On the belief merging by negotiation (2)

Available online at www.sciencedirect.com ScienceDirect Procedia Computer Science 35 (2014) 147 – 155 18th International Conference on Knowledge-Based and Intelligent Information & Engineering Systems - KES2014 On the belief merging by negotiation Trong Hieu Trana,∗, Quoc Bao Vob , Thi Hong Khanh Nguyenc a VNU University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam University of Technology, PO Box 218 Hawthorn VIC 3122 Victoria, Australia c Electric Power University, 235 Hoang Quoc Viet, Tu Liem, Hanoi, Vietnam b Swinburne Abstract Belief merging is an active research field with many important applications Most existing work addresses the belief merging issue using a centralised approach In this paper, we investigate a distributed approach to the problem of belief merging The contribution of this paper is two-fold: (i) we develop a negotiation-based model for belief merging, and (ii) we investigate the computational complexity of the belief merging problem within the proposed framework Through the proposed model of negotiation-based belief merging, we will present and discuss several significant logical properties and computational complexity results c 2014 The © The Authors Authors Published Publishedby byElsevier ElsevierB.V B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/) Peer-review under responsibility of KES International Peer-review under responsibility of KES International Keywords: Computational complexity; Belief merging; Knowledge integration; Negotiation Introduction Belief merging has emerged as an active research topic with important applications in many fields of Computer Science The main goal of belief merging problem is to obtain the common beliefs from several belief bases It is applied in database integration 1,2 , information retrieval 3,4,5,6 , sensor data fusion , coordination in multi-agent 8,9,10 , and multimedia systems 11,12 Several approaches have been proposed for addressing the belief merging problem In general, they can be classified as either centralized approaches or distributed ones The centralized belief merging approaches constitute the major direction in the belief merging literature in which the merging process is considered an arbitration The typical approaches in this group include belief merging with arbitration operators proposed by Revesz 13 , belief merging with weighted belief bases by Lin 14 , belief merging with integrity constraints by Konieczny and P´erez 15 , belief merging in a possibilistic logic framework proposed by Benferhat et al 16 , and belief merging with stratified bases by Qi et al 17 The solutions induced in these approaches satisfy a number of rational axioms for belief merging However, these approaches are not without some shortcomings In particular, they require a mediator without taking into account the roles of the agents who provide the source of the belief bases to be merged and they assume that all belief bases are ∗ Corresponding author Tel.: +84-945-89-3663 ; fax: +84-4-3858-8817 E-mail address: hieutt@vnu.edu.vn 1877-0509 © 2014 The Authors Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/) Peer-review under responsibility of KES International doi:10.1016/j.procs.2014.08.094 148 Trong Hieu Tran et al / Procedia Computer Science 35 (2014) 147 – 155 completely provided up front These requirements are generally inapplicable in many multi-agent systems The distributed belief merging approaches aim to overcome the above mentioned shortcoming In these approaches, belief merging is considered as a game in which agents who possess the source belief bases are self-interested and may act strategically The agents merge their belief bases by following a pre-defined protocol in order to reach the consensuses among themselves 18,19,20,21,22,23,24 An approach inspired by two-stage belief revision operators are proposed by Booth 18,19 It has been subsequently enriched by identifying a family of merging operators by Konieczny (see 21 ) Another important approach is proposed by Zhang 20 in which a negotiation model is built for the set of agents’ demands represented by logical formulas The negotiation is carried out by first aligning