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3 Soundness/Completeness of Multiple-Agent Possibilistic Logic

3 Soundness/Completeness of Multiple-Agent Possibilistic Logic

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Reasoning with Multiple-Agent Possibilistic Logic



A Refutation Method by Linear Multiple Agent


In possibilistic logic, the linear resolution strategy for the procedure of refutation

by resolution, defined in [7], works in the same way as in classical logic, and

thanks to an A∗ -like search method (changing the sum of the costs into their

minimum), one can obtain the refutation having the strongest weight first, this

weight being the one of the formula we want to prove. Here, the (fuzzy) subsets

of agents play the role of weights, but they are not totally ordered, while the

weights in possibilistic logic are; this makes the problem more tricky (since the

costs in the A∗ -like algorithm will be computed from these weights). However,

the procedure can be adapted to multiple-agent logic.


Refutation by Linear Multiple Agent Resolution

Let Γ be a knowledge base composed of multiple agent formulas. Proving (a, A)

from Γ comes down to adding (¬a, All), in clausal form, to Γ and applying

the resolution rule repeatedly until producing (⊥, A). Clearly, it comes down to

getting the empty clause with the greatest subset of agents set(a, Γ ). Formally:

set(a, Γ ) = ∪{A|Γ |= (a, A)}

Refutation by resolution using a linear strategy can be expressed in terms of

tree search in a state space. A state (C0 C1 , ..., Ci ) is defined by a central clause

Ci and the sequence (C0 C1 , ..., Ci−1 ) of central clauses ancestors of Ci . For each

state of the search tree, a subset of agents is associated, playing the role of a

cost. It corresponds to the subset of agents of the latest generated central clause

s.t. set(Ci ) (short for set(Ci , Γ )) is associated with state (C0 C1 , ..., Ci ). The goal

is to find the states ending with the empty clause with the greatest subsets of

agents. An analogy with the search in the state space with costs is established

in the following way:

– The initial state S0 is defined by the initial central clause C0 with a cost equal

to set(C0 ),

– The cost associated with the arc (C0 C1 , ..., Ci ) → (C0 C1 , ..., Ci Ci+1 ) is the set

associated with Ci+1 ,

– The global cost of the path C0 → C1 → ... → Ci is the intersection of (setvalued) costs of the elementary arcs,

– The objective states are states (C0 C1 , ..., Ci ) such that Ci = (⊥, Ai ) with

Ai = ∅,

– The state (C0 C1 , ..., Cn ) is expanded by generating all resolvents of Cn authorized by the linear strategy.

Searching for a refutation with the greatest subsets of agents is then equivalent

to searching for a path with maximal cost from the initial state to the objective

states. However, many differences exist:


A. Belhadi et al.

– costs here are to be maximized not to be minimized. Indeed, the goal is to

find the greatest subset of agents who believe a formula.

– costs are not additive but they are combined using the intersection operator.

– since only partial order can be defined between subsets, several objective states

exist. The latter are then combined by the union operator.

– if an order exists between subsets, the greatest subset is considered and the

other path is never explored, unlike search in space states.

As for heuristic search in space states, the ordered search is guided by an evaluation function f calculated as follows: for each state S of the search tree,

f (S) = g(S) ∩ h(S) where g(S) is the path cost from the initial state to S, and

h(S) a cost estimation from S to an objective state.

The different steps of the refutation by resolution using a linear strategy,

presented by Algorithm 1, can be summarized in the following way:

Let R(Γ ) be the set of clauses that has been produced (using resolution)

from Γ . For each refutation using the clause C, for each literal l of C and in

order to obtain ⊥, the use of a clause C containing the literal ¬l is required. A

refutation expanded from C will have a cost less than or equal to:

H(l) =

{set(C ), C ∈ R(Γ ), ¬l ∈ C }

The cost of the path until the contradiction developed from the clause C is


h1 (C) =

{set(C ), C ∈ R(Γ ), ¬l ∈ C }

{H(l), l ∈ C} =


with S = (C0 , ..., C). An admissible evaluation function is obtained f1 (S) =

set(C) ∩ h1 (S). h1 (S) depends only on C. A sequence of evaluation functions

can be defined as follows:

h0 (C) = All;

fp (C) = set(C) ∩ hp (C); p ≥ 0

{fp (C ), C ∈ R(Γ ), ¬l ∈ C }; p ≥ 0

hp+1 (C) =


Example 1. Let Γ be a multiple-agent clausal knowledge base:

C1 : (¬a ∨ b, All); C2 : (a ∨ d, All);

C3 : (a ∨ ¬c, A); C4 : (¬d, A);

C5 : (¬d, B).

