A Framework for Merging Possibilistic Knowledge Bases

Abstract

Knowledge base merging is one of active research fields with a large range of applications in Artificial Intelligence. Most of the works in this research field has a lot of restrictions such as in the centralized approach, drowning effect, it is difficult to apply to interactive systems such as multi-agent systems as well as omitting knowledge in the merging process. The purpose of this paper is to focus on the integration of possibilistic knowledge bases in the propositional sense in propositional language. The integration is done through the special possibility distribution of possibilistic knowledge bases. To solve that problem, we introduce a new powerful argumentation framework for merging knowledge bases. In order to model this argument, firstly, the source knowledge bases of each agent are ranked in order of priority and the merging of prior knowledge bases into a priority knowledge base and then infer the final knowledge. An axiomatic model, including a set of rational and intuitive postulates is interested and discussed so that the merging result of knowledge bases needs to be satisfied.

1 Introduction

In the past decades, knowledge base merging has become a compelling field in many areas of cooperative information systems. These approaches have been introduced in many papers for merging conflict knowledge bases. It is applied in a large range of areas such as Distributed Database, Multiagent Systems, GroupWare and Distributed Information as Web context. The advantage of knowledge base merging approach is the richness of information so get more from different sources. This knowledge is often contradictory and use priority issues to resolve conflicts, as well as balancing the various factors to achieve the best fusion of inconsistent sources of information is the issue we look forward to.

Uncertainty is almost an attribute of information and knowledge. For this reason, the handling of uncertainty in inference systems has been a long-standing issue of artificial intelligence (AI). When the second expert system was introduced, a proposed setting for the representation of reliability, skepticism and certainty factor was proposed. Then, a series of new proposals have been developed for uncertainty including the theoretical belief function [15], possibilistic theory [16], probability inaccuracy [17], in which the possibilistic theory has emerged at the forefront of AI methods, challenging the maximum right of the representation and the logic installed.

Possibilistic logic is a weighted logic that allows to handle uncertainty (and it also models preferences). In addition, possibilistic logic of inconsistency by taking advantage of the classification for the set of formulas. These are the basic features of standard possibilistic logic. Thus, in the standard possibilistic logic, there are four ways to determine the inference end of a knowledge base in terms of total order [14]: semantic approach based on ranking of representations, the syntactic approach based on Modus Ponens and Weakening, a classical approach based on cuts and rational rejection. They are equivalent and yield the same result as inference. Possibilistic logic is an equivalent framework for determining the ranking of the possible world. In fact, any rank of performance can be represented by a possibilistic knowledge base.

From the promising features of the possibilistic logic evokes us to merging knowledge approaches. Approaches have been published as a two-stage knowledge integration process [12, 13], negotiated game solutions [18], stratified knowledge merging model [9,10,11, 19, 20], a merging model for weighted knowledge bases [8], a model for knowledge merging by argumentation [21], and knowledge merging by argumentation in possibilistic logic [1, 3, 23].

In this model, we propose a new argumentation framework for merging possibilistic knowledge bases. The contribution of this paper is twofold. First, we introduce a framework for merging possibilistic knowledge-based in which a common argument framework is applied in the knowledge base to achieve meaningful results in comparison with other knowledge base merging techniques for knowledge base merging in possibilistic logic such as [3,4,5,6,7]. Second, an axiomatic model, including rational and intuitive propositions for merging results is introduced and many logical attributes are discussed.

The rest of this paper is organized as follows: We review about possibilistic logic in Sect. 2. Knowledge integration operators for the prioritized belief bases, presented by possibilistic logic is presented in Sect. 3. Section 4 introduces a general argumentation framework and a model to merge knowledge bases. Postulates for knowledge base merging by argumentation and logical properties is introduced and discussed in Sect. 5. Some conclusions and future work are presented in Sect. 6.

2 Possibilistic Logic

The theory of possibility has been studied for several decades. Scenarios of uncertainty may use rules or theoretical possibilities. We consider a propositional language ${\mathcal{L}}$, based on the limited set of propositional variables denote to the Greek letters $\alpha ,\beta ,\gamma , \ldots$, the formulas denoted by the letters $\phi ,\psi ,\xi , \ldots ,$ and Ω are the finite set of representations of ${\mathcal{L}}$. The set of models of $\phi$ notation $\left[ \phi \right]$, which is a subset of Ω. We denote ${ \vdash }$ the classical syntactical inference and ${ \vDash }$ the classical semantic inference. The possibility distribution, denoted by π, is a function from Ω to [0, 1]. We also denote $\pi \left( \omega \right)$ the degree of compatibility of ω with real world available knowledge. When $\pi \left( \omega \right) = 1$, it means that ω is completely consistent with the available knowledge, while $\pi \left( \omega \right) = 0$ means that ω is not possible. $\pi \left( \omega \right) > \pi \left( {\omega '} \right)$ means that ω is more priority than $\omega '$.

