Beli(ϕ) = 1 and Beli(¬ϕ) = 0 i.e, i believes ϕ and does not believe ¬ϕ; • If V M (i, j, ϕ, 0, 1) then Beli(ϕ) = 0 and Beli(¬ϕ) = 1 i.e, i does not believe ¬ϕ and believes ϕ.

The first result is coherent with a result obtained in Demolombe’s model in which Tvalidi,j (ϕ) ∧ BiIJi ϕ → Biϕ and Tvalidi,j (ϕ) ∧ BiIJi ϕ → ¬Bi¬ϕ are theorems. However, Demolombe’s model does not model misinformers thus the second result cannot be compared. 3.1.2

Second case.

Here we assume that the information source j did not report ϕ. In this case, i’s beliefs can be modelled by the mass assignment mi:

One can check that mi is defined by: Theorem 2 Let Beli be the belief function associated with assignment mi. Then, mi = mV MCF ⊕ m¬Ijϕ mi(¬Ij ϕ ∧ ϕ) = fj mi(¬Ij ϕ ∧ ¬ϕ) = cj mi(¬Ij ) = (1 − cj − fj ) Beli(ϕ) = fj , Beli(¬ϕ) = cj

Consequently, when i knows that j did report ϕ and when V M (i, j, ϕ, vj , mj ) and CF (i, j, , ϕ, cj , fj ) then i believes ϕ more than ¬ϕ if and only if fj > cj i.e, its belief degree in j’s being a falsifier is greater that its belief degree in j’s being complete. Again this is not surprising.

Example 2 Suppose now that agent a reads M´et´eo-France web site to be informed about storms in the south

of France and that no forecast of storm is indicated on the site. Here we consider a propositional language whose “information letter” are: storm (there will be a storm in the south of France), beach (beach access will be open) and whose “reporting letters” are: IMF storm (M´et´eo-France forecats a storm in the south of France), IMF ¬storm (M´et´eo-France forecats no storm in the south of France).

Consider that σ = storm → ¬beach, i.e, in case of storm beach access will be closed.

Suppose that a trusts at degree 0.8 this web site for being complete for storm forecast and a trusts at degree

0.1 this site for being falsifier for storm forecast, i.e., CF (a, M F, storm, 0.8, 0.1).

Then, according to theorem 2, we have: Bela(storm) = 0.1 and Bela(¬storm) = 0.8. Consequently, a has a greater degree of belief in the fact that there will be no storm than in the fact that there will be one. We also have Bela(¬beach) = 0.1 and Bela(beach) = 0. Bela(beach) = 0 may look strange but it comes from the fact that a has no guarantee that beach acces will be open (even if there is no storm). Indeed, ¬storm → beach is not deducible from σ, i.e., nothing states that if there is no storm, beach access will be open.

As a consequence of theorem 2 we have: • If CF (i, j, ϕ, 1, 0), then Beli(ϕ) = 0 and Beli(¬ϕ) = 1 i.e, i does not believes ϕ and believes ¬ϕ; • If CF (i, j, ϕ, 0, 1), then Beli(ϕ) = 1 and Beli(¬ϕ) = 0 i.e, i believes ϕ and does not believe ¬ϕ.

The first result is coherent with a result obtained in Demolombe’s model in which Tcompletei,j (ϕ) ∧ Bi¬IJi ϕ → Bi¬ϕ and Tcompletei,j (ϕ)∧Bi¬IJi ϕ → ¬Biϕ are theorems. However, Demolombe’s model does not model falsifiers thus the second result cannot be compared. 3.2

There is a third agent between the agent and the source

We consider here that i is not in direct contact with the source of information k, but there is a go-between agent named j. The question is again to know to which extent agent i can believe information it gets. Here, we consider a propositional language whose “information letters” are: p, q... and whose “reporting letters” are Ikϕ, Ij Ikϕ, Ij ¬Ikϕ. which respectively mean k reported ϕ, j told that k reported ϕ, j told that k did not report ϕ.

