A Modal Framework for Relating Belief and Signed

A Modal Framework for Relating Belief and Signed Information Emiliani Lorini1 , Laurent Perrussel2 , Jean-Marc Th´evenin2 1

IRIT Toulouse - France [email protected] 2 IRIT - Universit´e de Toulouse Toulouse - France [email protected], [email protected]

Abstract. The aim of this paper is to propose a modal framework for reasoning about signed information. This modal framework allows agents to keep track of information source as long as they receive information in a multi-agent system. Agents gain that they can elaborate and justify their own current belief state by considering a reliability relation over the sources of information. The belief elaboration process is considered under two perspectives: (i) from a static point of view an agent aggregates received signed information according to its preferred sources in order to build its belief and (ii) from a dynamic point of view as an agent receives information it adapts its belief state about signed information. Splitting the notions of beliefs and signed statement is useful for handling the underlying trust issue: an agent believes some statement because it may justify the statement’s origin and its reliability.

1 Introduction An agent embedded in a multi-agent system gets information from multiple origins; it captures information from its own sensors or, it may receive messages issued by other agents through some communication channels. Based on this set of basic information the agent then defines its belief state and performs actions [25]. As long as it gets information, the agent has to decide what it should believe and also which beliefs are no longer available [14, 16, 4]. In order to decide which beliefs should hold, the agent needs some criteria. A common criterion consists of handling a reliability relation about information origins [10, 6]. According to its opinion about the reliability of the information source, the agent decides to adopt the received piece of information or not. Keeping track of information and its origin is a key issue for trust characterization. Agents can justify their beliefs: agent a believes ϕ because agent b has provided ϕ and b is reliable [19]. In addition, keeping track of agents involved in information broadcasting enables agents to evaluate, from their own point of view, whether they are all reliable, i.e. believable [21]. Although several works have been made in order to show how an agent can merge information issued from multiple origins [27, 23], very few works have focused on the

explicit representation of the origins of information [24, 19] in the context of BDI-based systems with communication actions. This explicit representation is necessary since it represents the underlying rationale of agents’ beliefs. The aim of this article is to propose a modal framework to describe agent’s belief state while preserving information source. The underlying purpose is to avoid the syntax dependency of the work proposed in [24]. We formalize the transition from information to belief and consider the dynamics of information, that is how the agent adapts its belief state about signed information with respect to new incoming information (messages). The dynamics is usually described in terms of performative actions based on KQML performatives [13] or speech acts [7, 8]. Hereafter, we propose to consider a tell action as a private announcement from one agent (the sender of the message) to a second agent (the receiver of the message). A private announcement enables to stress up how an agent “restricts” its belief state as it receives information. More precisely, it shrinks the space of possible information states with respect to information sent by its sources and then according to that space, it builds up its beliefs. The article is structured as follows: In section 2, we present the intuitive meaning of signed information and belief state. Next in section 3, we present the technical details of the modal logic framework. In section 4, we then formalize two intuitive and common policy for relating signed information and belief which consist in the adoption as belief of (i) information commonly signed by some set of agents and (ii) all consistent signed information. In section 5, we extend the modal framework with actions of the form “agent a tells to agent b that a certain fact ϕ is true”. In section 6, we apply our results on an example. We conclude the paper in section 7 by summing up the contribution and considering some open issues.

2 Setting the Framework Handling the source of information leads to the notion of signed statement, that is some statement is true according to some source. From a semantics perspective, we want to be able to represent for an agent, what are the possible states according to information received from each source (w.r.t. some initial state of affairs). 2.1 Representing signed statements Signed information can be represented through Kripke models using one accessibility relation per source of information. Relation denoted Sb describes all the states reachable from some initial state according to information issued from source b. The belief states of each agent, can in turn be represented using one accessibility relation per agent. Relation denoted Ba provides all the possible belief states agent a can reach from some initial state. Figure 1 represents signed states that can be reached from the four possible belief states of agent a, namely w1 , w2 , w3 and w4 , according to information issued from agents b and c. Modal statement Sign(b, p) stands for statement p is true in some state according to source b. Intuitively Sign(b, p) is true in state w if statement p holds in all states reachable from w through relation Sb . Let us have a look at the example presented