all the belief bases in their lowest priority layers and then iteratively removing the lowest layers of belief bases until the remaining layers are jointly consistent or a disagreement situation arises However, this approach suffers the drowning effect and is syntax sensitive In 22,24 , we propose a solution for belief merging by negotiation, which can overcome the drowning effect, but it is still syntax sensitive Moreover, our more recent work 23 can overcome both issues While the literature of centralised belief merging has been quite comprehensive addressing both the rationality of the merging operators and their computational complexity, there is a noticeable lack of results regarding the computational complexity of the distributed merging problems It is another aim of this work to address this issue In this paper, we propose a model for negotiation-based belief merging in which the merging process is organised into two stages In the first stage, the preferences over the set of all possible worlds are constructed from the stratified belief bases, based on several ordering strategies The second stage is a negotiation process which works on the preferences constructed in the first stage Then, several logical properties and complexity results for the proposed belief merging operator will be presented and discussed The rest of this paper is organized as follows Section provides some formal preliminaries A model for belief merging by negotiation is introduced in Section 3, in which we present a model for belief merging and a set of axioms to characterize the negotiation solutions Several computational complexity results are presented and discussed in Section Finally, Section concludes the paper with a discussion on the future work Formal preliminaries 2.1 Stratified knowledge base In this paper, we consider a propositional language L over a finite alphabet P and the constants { , ⊥} Symbol W is used to denote the set of possible worlds, where each possible world is a function from P to { , ⊥} A model of a formula φ is a possible world ω which makes φ true, written as ω φ With Φ being a set of formulas, Mod(Φ) denotes the set of models of Φ, i.e Mod(Φ) = {ω ∈ W |∀φ ∈ Φ(ω φ)} By abuse of notation, we use Mod(φ) instead of Mod({φ}) for some φ ∈ L We also use the symbol to denote the consequence relation, for example {φ, ψ} θ means θ is a logical consequence of {φ, ψ} Let be a binary relation on a non-empty set X ⊆ L The relation is a total pre-order on X if it satisfies the following properties: - ∀α ∈ X, α α;(Reflexivity) - ∀α, β, γ ∈ X, if α β and β γ then α γ;(Transitivity) - ∀α, β ∈ X, α β or β α (Totality) A stratified belief base, sometimes also called ranked knowledge base or prioritized knowledge base, is a belief base K together with a total pre-order on K Stratified belief base (K, ) can be equivalently defined as a sequence (K, ) = (S , , S n ), where each S i (i = 1, , n) is a non-empty set and for φ ∈ S i , ψ ∈ S j , φ ψ iff i ≤ j, i.e when i ≤ j each formula in S i is more reliable than the formulas of the stratum S j Each subset S i is called a stratum of K, and index i indicates the level of each formula of S i It is clear that each formula in S i is more reliable than any formula of the stratum S j for j ≥ i Given stratified belief bases (K, ) = (S , , S n ) and (K , ) = (S , , S m ), we say that these belief bases are equivalent, denoted by (K, ) ≡ (K , ), if m = n and S i ≡ S i for i = 1, , n Further, a belief set E = {(K1 , ), , (Kn , n )} is logically equivalent to a belief set E = {(K1 , ), , (Kn , n )}, denoted E ≡ E , if and only if there exists a permutation π on the set {1, , n} such that (Ki , i ) ≡ (Kπ(i) , π(i) ) for all i = 1, , n Additionally, given any set S , we use S to denote the cardinality of S Trong Hieu Tran et al / Procedia Computer Science 35 (2014) 147 – 155 2.2 Computational complexity In this section we briefly recall some computational complexity classes, including the complexity class Θ2p (see e.g., 25 ) We assume the reader familiarity with the classes P, NP and coNP Some important classes in the polynomial hierarchy are defined in the following The complexity classes Δkp , Σkp , and Πkp (k is a positive