Each possibilistic logic formula has the form $\left( {\phi_{i} ,\alpha_{i} } \right)$ such that $\phi_{i}$ is a propositional formula and $\alpha_{i} \, \in \,\left[ {0, 1} \right]$. The weight $\alpha_{i}$ is the degree of certainty or priority that the propositional formula $\phi_{i}$ is true, namely $N(\phi_{i} )\, \ge \,\alpha_{i} \, \ge \,0$, where N is a necessary measure. When the weight is 1, the possibilistic logic is the classical logic. A possibility distribution that allows two functions from ${\mathcal{L}}$ to [0, 1] called possibility and necessity measures, denoted by $\varPi$ and N, and defined by:

$$ \begin{aligned}\Pi \left( \phi \right) & = { \hbox{max} }\left\{ {\pi \left( \omega \right):\omega \, \in \,\Omega ,\,\upomega\,{ \vDash }\,\phi } \right\},{\text{and}} \\ N\left( \phi \right) & = 1 - \varPi \left( {\neg \phi } \right). \\ \end{aligned} $$

Constraints of the form $\prod \left( \phi \right)\, \ge \,\alpha$ can also be processed in logic, but they correspond to poor information, whereas $N\left( \phi \right)\, \ge \,\alpha \, \Leftrightarrow \,\prod \left( {\neg \phi } \right)\, \le \,1 - \alpha$ show that $\neg \phi$ somewhat can not and have more information.

A possibilistic formula $\left( {\phi ,\,\alpha } \right)$ consists of a propositional formula ϕ and a weight $\upalpha\, \in \,[0,1]$. A possibilistic knowledge base is a finite set of the formulas $P = \{ \left( {\phi_{i} ,\,\alpha_{i} } \right)| \,i = 1,\, \ldots , \,n\}$. We denote $P^{*}$ an aggregation knowledge base for P defined as follows: $P^{*} = \{ \phi_{i} |\left( {\phi_{i} ,\,\alpha_{i} } \right)\, \in \,P\}$. Obviously, a possibilistic knowledge base P is consistent if $P^{*}$ is consistent and vice versa.

By a semantic approach, with a possibilistic knowledge base, there are generally many possibility distributions π on the set of representation Ω such that the necessary measure is determined from this possibility distribution satisfying ${\rm N}(\phi_{i} ) \ge a_{i}$ with all formulas $\phi_{i}$. Among these possibility distributions, there is a special distribution, this probability distribution is found by the minimum specification principle, which is the maximum measure of entropy determined as follows:

Definition 1

[5, 23]. ∀ω ∈ Ω

$$ \pi_{P} (\omega ) = \left\{ {\begin{array}{*{20}c} {1 \;\text{if}\; \forall \left( {\phi_{i} ,\alpha_{i} } \right)\, \in \,P,\omega \,{ \vDash }\,\phi_{i} } \\ {1 - { \hbox{max} }\left\{ {\alpha_{i} :\left( {\phi_{i} ,\alpha_{i} } \right)\, \in \,P\,{\text{and }}\omega \,{ \nvDash }\,\phi_{i} } \right\}\;{\text{otherwise}}} \\ \end{array} } \right. $$

(1)

Example 1.

Suppose that

$$ P = \left\{ {\left( {a,0.8} \right),\left( {\neg c,0.7} \right),\left( {b \to a,0.6} \right),\left( {c,0.5} \right),\left( {c \to \neg b,0.4} \right)} \right\}. $$

According to Definition 1, we can determine the possibility distribution for P as follows:

$$ \begin{aligned}\uppi_{\text{P}} ({\text{a}}\neg {\text{b}}\neg {\text{c}}) & =\uppi_{\text{P}} ({\text{ab}}\neg {\text{c}}) = 0.5,\uppi_{\text{P}} \left( {\text{abc}} \right) =\uppi_{\text{P}} ({\text{a}}\neg {\text{bc}}) = 0.3\,{\text{and}} \\ \pi_{P} (\neg {\text{abc}}) & = \pi_{P} (\neg {\text{ab}}\neg {\text{c}}) = \pi_{P} (\neg {\text{a}}\neg {\text{bc}}) = \pi_{P} (\neg {\text{a}}\neg {\text{b}}\neg {\text{c}}) = 0.2 \\ \end{aligned} $$

Definition 2.