Fig. 1. Relating belief state and signatures

Figure 1. Statements p or ¬p and q or ¬q are mentioned between brackets above each world reachable through Sb or Sc in which they hold. Following Sb in state w1 agent a can only reach state w11 and statement p holds in this state. Consequently Sign(b, p) is true for in state w1 . Following Sc from state w1 , agent a can only reach state w12 where statement ¬p hold. Consequently, in state w1 Sign(c, ¬p) is true. In this framework, signed statements represent the rationales for beliefs. It is possible to interpret formulas such as Bel(a, Sign(b, p)) which stands for agent a believes that agent b signs statement p by checking that in all possible belief states of agent a Sign(b, p) is true. According to Figure 1, we get Bel(a, Sign(b, p) ∧ Sign(c, ¬p)) since Sign(b, p) and Sign(c, ¬p) hold in w1 , w2 , w3 and w4 . 2.2 Preferences over information sources In order to handle possibly mutually inconsistent signed statements, agents consider extra information stating which signed statement they prefer. Agents may determine themselves their preferences by considering the sources of information [10, 23], temporal aspects or the topics of the statements [9]. In this paper, for the sake of conciseness and following numerous contributions such as [6], we propose to consider extra information representing a preorder relation denoted 6 defined over the reliability of sources of information: a 6 b stands for a is at least as reliable as b and a < b stands for a is more reliable than b. We assume that the agents consider information about only one topic and handling competencies or different kinds of reliability (such as suggested in [5]) is out of the scope of this paper. According to the example presented Figure 1 agent a believes that agent b signs p and agent c signs ¬p. Considering extra information Bel(a, c < b) standing for agent a believes that agent c is more reliable than agent b, agent a should adopt statement ¬p as a belief. In semi-formal terms, we get that: Bel(a, (Sign(b, p) ∧ Sign(c, ¬p) ∧ c < b)) ⇒ Bel(a, ¬p)

(Adpt)

Using extra-information on the reliability of sources and considering signed statements rather than statements, the problem of belief change [14] is almost rephrased in terms

close to the ones used in belief merging [20, 18]. Reliability order over sources of information enables us to stratify signed information and then by merging this stratified information in a consistent way the agents get “justified” beliefs [3]. By splitting the two concepts of beliefs and signed statements, we are not limited to axiom schema (Adpt). It is possible to define multiple policies to define agent’s beliefs: lexicographic aggregation; focus on statements signed by some specific agents; evaluating the truthfulness of signed information (does the receiver believes that the sender believes the signed statement). This dichotomy also avoids defining belief state (private state) only by considering belief about other agent’s belief (also a private state). We adopt a different principle which enables to show the transition from a public characteristic (signed information and tell actions) to a private one (belief). 2.3 Representing tell statements Let us consider performative T ell(b, a, q) which stands for agent b tells to agent a that q is true. We interpret this performative as a private announcement [29] rather than with help of actions and transitions between states. After performative T ell(b, a, q) agent a believes that b signs q. Consequently, all belief states for which Sign(b, q) is false are no more reachable for agent a after T ell(b, a, q).

Fig. 2. Agent b tells q to agent a

Figure 2 illustrates how agent a’s belief state changes after receiving performative T ell(b, a, q). In the initial situation (the left part of the figure), agent a believes Sign(b, p) but does not believe Sign(b, q) since q does not hold in states w21 and w41 that can be reached through Sb from belief states w2 and w4 . After receiving agent b’s message (right part of the figure), belief state w2 and w4 are no longer reachable through Ba and agent a believes Sign(b, p) and Sign(b, q). Notice that agent a still believes Sign(c, ¬p) and still does not believe Sign(c, q) and Sign(c, ¬q). In other words, agent a’s beliefs about information signed by c have not changed. At this stage, using extra information Bel(a, c < b) agent a can aggregate its signed beliefs as follows: Bel(a, ¬p) and Bel(a, q). Indeed, statement q issued from b is not contradicted by the more reliable source c. Private announcements stress up the information gathering aspect: possible worlds accessible through relation Ba represent the ignorance of agent Ba and by shrinking this

set of possible believable worlds, we represent how agent a gains information. Notice that this way of handling the dynamics entails as a drawback that agent’s belief cannot always be consistent: updating a model might lead to a model where seriality cannot be guaranteed [29].