integer) are defined recursively as follows: - Δ0p = Σ0p = Π0p = P; p p = PΣk ; - Δk+1 p p = NPΣk ; - Σk+1 p p = coNPΣk - Πk+1 In addition, the class Θ2p = Δ2p [O(log n)] is the class of problems Δ2P = PNP which can be determined via a logarithmic number of calls to an NP oracle We can now define the hardness and the completeness of different complexity classes as follows: Definition Let C be a complexity class, and Q be a decision problem We say that Q is C-hard iff for any decision problem P belonging to the class C, there is a reduction from P to Q For instance, a decision problem can be shown to be NP-hard by identifying a reduction of the SAT (Boolean satisfiability) problem to this problem In general, in order to prove that a problem P is C-hard, we just need to prove that there exists a reduction of a known C-hard problem to P Definition Let C be a complexity class, and P be a decision problem We say that P is C-complete iff P ∈ C and P is C-hard C-complete problems are the most “difficult” problems in the complexity class C A model of negotiation for belief merging This section presents an axiomatic model for merging stratified belief bases by negotiation We introduce the concept of mapping solution, which maps the preferences of agents into the layers, as a vehicle to represent the belief states of agents and their attitudes towards negotiation situations The belief merging process in our model is divided into two stages In the first stage, the stratified belief bases of agents are mapped to their preferences In the second stage, a negotiation between the agents is carried out based on these preferences To this end, a set of rational axioms for negotiation-based belief merging is proposed and a negotiation solution which satisfies the proposed axioms is introduced Finally, the logical properties of a family of merging-by-negotiation operators are discussed We start the work in this section by considering a set of agents A = {a1 , , an }, where each agent has a stratified belief base (Xi , i ) in which Xi ⊆ LV , and relation i ⊆ Xi × Xi is a total pre-order A negotiation game is a sequence of stratified belief bases together with the integrity constraints presented logically equivalent to a formula The set of all negotiation games from the set of agents A in language LV is denoted by gA,LV The negotiation solution is defined as follows: Definition A negotiation solution is a function f : gA,LV → 2W \{∅} which maps each negotiation game to a non-empty subset of all possible worlds Note that we consider the negotiation solution of any negotiation game as a set of possible worlds instead of a single one 149 150 Trong Hieu Tran et al / Procedia Computer Science 35 (2014) 147 – 155 3.1 From stratified belief base to preferences In this section, we consider several ordering strategies from a given stratified belief base (K, ) = (S , , S n ) where {S , , S n } is a partition of K w.r.t the total preorder such that ∀φ ∈ S i , ∀ψ ∈ S j , φ ψ iff i < j as follows: +∞ i f ∀S i (ω |= S i ), −maxsat ordering 26 : let r MO (ω) = where ω ∈ W Then the maxsat ordering min{i : ω |= S i } otherwise maxsat on W is defined as: ω maxsat ω iff r MO (ω) ≤ r MO (ω ) −leximin ordering 27 : let K i (ω) = {φ ∈ S i : ω |= φ} Then the leximin ordering leximin on W is defined as: ω leximin ω iff K i (ω) = K i (ω ) for all i = 1, , n or there exists j ≤ n such that K j (ω ) < K j (ω) and K i (ω) = K i (ω ) for all i < j i f ω |= S i , −vector ordering : let vi (ω) = otherwise Then the vector ordering vector on W is defined as: ω vector ω iff vi (ω) = vi (ω ) for all i = 1, , n or there exists j ≤ n such that v j (ω ) < v j (ω) and vi (ω) = vi (ω ) for all i < j ω Given a preorder on W, the associated strict partial order ≺ is defined by ω ≺ ω iff ω ω but not ω An ordering Y is more specific than another X iff ω ≺X ω implies ω ≺Y ω We have the relation among the above ordering strategies as follows: Proposition Let ω, ω ∈ W, K be a stratified belief base The following relationships hold: 1) ω ≺maxsat ω implies ω ≺vector ω , 2) ω ≺maxsat ω implies ω ≺leximin ω 3.2 Negotiation on the preferences Clearly, given a stratified belief base and an ordering strategy, one can easily partition