Given a possibilistic knowledge base P and $\alpha \, \in \, \left[ {0, 1} \right]$, the α-cut of P is denoted by $P_{ \ge \alpha }$ and defined: $(P_{ \ge \alpha } \text{ = }\{ \phi \, \in \,P^{*} |\left( {\phi , \beta } \right)\, \in \,P, \,\beta \, \ge \,\alpha \}$). Similarly, a strict α-cut of P is denoted by $P_{ > \alpha }$ and defined: $\left( {P_{ > \alpha } = \left\{ {\phi \, \in \,P^{*} \left| {\left( {\phi , \beta } \right)\, \in \,P, \beta \, > \,} \right.\alpha } \right\}} \right)$.

Definition 3.

Possibilistic knowledge base $P_{1}$ is equivalent to possibilistic knowledge base $P_{2}$, written $P_{1} \equiv P_{2}$ if and only if $\pi_{{P_{1} }} = \pi_{{P_{2} }}$.

Definition 4.

The inconsistency degree of a possibilistic knowledge base P is as follows:

$$ Inc\left( P \right) = max\{ \alpha_{i} :P_{{ \ge_{{\alpha_{i} }} }} \,{\text{is inconsistent}}\} $$

(2)

The inconsistency degree of possibilistic knowledge base P is the maximal value $\alpha_{i}$ where $\alpha_{i} - cut$ of P is inconsistent. Conventionally, if P is consistent, then $Inc\left( P \right) = 0$.

Definition 5.

Given a possibilistic knowledge base P and $\left( {\phi , \alpha } \right) \, \in \,P, \left( {\phi , \alpha } \right)$ is a subsumption in P if:

$$ \left( {P \backslash \left\{ {\left( {\phi , \alpha } \right)} \right\}} \right)_{ \ge \alpha } \,{ \vdash }\,\phi $$

(3)

Further, $\left( {\phi , \alpha } \right)$ is a strict subsumption in P if $P_{ > \alpha } \,{ \vdash }\,\phi$.

The following lemma indicates that tautologies can be removed from possibilistic knowledge bases [2].

Lemma 1.

For $\left( {{ \top },a} \right)$ be a tautological formula in $P$ then P and $P' = P - \left\{ {\left( {{ \top },a} \right)} \right\}$ are equivalent.

Definition 6.

For a knowledge base P, the formula $\phi$ is a reasonable (reliable) consequence of P if

$$ P_{ > Inc\left( P \right)} \,{ \vdash }\,\phi $$

(4)

Definition 7.

For a possibilistic knowledge base P, the formula $\left( {\phi ,\,a} \right)$ is a possibilistic consequence of P, denoted P ${ \vdash }$_π (ϕ, a), if:

• $P_{ \ge a}$ is consistent

• $P_{ \ge a} \,{ \vdash }\,\phi$

• $\forall b\, > \,a,P_{ \ge b} \,{ \nvdash }\,\phi$

With any inconsistent possibilistic knowledge base P, all formulas with certainty degrees smaller than or equal to Inc(P) will be omitted in the merging process.

Example 2.

Continuing Example 1, obviously P is equivalent to

$$ P^{\prime} = \{ (\text{a},0.8),(\neg {\text{c}},0.7),({\text{b}} \to {\text{a}},0.6),({\text{c}},0.5)\} . $$

Formula (c → ¬b, 0.4) is missed because of $Inc\left( P \right) = 0.5$.

We have:

• Reasonable inferences of P: $\neg a, c \to a, b \to a$, …

• Possibilistic consequences of: $\left( {c \to a, \,0.7} \right), \left( {b \to a, \,0.6} \right)$, …

The two possibilistic knowledge bases P and $P^{\prime}$ are said to be equivalent, denoted $P \equiv_{s} P^{\prime}$, if and only if $\forall a \in \left( {0,1} \right], P_{ \ge a} \equiv P^{\prime}_{ \ge a}$. The two possibilistic knowledge profiles $\delta_{1}$ and $\delta_{2}$ are said to be equivalent $\left( {\delta_{1} \, \equiv_{s} \,\delta_{2} } \right)$ iff there is a bijection between them that each possibilistic knowledge base of $\delta_{1}$ is equivalent to its image in $\delta_{2}$.