Let us to consider the search of the greatest subset of agents who believe b.

Let then Γ be the set of clauses equivalent to Γ = Γ ∪{(¬b, All)}. C0 = (¬b, All)

as Γ − {C0 } is coherent. The only clause which contains the literal b is C1 (see

Fig. 1). The next state is then S1 = (C0 C6 ) with C6 : (¬a, All) and cost equal

to set(C0 ) ∩ set(C1 ) = set(C6 ) = All. Different paths with C2 and C3 exist from

this state. The evaluation function then will be calculated. The greatest set that

maximizes the evaluation function is All, because A ⊂ All. Effectively, taking

Reasoning with Multiple-Agent Possibilistic Logic


Fig. 1. Refutation tree of Example 1

into account this inclusion order, the path with the clause C3 is not explored. The

next state is then S2 = (C0 C6 C7 ) and has a cost set(C6 ) ∩ set(C2 ) = set(C7 ) =

All, with C7 : (d, All).

Several paths exist from this state. Those paths will be all explored because

they have incomparable evaluation functions, due to the partial order of subsets.

Let S3 = (C0 C6 C7 C8 ) be the next state. Its associated cost is set(C7 )∩set(C4 ) =

set(C8 ) = A. The clause C8 is a contradiction. So, the first objective state is


When dealing with the clause C5 , the next state is then S4 = (C0 C6 C7 C9 )

having the cost set(C7 ) ∩ set(C5 ) = set(C9 ) = B. The clause C9 is a contradiction. The last objective state is then reached. Thus Γ |= (b, A ∪ B).


Refutation by Linear Possibilistic Multiple Agent Resolution

In multiple-agent possibilistic logic, the gradual subset weakening states that if

β/B ⊆ α/A then (c, α/A) (c, β/B). The inclusion F ⊆ G between two fuzzy

subsets F and G of a referential U is classically defined by ∀u ∈ U, F (u) ≤ G(u).

In particular, if U = All, then α/A ⊇ β/B if and only if A ⊇ B and α ≥ β.

The goal is then to find a given formula with the greatest subset of agents

with the greatest certainty degree. Obviously, the union of two partial results

(⊥, α/A) and (⊥, β/B) should be taken if α > β and A ⊂ B. These observations

are used to directly extend the procedure of the previous section.

Example 2. Let Σ be a multiple-agent possibilistic knowledge base composed by

the following clauses:

C1 : (¬a ∨ b, 0.8/All)

C2 : (a ∨ d, 0.7/All)

C3 : (a ∨ ¬c, 0.9/A)


A. Belhadi et al.

Algorithm 1. Multiple agent refutation by resolution using linear strategy


Open ← {S0 }; Closed ← {S0 }; bset = ∅

while Open = ∅ do

Select a state Sn in Open maximizing f

if Sn is an objective state then

bset = bset ∪ Sn


Explore the node Sn by creating the set En of produced states.

if In the set En there are subsets included in other then

remove them from En

end if

En ← En \ Closed

Open ← (Open − {Sn }) ∪ En

Closed ← Closed ∪ {Sn }

calculate f for each new state of Open

end if

end while

if Open = ∅ then



display bset

end if


C4 : (¬d, 0.4/A)

C5 : (¬d, 0.3/B)

Note that the propositional knowledge base Σ ◦ coincides with Γ ◦ in the

example of Sect. 3. The problem is to find the greatest subset of agents who

believe b with the greatest certainty degree.

Let then Σ be the set of clauses equivalent to Σ = Σ ∪ {(¬b, 1/All)}. As

depicted in Fig. 2, let us take C0 = (¬b, 1/All) because Σ − {C0 } is coherent. As

the classical projection of Σ is the same as Γ , the next state is then S1 = (C0 C6 )

and the associated cost is f set(C0 ) ∩ f set(C1 ) = f set(C6 ) = 0.8/All. Different

paths starting with C2 and C3 exist from this state. However, unlike in the

previous example, both paths will be explored because the fuzzy set 0.9/A is not

included in the fuzzy set 0.7/All. Using C2 , let S2 = (C0 C6 C7 ) be the next state

with cost f set(C6 ) ∩ f set(C2 ) = f set(C7 ) = 0.7/All.

Several paths exist from this state using C4 or C5 . Let S3 = (C0 C6 C7 C8 ) be

the next state using C4 . Its associated cost is f set(C7 ) ∩ f set(C4 ) = f set(C8 ) =

0.4/A. The clause C8 is a contradiction. The first objective state is then reached.