W into the classes of possible worlds (W1 , , Wk ) where Wi ∅, i = 1, , n Therefore, for each possible world we can determine the unique class which contains this possible world We define the index function as follows: Definition Given a total preorder N+ , where for any ω, ω ∈ X: on W and X ⊆ W, the index function l of over X is defined as: lX : W → 1) lX (ω) = if ω ∈ min(X, ), 2) lX (ω) = lX (ω ) iff ω ω and ω 3) lX (ω) ≤ lX (ω ) iff ω ω, ω, 4) If ω ≺ ω then there exists ω ∈ X such that lX (ω ) = lX (ω) + and if ω ≺ ω then there exists ω ∈ X such that lX (ω ) = l (ω) − 5) ∀ω ∈ W, ω X, lX (ω ) = max{lX (ω )|ω ∈ X} + We use the index function lX (ω) to indicate the index of class that ω belongs to w.r.t the constraint X and relation , i.e lX (ω) = i indicates ω ∈ Xi Note that the indexes are consecutive integers up from 1, and the lower the index a possible world has, the more preferred it is, i.e formally, given ω, ω ∈ W, lX (ω) ≤ lX (ω ) iff ω ω Here, we define the solution mapping of a negotiation problem built from the set of preferences { , , n } achieved by the stratified belief bases and the ordering strategies, and a set C of models of the integrity constraint μ, i.e C = Mod(μ) and C is called the feasible set of the negotiation problem, as follows: Definition Given a negotiation problem G = (C, , , n ) where C ⊆ W and , , n are the preferences of agents a1 , , an respectively, a solution mapping of G is a function defined as: mG : W → Nn where mG (ω) = (lC (ω), , lC n (ω)) for any ω ∈ W Because the index of each possible world in a preference is unique, we have the following proposition: Trong Hieu Tran et al / Procedia Computer Science 35 (2014) 147 – 155 Proposition For each negotiation problem G, the solution mapping mG is unique Now, we present a set of axioms to characterize the negotiation solutions Firstly, the Pareto Efficiency axiom can be formulated in our model as follows: f (G) PE If G = (C, , , n ) is a negotiation problem with ω ∈ C, ω ∈ W and mG (ω) < mG (ω ) then ω Note that the Pareto efficiency we mention here is the Strong Pareto Efficiency It states that a solution is Pareto efficient if no one can improve its utility without causing another utility to be worse off Next, the Independence of Irrelevant Alternatives axiom can be formulated in our model as follows: IIA If G1 = (C1 , , , n ) and G2 = (C2 , , , n ) are negotiation problems with C2 ⊆ C1 and f (G1 ) ⊆ C2 then f (G1 ) = f (G2 ) The Symmetry axiom can be formulated as follows: SYM If G = (C, , , n ) and Gπ = (C, π(1) , , π(n) ) are negotiation problems with π being any permutation on {1, n} then mG (ω) = (mGπ (ω))π Obviously, the Invariant to equivalent utility representations axiom is applied to Affine spaces, while in this paper, we are working on ordinal spaces, thus it is omitted The Upper bound axiom can be formulated as follows: UB Given a negotiation problem G = (C, , , n ) and two possible outcomes ω1 , ω2 ∈ C If max mG (ω1 ) < max mG (ω2 ) then ω2 f (G) We say that ω1 , ω2 ∈ W is upper bound equal iff max mG (ω1 ) = max mG (ω2 ) The Upper bound axiom ensures that the negotiation process will be terminated immediately when an agreement is reached The Majority axiom can be formulated as follows: MA Given a negotiation problem G = (C, , , n ) and outcomes ω1 , ω2 ∈ C that are upper bound equal, if {i : ω1 i ω2 } < {i : ω2 i ω1 } then ω1 f (G) We also say that ω1 , ω2 ∈ W are majority equal iff ω1 , ω2 are upper bound equal and {i : ω1 i ω2 } = {i : ω2 i ω1 } The Majority axiom states that if two feasible worlds ω and ω are upper bound equal, whichever one is voted by the larger number of participants, is preferred to be the solution Although the majority property is studied in a wide range of works in Social Choice as well as Decision-making, it is usually criticized by being affected by the voting paradox However, it is not a serious problem in our work because if the paradox happens, we can take all the feasible worlds as the outcomes Lastly, the Lower bound axiom can be formulated as follows: LB Given a negotiation problem G = (C, , , n ) and two possible outcomes ω1 , ω2 ∈ C If ω1 and ω2 are majority equal and mG (ω1 ) < mG (ω2 ) then ω1 f (G) The Lower