3 A Merging Operator in Possibilistic Logic

In this paper, the knowledge bases of the agents are arranged in order of priority. Merging knowledge is organized in rounds, at each round, agents submit their most preferred beliefs, depend on the current state of argumentation process, some arguments may be eliminated and others are added to the set of accepted beliefs.

We consider a merging process for priority knowledge base by argumentation as follows:

• 1. Each agent arranges its knowledge bases in the order of priority, the more important knowledge is, the higher priority it has. Argumentation process is held in multiple rounds.

• 2. At each round, the agents give concurrently their knowledge.

  • If all the given knowledge bases are jointly consistent with the temporary results in the previous rounds, they will be given an accept set (temporary output).

  • If the knowledge of some of the joint agents conflicts with the temporary result, the knowledge of these agents is ignored and the remaining knowledge is added to the temporary result set.

  • If an agent proposes a knowledge and other agent can defeat it, this knowledge will be rejected.

• 3. The argument process will end when no agent has made any arguments. The final temporary result set will be the result of merging process.

We know that each agent corresponds to a knowledge base with a finite set of formulas, i.e., a finite set of arguments, so that the process of argumentation always ends. In addition, agents encourage their knowledge base to prioritize and submit priority arguments as soon as possible.

A possibilistic merging operator, denoted by $\oplus$ is a function from $\left[ {0, 1} \right]^{n} \;{\text{to}}\; \left[ {0, 1} \right]$. Operator $\oplus$ is used to aggregate the degree of certainty associated with the knowledge provided by different sources. Let ${\mathcal{P}} = \left\{ {P_{1} ,\, \ldots ,P_{n} } \right\}$ be set of n possibilistic knowledge base (may be jointly inconsistent). The result of knowledge base merging P using $\oplus$, denoted ${\mathcal{P}}_{ \oplus }$ is defined as follows:

Definition 8

[4]. (Aggregated base) Let ${\mathcal{P}} = \left\{ {P_{1} ,\, \ldots ,P_{n} } \right\}$ is a set of possibilistic knowledge bases and $\oplus$ a merging operator. The result of merging ${ \mathcal{P}}$ with $\oplus$ is defined by:

$$ {\mathcal{P}}_{ \oplus } = \left\{ {\left( {H_{j} , \oplus \left( {x_{1} ,\, \ldots ,x_{n} } \right)} \right):j = 1,\, \ldots ,n} \right\} $$

where $H_{j}$ are disjunctions of size J between formulas taken from different $P_{i} \left( {i = 1,\, \ldots ,n} \right)$ and $x_{i}$ is either equal to $a_{i}$ or to 0 based respectively on $\phi_{i}$ belongs to $H_{j}$ or not.

For the two knowledge bases $P_{1}$ and $P_{2}$ and the integration operator $\oplus$, the semantic combination rule combines the possibility distributions $\pi_{{P_{1} }}$ and $\pi_{{P_{2} }}$ using $\oplus$ where $\pi_{ \oplus } \left( \omega \right) = \pi_{{P_{1} }} \left( \omega \right) \oplus \pi_{{P_{2} }} \left( \omega \right)$. Its syntactical counterpart is the following possibilistic knowledge bases [4]:

(5)

Two attributes for $\oplus$ are recognized in this definition are as follows [2, 7]:

• 1. $\oplus (0,\, \ldots ,\,0) = 0$

• 2. ${\text{If}}\, a_{i} \ge b_{i} \;{\text{for}}\;{\text{all}} \;i = 1,\, \ldots ,n\; {\text{then}} \; \oplus \;\left( {a_{1} ,\, \ldots ,a_{n} } \right) \ge \oplus \left( {b_{1} ,\, \ldots ,b_{n} } \right)$

The first attribute says that if the formula does not appear in any knowledge base, it does not appear in the merging result. The second attribute is the attribute of complete agreement (called monotonic property) which means that if formula ϕ is more reliable (or preferred to) to formula ψ, then the result of the merging also prefers ϕ to ψ.

Example 3.