3 Formal Framework In this section, we focus on signatures, beliefs and preferences; tell actions will be introduced later. The proposed language for reasoning about these three notions is a restricted first order language which enables quantification over agent ids. Quantification allows agents to reason about anonymous signatures. Logical language L is based on doxastic logic. Modal operator Bel represents beliefs: Bel(a, ϕ) means agent a believes L-formula ϕ. Modal operator Sign represents signed statements: Sign(t, ϕ) means t (an agent id or a variable of the agent sort) signs statement ϕ. In order to represent an agent’s opinion about reliability, we introduce the notation a 6 b which stands for: agent a is said to be at least as reliable as b. Definition 1 (Syntax of L). Let P be a finite set of propositional symbols. Let A be a finite set of agent ids. Let V be a set of variables s.t. A ∩ V = ∅. Let T = A ∪ V be the set of agent terms. The set of formulas of the language L is defined by the following BNF: ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | Sign(t, ϕ) | Bel(a, ϕ) | ∀xϕ | t 6 t′ where p ∈ P, t ∈ T , t′ ∈ T , a ∈ A and x ∈ V. Writing a < b stands for a is strictly more reliable than b: a 6 b ∧ ¬(b 6 a). Operators → and ∃ are used according to their usual meaning. 3.1 Axiomatics Axiomatization of logic L includes all tautologies of propositional calculus. Table 1 details the axioms and inference rules describing the behavior of belief, signed statement and reliability. These axioms and inference rules are standard and follow KD45 logic for the behavior of signed statement and K45 logic for the behavior of belief; let us again stress that axiom D could not hold because of public announcement (see section 5). Notice axiom schema (T o6 ) which reflects that reliability relations have to be believed as total. Let ⊢ denotes the proof relation. 3.2 Semantics The semantics of L-formulas is defined in terms of possible states and relations between states [12]. Those relations respectively represent the notion of signatures and beliefs as discussed in section 2. In each state, propositional symbols are interpreted and total preorders representing agents’ reliability are set.

(KS ) Sign(a, ϕ → ψ) → (Sign(a, ϕ) → Sign(a, ψ)) (DS ) Sign(a, ϕ) → ¬Sign(a, ¬ϕ) (KB ) Bel(a, ϕ → ψ) → (Bel(a, ϕ) → Bel(a, ψ)) (4B ) Bel(a, ϕ) → Bel(a, Bel(a, ϕ)) (5B ) ¬Bel(a, ϕ) → Bel(a, ¬Bel(a, ϕ)) (R6 ) t 6 t (T r6 ) t 6 t′ ∧ t′ 6 t′′ → t 6 t′′ (T6 ) t 6 t′ ∨ t′ 6 t (T o6 ) Bel(a, t 6 t′ ) ∨ Bel(a, t′ 6 t) (M P ) From ϕ and ϕ → ψ infer ψ (G)

From ϕ infer ∀tϕ

(NS ) From ϕ infer Sign(t, ϕ) (NB ) From ϕ infer Bel(a, ϕ) Table 1. Logic L axioms and inference rules

Definition 2 (Model). Let M be a model defined as a tuple: [ [ hW, Si , Bi , I, i i∈A

i∈A

where W is a set of possible states. Si ∈ W × W is an accessibility relation representing signatures, Bi ∈ W × W is an accessibility relation representing beliefs. I is an interpretation function of the propositional symbols w.r.t. each possible state, I : W × P 7→ {0, 1}.  is a function which represents total preorders; these preorders are specific to each state, that is : W 7→ 2A×A . A variable assignment is a function v which maps every variable x to an agent id. A t-alternative v ′ of v is a variable assignment similar to v for every variable except t. For t ∈ T , [[t]]v belongs to A and refers to the assignment of agent terms w.r.t. variable assignment v, such that: if t ∈ A then [[t]]v = t

if t ∈ V then [[t]]v = v(t)

We define the satisfaction relation |= with respect to some model M , state w and variable assignment v as follows. Definition 3 (|=). Let M be a model and v be a variable assignment: v : V → A. M satisfies an L-formula ϕ w.r.t. a variable assignment v and a state w, according to the following rules: – M, v, w |= t 6 t′ iff ([[t]]v , [[t′ ]]v ) ∈ (w). – M, v, w |= p iff p ∈ P and I(w, p) = 1. – M, v, w |= Sign(t, ϕ) iff M, v, w′ |= ϕ for all w′ s.t. (w, w′ ) ∈ S[[t]]v