With the path using the clause C5 , the next state is then S4 = (C0 C6 C7 C9 )

with the cost f set(C7 ) ∩ f set(C5 ) = f set(C9 ) = 0.3/B. The clause C9 is a

contradiction. An objective state is then reached.

The development of the path with the clause C3 induces the next state S5 =

(C0 C6 C10 ) with the cost f set(C6 ) ∩ f set(C3 ) = f set(C10 ) = 0.8/A. The clause

Reasoning with Multiple-Agent Possibilistic Logic


Fig. 2. Refutation tree of Example 2

C10 is not a contradiction and there is no clause containing a literal c so no

objective state is reached here. Thus Σ |= (b, 0.4/A ∪ 0.3/B).


Experimental Study

In order to analyse the behaviour of the proposed approach, the proposed algorithms were implemented with Java and intensive experiments have been performed. For this purpose, several consistent knowledge bases, including multipleagent knowledge bases and possibilistic multiple-agent knowledge bases, have

been generated by varying the number of clauses. For each case of the following

experiments, the execution time of the algorithm is evaluated in seconds. The

number of Booleanvariables is set to 30 and the number of groups of agents is

set respectively to 5, 10 and 15 by setting to 20 the number of agents.

1. Results with multiple-agent knowledge bases:

Figure 3 shows the behaviour of refutation algorithm by varying the number

of clauses from 5000 to 50000. According to the obtained results, we notice

that the execution time increase proportionally to the number of clauses.

2. Results with multiple-agent possibilistic knowledge bases:

Figure 4 shows the behaviour of refutation algorithm by varying the number

of clauses from 5000 to 50000. According to Fig. 4, we notice also that the

execution time is increased by rising the number of clauses.

3. Comparison between refutations by linear multiple agent resolution

and by linear possibilistic multiple agent resolution:

In order to compare both approaches, other experiments have been carried

out, using large bases containing 50000 clauses, 30 variables and 15 groups

of agents. By varying the number of agents from 25 to 200, Fig. 4 reveals

us that the execution time of refutation by linear possibilistic multiple agent

resolution is only slightly greater than the execution time of refutation by

linear multiple agent resolution.


A. Belhadi et al.

Fig. 3. Execution time of the refutation algorithm for large multiple agent bases.

Fig. 4. Execution time of the algorithm for large possibilistic multiple-agent bases

Discussion. The obtained results allow us to estimate the performance of the

proposed approach, which depends on the number of agent groups. Indeed, the

execution time linearly increases with the number of clauses, but it increases

exponentially with the number of variables. Whereas, when the number of group

of agents increases, the execution time increases exponentially (but it linearly

increases with the number of agents if their subsets are given in extension)1 . This

can be explained by the way of the refutation tree is constructed, which is based

on the suitable clauses. Moreover, each branch of the tree represents one suitable

clause for the literal to be deduced. The results also confirm that the execution

time of the refutation algorithm for possibilistic multiple-agent knowledge bases


It should be noticed that a base Σ = {(a1 , α1 /A1 ), ..., (an , αn /An )} can be equivalently rewritten as a collection of at most 2n possibilistic logic bases, each of them

associated with an element of the partition of All induced by the Ai ’s. However, it is

in generally computationally better to handle the initial base in a global way using

the procedure described in this paper.

Reasoning with Multiple-Agent Possibilistic Logic


Fig. 5. Comparison between multiple-agent logic and possibilistic multiple-agent logic

in terms of computational time

is slightly greater than the one obtained for multiple-agent knowledge bases.

This is due to the fact that the construction of the refutation tree with fuzzy

sets of agents consumes more time than the construction of refutation trees with

crisp groups of agents.



This paper has investigated a multiple-agent logic. From a representation point

of view, this multiple-agent logic allows us to represent beliefs of groups of agents

and its possibilistic extension handles fuzzy subsets of agents, thus integrating

certainty levels associated with agent beliefs. From a reasoning point of view, we

proposed a refutation resolution based on linear strategy for the multiple logic

and its possibilistic extension. An experimental study was conducted to evaluate

the proposed algorithms. It shows the tractability of the approach.

One may think of several extensions. On the one hand, the multiple agent

extension of the Boolean generalized possibilistic logic [5] would allow us to

consider the disjunction and the negation of formulas like (p, A), and to express

quantifiers in propositions such as “at most the agents in subset A believe p”. On

the other hand, one might also take into account trust data about information

transmitted between agents [6,12]. For instance, assume agent a trusts agent

b at level θ, which might be written (b, θ/a), assimilating a, b to propositions.