Let $P_{1} = \left\{ {\left( {\phi \,{\bigvee }\,\psi ,\,0.9} \right),\left( {\neg \phi ,\,0.8} \right),\left( {\xi ,\,0.1} \right)} \right\}$

và $P_{2} = \left\{ {\left( {\neg \psi ,\,0.7} \right),\left( {\phi ,\,0.6} \right)} \right\}$. For $\oplus$ be the probabilistic sum defined by

$\oplus \left( {a,b} \right) = a + b {-} ab$. Following Definition 8 and formula (5), we get:

$$ \begin{aligned} {\mathcal{P}}_{ \oplus } & = \left\{ {\left( {\phi \vee \psi , 0.9} \right),\left( {\neg \phi ,0.8} \right),\left( {\xi , 0.1} \right)} \right\}\, \cup \,\left\{ {\neg \psi ,0.7 ),\left( {\phi ,0.6} \right)} \right. \\ & \quad \cup \,\{ \left( {\phi ,0.97} \right),\left( {\phi \vee \psi , 0.96} \right),\left( {\neg \phi \vee \neg \psi ,0.94} \right),\,\left( {\xi \vee \neg \psi , 0.73} \right), \left( {\xi \vee \phi ,0.64} \right) \\ \end{aligned} $$

Which is equivalent to

$$ \left\{ {\left( {\phi \vee \psi , 0.96} \right),\left( {\neg \phi \vee \neg \psi , 0.94} \right),\,\left( {\neg \phi ,0.8} \right),\left( {\xi \vee \neg \psi ,0.73} \right),\left( {\neg \psi ,0.7} \right), \left( {\xi \vee \phi ,0.64} \right),\left( {\phi ,0.97} \right),\left( {\xi ,0.1} \right)} \right\} $$

Lemma 2, rewritten ${\mathcal{P}}_{ \oplus }$ given in Definition 8, will be useful in the rest of the paper, but first we give the following definition:

Definition 9.

(Existence of results) Let P be the knowledge base. The formula $\left( {\phi ,\,a} \right)$ is a consequence of P, denoted $P\,{ \vdash }\,\left( {\phi ,a} \right)$ if and only if:

• 1. $\exists P^{\prime}\, \subseteq \,P$ such that $P^{\prime}\,{ \vdash }_{\pi } \,\left( {\phi ,a} \right)$,

• 2. $P^{\prime}$ is consistent,

• 3. $a = { \hbox{min} }\left\{ {a_{i} :\left( {\phi_{i} ,a_{i} } \right)\, \in \,P^{\prime}} \right\}$,

• 4. $P^{\prime}$ is a minimal for set inclusion,

• 5. ${\nexists }P^{\prime\prime}\, \subseteq \,P$ satisfying the above conditions with $P^{\prime\prime}\,{ \vdash }_{\pi } \,\left( {\phi ,b} \right)$ and b > a.

This definition focuses on the foundation of the most preferred formulas.

Example 4.

Let P = {(ϕ ∨ ψ, 0.9), (¬ϕ, 0.7), (ξ ∨ ψ, 0.6), (¬ξ, 0.5)}.

Then $P\,{ \vdash }\,\left( {\phi \,{\bigvee }\,\psi ,\,0.9} \right)$, $P\,{ \vdash }\,\left( {\neg \phi ,\,0.7} \right)$ and $P\,{ \vdash }\,\left( {\psi ,\,0.7} \right)$. However, $P\,{ \vdash }\,\left( {\neg \psi ,\,0} \right)$.

4 Knowledge Merging by Argumentation in Possibilistic Logic

In this section, we consider an implementation of general framework above in order to solve the inconsistencies occur when we combine belief bases $\left( {P_{1} ,\, \ldots .,\, P_{n} } \right)$. Let us start with the concept of argument [1, 22].

Definition 10.

A set of arguments ${\mathcal{A}}$ is said to be conflict-free if there is no argument $A$ and $B$ in ${\mathcal{A}}$ where $A$ attacks $B$.

Definition 11.

An argumentation framework is a set of $\left\langle {{\mathcal{A}},\,{\mathcal{R}},\,{ \succcurlyeq }} \right\rangle$ where ${\mathcal{A}}$ is a finite set of arguments, ${\mathcal{R}}$ is a binary relation representing the relation among the arguments in ${\mathcal{A}}$, and ${ \succcurlyeq }$ is a preorder on ${\mathcal{A}} \times {\mathcal{A}}$. We also use $\succ$ to represent the strict order.

An argument is a message that the agents want to interact with many other agents. So, when arguments arise, there are states between arguments, that is, support, attack, and rejection.

Definition 12.

(Support) Given two arguments $A_{1} = \left\langle {P_{1} , \,a_{1} } \right\rangle$, $A_{2} = \left\langle {P_{2} , \,a_{2} } \right\rangle$ and $A_{2}$ is a dialogue of $A_{1}$, if the knowledge set $P_{1}$ contains sentence $a_{2}$, then $P_{2}$ is called a the dialogue support $A_{1}$.