bound axiom ensures the solution is fair in the sense that the difference between the best and the worst is minimal Given a set of possible outcomes S , we use max(S , #) to denote the subset of possible outcomes of S which is most supported by agents w.r.t cardinality Formally, we have: max(S , #) = {ω ∈ S : ω ∈ S ( {i : ω i ω} < {i : ω i ω } )} We also denote G as the set of all negotiation problems Now, we show the possibility of the set of the above axioms by pointing out a solution based on the idea of the well-known egalitarian solution as follows: Theorem Let f G : G → 2W \{∅} be a negotiation solution, where - f G ((C, , , n )) = arg maxω∈LS min(mG (ω)), where - LS = max(BS , #), where - BS = arg minω∈C (max(mG (ω))) A negotiation solution f : G → 2W \{∅} satisfies U B, MA and LB iff f = f G We also see the relation between the negotiation solution f G and the axioms IIA, PE, S Y M as follows: Proposition The negotiation solution f G satisfies IIA, PE, and SYM 151 152 Trong Hieu Tran et al / Procedia Computer Science 35 (2014) 147 – 155 3.3 Logical properties Given a negotiation game G = ({(Ki , i )|ai ∈ A}, μ) ∈ gA,LV , iXi is the preference of agent on W according to an ordering strategy Xi ∈ { maxsat , vector , leximin }, and X = {X1 , , Xn } Let ΔμX (G) be a belief merging operator such that Mod(ΔμX (G)) = f G ((Mod(μ), 1X1 , , nXn )) We call such operators the Negotiation-based Merging operators We need to modify some postulates for belief merging with integrity constraints ( 28 ) to accommodate the merging on the stratified knowledge bases In particular, postulates (IC2) and (IC3) should be modified as follows: (IC2’) Let ∧G = ∧ai ∈A ∧φ∈Ki φ, if ∧G ∧ μ is consistent then ΔμX (G) ≡ ∧G ∧ μ (IC3’) Given two negotiation games G = ({(Ki , i )|ai ∈ A}, μ) and G = ({(Ki , i )|ai ∈ A}, μ ), (G, G ∈ gA,LV ), if μ ≡ μ and there exists a permutation π on {1, , n} such that (Ki , i ) ≡ (Kπ(i) , π(i) ) and Xi = Xπ(i) for all i ∈ {1, , n} then ΔμX (G) ≡ ΔμX (G ) Proposition If ΔμX (G) is a Negotiation-based Merging operator, then ΔμX (G) satisfies (IC0), (IC1), (IC2’), (IC5), (IC7), (IC8) If Xi ∈ { maxsat , vector } for all i then ΔμX (G) also satisfies (IC3’) We also have the relation between negotiation solutions according to the ordering strategies as follows: Proposition Given a negotiation game G = ({(Ki , Xi is more specific than Xi for all i = 1, , n, then f G ((Mod(μ), X1 , , Xn n )) ⊆ f G ((Mod(μ), X1 , , i )|ai ∈ A}, μ) ∈ gA,LV , if Xi , Xi ∈ { maxsat , vector , leximin } and Xn n )) Computational complexity of belief merging by negotiation In this sub-section, we discuss the computational complexity of the family of belief merging operators by negotiation Firstly, we consider the computational complexity of stratifying W from a stratified belief base and an ordering strategy According to 29 , the problems for logical preference representation languages need to be taken into account as follows: Definition ( 29 ) Given a stratified belief base (K, ), an ordering strategy X, a formula φ and two interpretations ω and ω • The COMPARISON problem determines whether ω X ω • The NON-DOMINANCE problem determines whether ω is non-dominated by ω ≺X ω X, i.e there is not ω such that We have a proposition for the ordering strategies presented in Subsection 3.1 as follows: Proposition Let (K, ) be a stratified belief base and X be an ordering strategy For X ∈ {maxsat, leximin, vector} we have: - COMPARISON is in P; - NON-DOMINANCE is coNP − complete In order to stratify W, we need to take into account the problem determining all non-dominated interpretations This problem is computationally much harder than the NON-DOMINANCE problem To simplify the computation of our merging operators, we assume that W is stratified from each stratified knowledge base during an off-line preprocessing stage Let f be a negotiation solution, and we define the decision problem MER NEGO( f ) as follows: Input: a tuple E, μ, φ, X where E = {(K1 , ), , (Kn , n )} is a belief set of stratified belief bases, μ and φ are formulas, X = {X1 , , Xn } is set of ordering strategies (Xi is attached to Ki , respectively) Trong Hieu Tran et al / Procedia Computer Science 35 (2014) 147 – 