Then together with (p, α/b) (agent b is certain at level α that p is true), it would

enable us to infer (p, min(α, θ)/a) [2].


1. Belhadi, A., Dubois, D., Khellaf-Haned, F., Prade, H.: Multiple agent possibilistic

logic. J. Appl. Non-Class. Logics 23(4), 299–320 (2013)

2. Belhadi, A., Dubois, D., Khellaf-Haned, F., Prade, H.: Reasoning about the opinions of groups of agents. In: 11th Europe Workshop on Multi-Agent Systems

(EUMAS 2013), Toulouse, France, 12–13 December (2013). https://www.irit.fr/

EUMAS2013/Papers/eumas2013 submission 68.pdf


A. Belhadi et al.

3. Belhadi, A., Dubois, D., Khellaf-Haned, F., Prade, H.: Algorithme d’infrence

pour la logique possibiliste multi-agents. In: Actes Rencontres francophones sur

la logique floue et ses applications (LFA 2014), Cargese, France, 22–24 October,

pp. 259–266. C´epadu`es (2014)

4. Belhadi, A., Dubois, D., Khellaf-Haned, F., Prade, H.: Lalogique possibiliste multiagents: Une introduction. In: Actes Rencontres francophones sur la logique floue

et ses applications (LFA 2015), Poitiers, France, 5-6 November, pp. 271–278.

C´epadu`es (2015)

5. Dubois, D., Prade, H., Schockaert, S.: Stable models in generalized possibilistic

logic. In: Brewka, G., Eiter, Th., McIlraith, S.A. (eds.) Proceedings of the 13th

International Conference on Principles of Knowledge Representation and Reasoning (KR 2012), Roma, June 10–14, pp. 519–529. AAAI Press (2012)

6. Cholvy, L.: How strong can an agent believe reported information? In: Liu, W.

(ed.) ECSQARU 2011. LNCS, vol. 6717, pp. 386–397. Springer, Heidelberg (2011)

7. Dubois, D., Lang, J., Prade, H.: Theorem proving under uncertainty - a possibility

theory-based approach. In: McDermott, J.P. (ed.) Proceedings of the 10th International Joint Conference on Artificial Intelligence (IJCAI 1987), Milan, August,

pp. 984–986. Morgan Kaufmann (1987)

8. Dubois D., Lang J., Prade H.: Possibilistic logic. In: Gabbay, D.M., Hogger, C.J.,

Robinson, J.A., Nute, D. (eds.) Handbook of Logic in Artificial Intelligence and

Logic Programming, vol. 3, pp. 439–513. Oxford University Press (1994)

9. Dubois, D., Prade, H.: Possibilistic logic: a retrospective and prospective view.

Fuzzy Sets Syst. 144, 3–23 (2004)

10. Dubois D., Prade H.: Extensions multi-agents de la logique possibiliste. In: Proceedings of the Rencontres Francophones sur la Logique Floue et ses Applications

(LFA 2006), Toulouse, 19–20 October, pp. 137–144. C´epadu`es (2006)

11. Dubois, D., Prade, H.: Toward multiple-agent extensions of possibilistic logic. In:

Proceedings of the IEEE International Conference on Fuzzy Systems (FUZZ-IEEE

2007), London, 23–26 July, pp. 187–192 (2007)

12. Gutscher, A.: Reasoning with uncertain and conflicting opinions in open reputation

systems. Electron. Notes Theor. Comput. Sci. 244, 67–79 (2009)

Incremental Preference Elicitation

in Multi-attribute Domains for Choice

and Ranking with the Borda Count

Nawal Benabbou1(B) , Serena Di Sabatino Di Diodoro1,2 , Patrice Perny1 ,

and Paolo Viappiani1


Sorbonne Universit´es, UPMC Univ Paris 06 and CNRS,

LIP6, UMR 7606, Paris, France




Department of Electronics and Information (DEIB),

Politecnico di Milano, Milan, Italy

Abstract. In this paper, we propose an interactive version of the Borda

method for collective decision-making (social choice) when the alternatives are described with respect to multiple attributes and the individual

preferences are unknown. More precisely, assuming that individual preferences are representable by linear multi-attribute utility functions, we

propose an incremental elicitation method aiming to determine the Borda

winner while minimizing the communication effort with the agents. This

approach follows the recent work of Lu and Boutilier [8] relying on the

minimax regret as a criterion for dealing with uncertainty in the preferences. We show that, when preferences are expressed on a multi-attribute

domain and are additively separable over attributes, regret-based incremental elicitation methods can be made more efficient to determine or

approximate the Borda winner. Our approach relies on the representation

of incomplete preferences using convex polyhedra of possible utilities and

is based on linear programming both for minimizing regrets and selecting informative preference queries. It enables to incrementally collect

preference judgements from the agents until the Borda winner can be

identified. Moreover, we provide an incremental technique for eliciting a

collective ranking instead of a single winner.