In this section, the mass assigment mV MCF is defined by:

Definition 9

mV MCF = mV M(i,j,Ikϕ,vj,mj) ⊕ mCF (i,j,Ikϕ,cj,fj) ⊕ mV M(i,k,ϕ,vk,mk) ⊕ mCF (i,k,ϕ,ck,fk)

This assignment represents the beliefs of i as regard to agent j being valid, complete, a misinformer or a falsifier for information Ikϕ and as regard to agent k being valid, complete, a misinformer or a falsifier for information ϕ. 3.2.1

First case.

Assume that j reported that k told it ϕ. In this case, i’s beliefs are modelled by the mass assignment mi defined by: Theorem 3 Let Beli be the belief function associated with assignment m.

mi = mV MCF ⊕ mIjIkϕ Beli(ϕ) = vk.fk + vk.vj − vk.fk.vj − vk.fk.mj + fk.mj and Beli(¬ϕ) = mk.ck + mk.vj − mk.ck.vj + ck.mj − mk.ck.mj • If V M (i, j, Ikϕ, 1, 0) and V M (i, k, ϕ, 1, 0) then Beli(ϕ) = 1 and Beli(¬ϕ) = 0. • If V M (i, j, Ikϕ, 1, 0) and V M (i, k, ϕ, 0, 1) then Beli(ϕ) = 0 and Beli(¬ϕ) = 1. • If V M (i, j, Ikϕ, 0, 1) and CF (i, k, ϕ, 1, 0) then Beli(ϕ) = 0 and Beli(¬ϕ) = 1. • If V M (i, j, Ikϕ, 0, 1) and CF (i, k, ϕ, 0, 1) then Beli(ϕ) = 1 and Beli(¬ϕ) = 0.

The first result could be provided in Demolombe’s model if the inform operator was defined by omitting the agent which receives the information. In this case, we would have T validi,k(ϕ) = Bi(Ikϕ → ϕ). Thus T validi,j (Ikϕ) ∧ T validi,k(ϕ) ∧ Ij Ikϕ → Biϕ and T validi,j (Ikϕ) ∧ T validi,k(ϕ) ∧ Ij Ikϕ → ¬Bi¬ϕ would be theorems. As for the last three results, they cannot be compared since Demolombe’s framework does not model misinformers nor falsifiers. 3.2.2

Here, we assume that j does not tell that k reported ϕ. In this case, i’s beliefs are modelled by: mi = mV MCF ⊕ m¬IjIkϕ Theorem 4 Let Beli be the belief function associated with assignment m.

Beli(ϕ) = vk.ck.fj + vk.fk + vk.(1 − ck − fk).(1 − cj ) + mk.fk.cj + (1 − vk − mk).fk and

Beli(¬ϕ) = vk.ck.cj + mk.ck + mk.fk.(1 − cj ) + mk.(1 − ck − fk).fj + (1 − vk − mk).ck.cj Consequently, • If CF (i, j, Ikϕ, 1, 0) and CF (i, k, ϕ, 1, 0) then Beli(ϕ) = 0 and Beli(¬ϕ) = 1. • If CF (i, j, Ikϕ, 1, 0) and CF (i, k, ϕ, 0, 1) then Beli(ϕ) = 1 and Beli(¬ϕ) = 0. • If CF (i, j, Ikϕ, 0, 1) and V M (i, k, ϕ, 1, 0) then Beli(ϕ) = 1 and Beli(¬ϕ) = 0. • If CF (i, j, Ikϕ, 0, 1) and V M (i, k, ϕ, 0, 1) then Beli(ϕ) = 0 and Beli(¬ϕ) = 1.

Again, the first result could be provided in Demolombe’s model if the inform operator was defined by omitting the agent which receives the information. In this case, we would have T completei,k(ϕ) = Bi(ϕ → Ij ϕ). Thus T completei,j (Ikϕ) ∧ T completei,k(ϕ) ∧ ¬Ij Ikϕ → Bi¬ϕ and T validi,j (Ikϕ) ∧ T validi,k(ϕ) ∧ Ij Ikϕ → ¬Biϕ would be theorems. As for the last three results, they cannot be compared since Demolombe’s framework does not model misinformers nor falsifiers.