Definition 13.

(Attack) For two arguments $A_{1} = \left\langle {P_{1} , a_{1} } \right\rangle$, $A_{2} = \left\langle {P_{2} , a_{2} } \right\rangle$, and $A_{2}$ is a dialogue of $A_{1}$, if the knowledge set $P_{1}$ does not contain sentence $a_{2}$, then $A_{2}$ is called an attack dialogue $A_{1}$.

Definition 14.

(Rejected) For two arguments $A_{1} = \left\langle {P_{1} , a_{1} } \right\rangle$, $A_{2} = \left\langle {P_{2} , a_{2} } \right\rangle$, and $A_{2}$ is a dialogue of $A_{1}$, if sentence $a_{2}$ is not equal to the $a_{1}$, then the dialogue $A_{2}$ is called reject $A_{1}$.

Definition 15.

(Dialogue) If there is any relationship that supports attack or rejection between arguments, then it is called a dialogue.

Definition 16.

Each argument is a pair $\left\langle {K,k} \right\rangle$, where k is a formula and K is set of formulas, in which:

• 1. ${\text{K}} \subseteq {K}^{*} ,$

• 2. K ${ \vdash }$ k,

• 3. K is consistent and K is minimal w.r.t. set inclusion.

K is the support and k is the conclusion of this argument. We denote ${\mathcal{A}}\left( {\mathcal{K}} \right)\text{ }$ the set of all arguments constructed from ${\mathcal{K}}$.

Definition 17.

Let ${\text{X and Y}}$ be two arguments in ${\mathcal{X}}$.

• Y attacks X if ${\text{Y}}\,{ \succcurlyeq }\,{\text{X}}$ and ${\text{Y }}\,{\mathcal{R} }\,{\text{X}}$.

• If ${\text{Y }}\,{\mathcal{R} }\,{\text{X}}$ but ${\text{X}}\, \succ \,{\text{Y}}$ then ${\text{X}}$ can defend itself.

• A set of arguments ${\mathcal{A}}$ defends ${\text{X}}$ if ${\text{Y}}$ attacks ${\text{X}}$ then there always exists ${\text{Z}}\, \in \,{\mathcal{A}}$ and Z attacks Y.

Assume that $P_{1}$ and $P_{2}$ are in conflict and if they are equally prioritized, then the merging result neither infer $P_{1}$ nor $P_{2}$. The question now is ‘‘How do two representations of knowledge base have the same priority in the possibilistic logic?’’. We first need to introduce the concept of a priority degree of a sub-knowledge base.

Definition 18.

Let ${\mathcal{A}}$ be a sub-knowledge base of P. We define its priority degree, denoted $Deg_{B} \left( {\mathcal{A}} \right)$, by:

$$ Deg_{P} \left( {\mathcal{A}} \right) = { \hbox{min} }\left\{ {a:\left( {\phi ,a} \right)\, \in \,{\mathcal{A}}\,{ \bigcap }\,P} \right\} $$

and

$$ Deg_{P} \left( {\mathcal{A}} \right) = 1 {\text{if }}{\mathcal{A}}\,{ \bigcap }\,P \,{\text{is}}\,{\text{empty}} $$

This definition implies that $Deg_{P} \left( {\mathcal{A}} \right)$ is equal to the weight of the lowest priority formulas in ${\mathcal{A}}$. Now, we define the preference between two knowledge bases as follows:

Definition 19.

$P_{1}$ is said to be more prioritized than $P_{2}$ if for every conflict ${\mathcal{E}}$ in $P_{1} \,{\bigcup }\,P_{2}$, we have: $Deg_{{P_{1} }} \left( {\mathcal{E}} \right)\, > \,Deg_{{P_{2} }} \left( {\mathcal{E}} \right)$

That is, $P_{1}$ is more prioritized than $P_{2}$ if the least priority (i.e. least weighted) of the formulas in each conflict ${\mathcal{E}}$ between $P_{1}$ and $P_{2}$ is $P_{2}$.

Let ${\mathcal{L}\mathcal{P}}\left( {\mathcal{E}} \right)$ be the set of least prioritized formulas in ${\mathcal{E}}$.