155 Question: Does f (E, μ, X) |= φ hold? Theorem Let X = {X1 , , Xn }, where Xi ∈ {maxsat, leximin, vector} (i=1, ,n) We have MER NEGO( f ) is Θ2p − complete Proof: • Membership of MER NEGO( f ) in Θ2p is demonstrated by the following algorithm: Determine the smallest k such that for all ω |= μ, l i (ω) ≤ k by using binary search, resulting in O(log n) calls to a S AT − oracle Determine the largest m such that for all ω |= μ, max(mod(ω), #) ≤ m, resulting in O(log n) calls to a S AT − oracle Determine the largest n such that for all ω |= μ, l i (ω) ≥ n by using binary search, resulting in O(log n) calls to a S AT − oracle Determine whether an interpretation obtained by applying in sequence the above steps is a model of φ by one call to a S AT − oracle • Hardness is proved by using the Θ2p − complete problem UOCS AT 30 stated as follows: Given a set of clauses Φ, decide if all truth-assignments that satisfy a maximum number of clauses in Φ always satisfy the same set of clauses in Φ We will prove that UOCS AT can be reducible to MER NEGO in polynomial time as follows Suppose we have an instance of UOCS AT Φ = {φ1 , , φm } over the variables p1 , , pn Let c1 , , cm , cm+1 , cm+2 be new variables that have not occurred in Φ We define: - K = {{c1 }, , {cm }, {cm+1 }, {¬cm+2 }}, - μ = (φ1 ∨ ¬c1 ) ∧ ∧ (φm ∨ ¬cm ) ∧ cm+1 ∧ cm+2 It is easy to see that by any ordering strategy in {maxsat, leximin, vector} each belief base Ki of K classifies W into two layers, one being the set of models of Ki and the other the set of the rest Moreover, for all ω |= μ, we also have ω |= cm+1 and ω |= ¬cm+2 Hence, for all ω ∈ W we always have max(l i (ω)) = = l m+2 (ω) and min(l i (ω)) = = l m+1 (ω) Therefore, for all ω |= μ, ω ∈ f ((Mod(μ), , , m+2 )) if and only if ω ∈ max(Mod(μ), #) if and only if ω satisfies a subset with maximal number of elements of Φ Similarly, we consider c1 , , cm be new variables that have not occurred in Φ and - K = {{c1 }, , {cm }, {cm+1 }, {¬cm+2 }}, - μ = (φ1 ∨ ¬c1 ) ∧ ∧ (φm ∨ ¬cm ) ∧ cm+1 ∧ cm+2 We also have: for all ω |= μ , ω ∈ f ((Mod(μ ), , , m , m+1 , m+2 )) if and only if ω ∈ max(Mod(μ ), #) if and only if ω satisfies a subset with maximal number of elements of Φ Now we define: - K∗ = K K, ∗ - μ =μ∧μ , - φ = (c1 ≡ c1 ) ∧ ∧ (cm ≡ cm ) Clearly, f ((Mod(μ∗ ), ∗1 , ∗m+2 )) “merges” each model on f ((Mod(μ), , m+2 ) on Var(K ) Var(μ ) with a model on f ((Mod(μ ), , m , m+1 , m+2 )) on Var(K ) Var(μ ) Therefore f (K ∗ , μ∗ , X) |= φ if and only if all truth assignments that satisfy a maximum number of clauses in Φ always satisfy the same set of clauses in Φ Thus, MEG NEGO is θ2p − hard Therefore, MEG NEGO has θ2p membership and MEG NEGO is θ2p − hard It concludes that MEG NEGO is θ2p − complete 153 154 Trong Hieu Tran et al / Procedia Computer Science 35 (2014) 147 – 155 Conclusion This paper proposed a two-stage model-based approach for belief merging by negotiation The first stage lets each agent build its own preference on the set of possible outcomes from its stratified belief base and an ordering strategy The second stage allows the agents to negotiate with each other based on the constructed preferences to reach agreement as the result of merging A set of rational axioms for merging by negotiation is proposed and analyzed and a negotiation solution that satisfies these axioms is identified Especially, several significant computational complexity results are also presented, evaluated, and discussed Although this paper has addressed the problem of belief merging by negotiation in both axiomatic and constructive models, it is necessary to consider the strategic model in the negotiation stage Moreover, the complexity of the merging process in the presence of strategic behaviours is also an interesting topic to be investigated We will explore these open issues in future works Acknowledgements This study was fully supported by Science and Technology Development Fund (B) from Vietnam National University, Hanoi under grant number QG.14.13 (2014-2015) References de Amo, S., Carnielli, W.A., Marcos, J A logical framework for integrating inconsistent information in multiple databases In: FoIKS 2002; 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