The following example illustrates the first item.

Example 3 Let us consider three agents denoted me (me), n (my neighbour) and M F (M´et´eo-France web site).

Suppose that my neighbour, who regularly reads M´et´eo-France web site, does not tell me that a storm is forecasted in the south of France.

Suppose that I trust my neighbour to be complete (he always tell me what he reads on M´et´eo-France web site)

i.e.in particular we have CF (me, n, IMF storm, 1, 0).

Suppose that I also trust M´et´eo-France to be complete relatively to storms forecats (they always indicate forecasted storms) i.e., we have: and CF (me, M F, storm, 1, 0).

In this case, we get Belme(storm) = 0 and Belme(¬storm) = 1 i.e., I can conclude that there will be no storm. 3.2.3

Third case

Here, we assume that j reports that k did not report ϕ. In this case, i’s beliefs are modelled by:

We do not detail this case but we think that the reader gets the idea. Here, we assume that j does not tell that k did not report ϕ. In this case, i’s beliefs are modelled by: mi = mV MCF ⊕ mIj¬Ikϕ mi = mV MCF ⊕ m¬Ij¬Ikϕ

We do not detail this case but the reader can easily imagine the kind of conclusions we can derive. 4

Concluding remarks

To which extent an agent can believe a piece of information it gets from an information source? This was the question addressed in this paper in which we have provided a model for expressing the relations between the trust an agent puts in the sources and its beliefs about the information they provide. This model is based on Demolombe’s model and extends it by considering information sources that can report false information. Furthermore, this model is defined in the logical belief function theory allowing degrees of trust to be modelled in a quantitative way and allowing the agent to consider integrity constraints. We have shown that not only this model can be used when the agent directly gets information from the source but it can also be used when the agent gets second hand information i.e., when the agent is not directly in contact with the source.

Notice that the belief function theory has already been used in problems related to trust management. For instance, subjective logic, a specific case of belief function theory, has been used in [ 11 ] for trust network analysis. Notice however that trust network analysis is not the problem addressed here. [ 6 ], already mentionned in the introduction is another paper which uses belief function for estimating the plausibility of a piece of information emitted by sources which may be unrelevant or untruthful. But the case of second hand information is not addressed. More recently [17] defines a formalism based on description logic and belief function theory to reason about uncertain information provided by untrustworthy sources. However, in this work, a source is given a single degree. This degree, called the “degree of trust”, is used for discounting the information it provides. It looks close to the “degree at which an agent trusts a source for being valid” as introduced here but a formal comparision has to be done. Furthermore again, the case of second hand information is not addressed.

As to the use of the logical belief function theory, a question concerns the choice of the combination rule. Here, we have chosen the logical DS rule of combination which is the reformulation of the most classical rule. But can the other rules of combination be used ? Or more precisely, in which cases, should we use them instead? This is an important question to be answered in the future.

The model defined here would be more general if it could define information sources properties according to sets of propositions and not to a given proposition. For instance, we could express that an agent is valid for any proposition related to the topic “weather forecasts” and complete for any proposition related to the topic “south of France”. That would imply that this agent is valid and complete for any information related to weather forecat in the south of France. However, such a reasoning has to be formally defined and describing these topics by means of an ontology is a possible solution.

Finally, this work could be extended by considering the case when information sources provide uncertain information like [17] does. This would allow us to deal with the case when an agent reports that some fact is highly certain or when an agent reports that another agent has reported that some fact was fairly certain. Modelling this kind of uncertainty and mixing it with the one which is introduced in this paper would allow us to deal with more complex cases.

Acknowledgements. I thank the anonymous referees whose pertinent questions led me to improve this paper. [19] STANAG 2511 Standardization Agreement 2511. Intelligence reports, NATO (unclassified), january 2003.

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