Two possibilistic knowledge bases $P_{1}$ and $P_{2}$ are said to have the same priority if for every conflict ${\mathcal{E}}$ in $P_{1} \,{\bigcup }\,P_{2}$, there exists at least one formula in ${\mathcal{L}\mathcal{P}}\left( {\mathcal{E}} \right)$ in $P_{1}$, and at least one formula in ${ \mathcal{L}\mathcal{P}}\left( {\mathcal{E}} \right)$ belongs to $P_{2}$.

Example 6.

Let $P_{1}$ and $P_{2}$ be two possibilistic bases defined as follows:

$$ \begin{array}{*{20}c} {P_{1} = \left\{ {\phi \, \vee \,\psi \, \vee \,\xi ,0.9,\neg \psi ,0.7,\neg \delta ,0.5} \right\}\,{\text{and}}} \\ {P_{2} = \left\{ {\left( {\neg \phi ,0.8),(\neg \xi ,0.7} \right),\left( {\xi \, \vee \,\delta ,0.5} \right),\left( {\sigma \, \vee \,\psi ,0.4} \right)} \right\}} \\ \end{array} $$

There are two conflicts in $P_{1} \,{\bigcup }\,P_{2}$:

$$ \begin{array}{*{20}c} {{\mathcal{E}}_{1} = \left\{ {\phi \vee \psi \vee \xi ,\neg \phi ,\neg \xi ,\neg \psi } \right\} , {\text{and}}} \\ {{\mathcal{E}}_{2} = \left\{ {\neg \xi ,\xi \vee \delta ,\neg \delta } \right\}} \\ \end{array} $$

We have $Deg_{{P_{1} }} \left( {{\mathcal{E}}_{1} } \right) = Deg_{{P_{2} }} \left( {{\mathcal{E}}_{2} } \right) = 0.7$ and $Deg_{{P_{1} }} \left( {{\mathcal{E}}_{2} } \right) = Deg_{{P_{2} }} \left( {{\mathcal{E}}_{2} } \right) = 0.5$. Then $P_{1}$ and $P_{2}$ are equal priority.

However, if we have $P^{\prime}_{2} = \left\{ {\left( {\neg \phi ,0.8),(\neg \xi ,0.6} \right),\left( {\xi \, \vee \,\delta ,0.4} \right),\left( {\sigma \, \vee \,\psi ,0.4} \right)} \right\}$ then $P_{1}$ is more prioritized than $P^{\prime}_{2}$.

From $Deg_{{P_{1} }} \left( {{\mathcal{E}}_{1} } \right)\, > \,Deg_{{P_{2} }} \left( {{\mathcal{E}}_{1} } \right)$ and $Deg_{{P_{1} }} \left( {{\mathcal{E}}_{2} } \right)\, > \,Deg_{{P_{2} }} \left( {{\mathcal{E}}_{2} } \right)$.

Lemma 2.

Let ${\mathcal{P}}_{ \oplus }$ be the merging result ${\mathcal{P}} = \left\{ {P_{1} ,\, \ldots .,P_{n} } \right\}$ with $\oplus$. Then ${\mathcal{P}}_{ \oplus }$ is equivalent to

$$ \{ \left( {\phi , \oplus \left( {a_{1} , \ldots ,a_{n} } \right)} \right):\phi \, \in \,{\mathcal{L}}\,{\text{and}}\,P_{i} \,{ \vdash }\,\left( {\phi ,a_{i} } \right)\} $$

Now we define the merging result. This corresponds to the possibility distributions consequences of ${\mathcal{P}}_{ \oplus }$:

Definition 20.

(Result of merging) The merging result is:

$$ {\mathcal{T}} = \left\{ {\left( {\phi_{i} ,a_{i} } \right)|{\mathcal{P}}_{ \oplus } \,{ \vdash }_{\pi } \,\left( {\phi_{i} ,a_{i} } \right)} \right\} $$

5 Postulates and Logical Properties

In this section we give the postulates and the logical properties and focus only on the set of reasonable results without putting the weights into the inference. That is, a set of logical properties that this argumentation framework needs to satisfy.

• CON (Consistent): $\oplus \left( {{\mathcal{P}}_{ \oplus } } \right)$ is consistent.

In possibilistic logic does not require every knowledge base consistently. So, we do not need to require that all knowledge base merging be consistent.

• ADI (Additional Information):

If $P_{1} \, \cup \, \ldots \, \cup \,P_{n}$ is consistent then $\oplus \left( {{\mathcal{P}}_{ \oplus } } \right) \equiv \oplus \left( {P_{1} \, \cup \, \ldots \, \cup \,P_{n} } \right)$.