– M, v, w |= Bel(a, ϕ) iff M, v, w′ |= ϕ for all w′ s.t. (w, w′ ) ∈ Ba – M, v, w |= ∀tϕ iff for every t-alternative v ′ , M, v ′ , w |= ϕ. We write |= ϕ iff for all M , w and v, we have M, v, w |= ϕ. The semantics for operators ¬, →, ∨, ∧ and ∃ is defined in the standard way. Let us now detail the constraints that should operate on the model. We require that signature has to be consistent which entails that all relations Si have to be serial. Belief operator as well as signature operator are K45 operator and thus all Bi and Si are transitive and euclidean. Interwoven relations between signatures and beliefs are detailed in the next section. Constraining the Reliability Relations We assume that every agent holds belief about reliability without any uncertainty. That is, agent’s beliefs about reliability can be represented as a total preorder. We propose to handle multiple preorders by indexing reliability with worlds. However, the aggregation of the preorders associated to all the believable worlds of one agent (which are total) must lead to a total preorder. This will then help the agent to aggregate all signed statements. In other words, we require that the integration (or merging) of signed statements should be based on an underlying total preorder over statements (as it is commonly assumed in the belief revision and merging areas—see [14, 17, 18]). Formally we introduce the two following constraints: 1. for all states w, t  (w) t′ or t′  (w) t and, 2. suppose w′ s.t. (w, w′ ) ∈ Bi and t  (w′ )t′ , then for all states w′′ s.t. (w, w′′ ) ∈ Bi , t  (w′′ ) t′ . The first constraint enforces that preorders are total in all states, which reflects axiom schema (T6 ). The second constraint expresses that totality should hold in all the belief states of one agent, which reflects axiom schema (T o6 ). Moreover, preorder definition entails that reflexivity and transitivity hold. We conclude the section by giving the results about soundness and completeness. Theorem 1. Logical system L is sound and complete. Proof. Soundness is straightforward. The completeness proof is mainly based on [12]. The proof is based on the definition of a canonical model which is itself built upon the definition of maximal consistent sets. A maximal consistent set is defined as follows: let ϕ0 , ϕ1 , · · · be an infinite sequence of L-formulas. W.r.t. the sequence ϕ0 , ϕ1 , · · · , a maximal and consistent set T is built s.t. T = ∪i∈0···∞ T i where: (i) if ϕi 6= ∀tϕ(t) then Ti = Ti−1 ∪ {ϕi } if Ti−1 ∪ {ϕi } is consistent and Ti = Ti−1 ∪ {¬ϕi } otherwise and (ii) if ϕi = ∀tϕ(t) then Ti = Ti−1 ∪ {ϕi } if Ti−1 ∪ {ϕi } is consistent and Ti = Ti−1 ∪ {¬ϕi } ∪ {¬ϕ(t¯)} otherwise (t¯ is a new variable). Using the set of maximal consistent sets, we then define the canonical model M c : M c = hW c , ∪i∈A Si , ∪i∈A Bi , I, i s.t. (i) W c is the set of maximal consistent sets, (ii) I(w, p) = 1 if p ∈ w and I(w, p) = 0 otherwise, (iii) a  (w)b iff a 6 b ∈ w, (iv) (w, w′ ) ∈ Ba iff {ϕ | Bel(a, ϕ) ∈ w} ⊆ w′ (idem for Sa ). Using that canonical model, it is routine to prove that M c , w |= ϕ iff ϕ ∈ w by assuming that property M c , w |= ϕ′ iff ϕ′ ∈ w is satisfied for every subformula ϕ′ of ϕ. Next, we conclude the proof by showing that M c is a model for logic L, that is all Sa are serial, all Ba are transitive and euclidean and all  are total preorders (because of the axioms (R6 ), (T r6 ) and (T o6 )). Completeness is then proved.

4 Linking Signatures and Beliefs There are multiple ways to switch from information to beliefs. These different ways may follow principles issued from the belief merging principle [20, 18, 6] or epistemic attitudes such as trust [21, 19]. As previously mentioned, we do not require that an agent has to believe that other agents believe in information they provide. This is a key issue when information is propagated from one agent to another. At some stage, an agent may just broadcast some information without committing to that information in terms of belief. Hereafter, we present two different policies showing how an agent switches from signed information to belief: the first one consists of aggregating in an incremental way signed statements that are commonly signed by a subset of agents which are equally reliable; the second policy consists of accepting as beliefs statements which are signed by one agent and not contradicted by the other agents. 4.1 Ranking agents The two policies for aggregating signed statements require that these signed statements are considered in an incremental way; that is “from the most reliable to the less reliable statements”. Agents can be ranked since we always consider a total preorder. The agents which are equally reliable are gathered in the same group and the groups can then be ranked. Agents are ranked as follows. At first, we characterize the most reliable set of agents denoted as set C1 with the help of the following formula: a ∈ C1 =def ∀t(a 6 t) The formula characterizing members of C1 can then be used for characterizing membership to a set Ci such that i > 1. a ∈ Ci =def



^

j∈1...(i−1)


¬(a ∈ Cj ) ∧ ∀t (

^

j∈1...(i−1)


¬(t ∈ Cj )) → (a 6 t)

It means that all agents belonging to a set Ci are equally reliable and for all a ∈ Ci , b ∈ Cj if i