The ADI says that in each round of arguments, if all the information from the agents submitted simultaneously does not cause conflicts, they will be added to the results of the merging. Such a request can be satisfied if the merging operator ensures that each agent has at least one knowledge base contributing to the merging result. This defines as follows:

Definition 21.

$\oplus$ is called a conjunctive operator if $\oplus \left( {a_{1} ,\, \ldots ,a_{n} } \right)\, > \,0$ with $\forall a_{i} \, > \,0$.

EQU (Equality):

$\oplus \left( {{\mathcal{P}}_{ \oplus } \left( {\left\{ {P_{1} ,\, \ldots ., P_{n} } \right\}} \right)} \right) = \, \oplus \left( {{\mathcal{P}}_{ \oplus } \left( {\left\{ {P_{\pi \left( 1 \right)} ,\, \ldots ., P_{\pi \left( n \right)} } \right\}} \right)} \right)$, where π be a permutation on {1, …, n}.

The EQU ensures that all agents are treated fairly in the process of argumentation.

CAU (Cautiousness): If $P_{1}$ and $P_{2}$ are inconsistent and $P_{1}$ and $P_{2}$ have the same priority, then

$$ \oplus \left( {{\mathcal{P}}_{ \oplus } } \right)\,{ \nvdash }\, \oplus \left( {P_{1} } \right)\,{\text{and}}\, \oplus \left( {{\mathcal{P}}_{ \oplus } } \right)\,{ \nvdash }\, \oplus \left( {P_{2} } \right). $$

The idea in the CAU postulate is that when we merge two conflict knowledge bases, the result of merging does not give preference to any knowledge base. This requirement is natural in propositional logic from formulas and there is no way of giving preference between them. Therefore, this cannot be true in the possibilistic logic framework.

• REA (Reaccept): If $\oplus \left( {{\mathcal{P}^{\prime}}_{ \oplus } } \right) \,{\bigcup }\, \oplus \left( {{\mathcal{P}^{\prime\prime}}_{ \oplus } } \right)$ is consistent then $\oplus \left( {{\mathcal{P}}_{ \oplus } } \right)\,{ \vdash }\, \oplus \left( {{\mathcal{P}^{\prime}}_{ \oplus } } \right)\,{\bigcup }\, \oplus \left( {{\mathcal{P}}_{ \oplus }^{''} } \right)$

If the consequence of the two subgroups is consistent, the result of knowledge integration is the result of this combination.

• P_Maj (Majority): $\forall P^{\prime},\exists n, \oplus \left( {\left( {{\mathcal{P}}\,{ \sqcup }\,P^{'n} } \right)_{ \oplus } } \right)\,{ \vdash }\, \oplus \left( {P^{\prime}} \right)$

Intuitively, majority is related to the idea of reinforcement, that is if the formulas have the same weight $a$ from two different agents, they will gain a larger weight in the integration result.

• P_Arb(Arbitration): $\forall P^{\prime},\forall n, \oplus \left( {\left( {{\mathcal{P}}\,{ \sqcup }\,P^{'n} } \right)_{ \oplus } } \right)\, \equiv \, \oplus \left( {\left( {{\mathcal{P}}\,{ \sqcup }\,P^{\prime}} \right)_{ \oplus } } \right)$

This postulate states that, if we combine n knowledge bases with the same weight, then the integrated result holds only one. That is to say that the postulate arbitration ignores redundancies.

From the propositions satisfying the merging operator, in comparison with the proposed knowledge merging framework by argumentation, we propose the following theorem.

Theorem 1.

The Argumentation knowledge base merging solution satisfies CON, ADI, EQU, REA and P_Arb attributes, and does not satisfy CAU and P_Maj.

6 Conclusion

In this paper, a framework for merging possibilistic knowledge bases by argumentation is introduced and discussed. The key idea in this work that we presented an argumentation-based framework for resolving conflicts between knowledge bases in a prioritized case of possibilistic logic framework. The proposed approach is different from the classical way used in the literature to deal with conflicting multiple sources information. The main result of the work presented in this paper is that the argumentation framework base on merging operator defined and is an interesting problem to merge the bases in multi-agent systems. In such a system, each agent has its own base which may conflict with the bases of the other agents. A set of postulates is introduced and logical properties are mentioned and discussed. They assure that the proposed model is sound and complete. However, our approach is still affected by drowning effect. In the future, we will propose a merging approach based on multiple operators, combining consistent and conflict information using different operators and evaluation of computational complexities of knowledge base merging operators.