Voting is an effective method for collective decision-making, used in political

elections, technical committees, academic institutions. Recently, interest in voting has increased in computer science, given the possibility offered by online web

systems to support voting protocols, or protocols inspired by voting, for group

decision-making (for example, for scheduling a meeting). In many real situations,

however, it may be necessary to reason with partial preferences, as some preferences are not available and too expensive to obtain (with respect to a cognitive

or economic cost). This observation has motivated a number of recent works on

c Springer International Publishing Switzerland 2016

S. Schockaert and P. Senellart (Eds.): SUM 2016, LNAI 9858, pp. 81–95, 2016.

DOI: 10.1007/978-3-319-45856-4 6


N. Benabbou et al.

social choice with partial preferences, e.g., [2–6,8,9,12]. In this research stream,

typical questions concern the determination of possible and necessary winners,

the selection of preference queries to ask to the agents for further eliciting preferences, the approximation of optimal solutions or the determination of robust

recommendations based on the available preference information.

Acquiring agents’ preferences is expensive (with respect to time and cognitive cost). It is therefore essential to provide techniques that allow to reason

with partial preference information, and that can effectively elicit the most relevant part of preferences to make a decision. Adaptive utility elicitation [1,10,11]

tackles the challenges posed by preference elicitation by representing the system knowledge about the agents’ preferences in the form of a set of admissible

utility functions. This set includes all functions compatible with the preferences

collected so far, and is updated following agents’ responses. In this way, one can

often make good (or even optimal) recommendations with sparse knowledge of

the users’ utility functions.

The aim of this paper is to introduce an adaptive utility elicitation procedure in the context of voting, for the fast determination of a Borda winner or a

social ranking based on the Borda score, and to test the practical efficiency of

this procedure. In particular, we extend the work of [8] to the multi-attribute

case. Multiple attributes may appear in well-known collective decision problems such as committee elections or voting in multi-issue domains [7]. In these

cases, attributes are boolean and represent elementary decisions on candidates

or issues. More generally, the multi-attribute case occurs when the alternatives of

a collective decision problem are described by different features, non-necessarily

boolean. Individual preferences are assumed here to be representable by a linear

function of the attribute values. Since utilities are decomposable over attributes,

a set of preference statements formulated by an agent on some pairs of alternatives will possibly allow to infer other preference statements with respect to

other pairs, without asking them explicitly. We show in the paper how this type

of inference mechanism can be implemented using mathematical programming

to reduce the number of queries and speed-up the determination of a necessary

Borda winner.

The paper is organized as follows: in Sect. 2, we introduce the basic framework for voting on multi-attribute domains. Then, we present the minimax regret

decision criterion as a useful tool for decision under uncertainty and preference elicitation. In Sect. 3, we introduce a new method based on mathematical

programming to minimize regrets based on the Borda count. Section 4 deals

with preference elicitation for the Borda count; we introduce different strategies

for generating preference queries and compare them experimentally. Finally, in

Sect. 5, we extend the approach to ranking problems based on the Borda score

and provide additional numerical tests to evaluate the efficiency of our approach

in ranking.

Incremental Preference Elicitation in Multi-attribute Domains



Social Choice in Multi-attribute Domains

with Incomplete Preferences

We consider a set of n voters or agents and a set X of m alternatives (candidates, options, items), characterized by a finite set of q attributes or criteria; an

alternative is associated to a vector x = (x1 , . . . , xq ) where each xk represents

the value of an attribute k or a performance with respect to a given point of


Individual preferences are assumed here to be represented by linear utilities


of the form ui (x) = k=1 ωki xk , where ω i = (ω1i , . . . , ωqi ) is a vector of weights

characterizing the preferences of agent i. Hence, an alternative x is as least as



good as y for agent i whenever k=1 ωki xk ≥ k=1 ωki yk . Our framework can be

used to address two different cases: a multi-criteria decision setting or a multiattribute utility where the utility is defined as the weighted sum of attribute

values. Formally, these preferences are defined by the following relation i :





ωki (xk − yk ) ≥ 0



A preference profile

1 , . . . , n of an election is therefore completely characterized by the weight vectors ω 1 , . . . , ω n (each associated with an agent). We can

now define the Borda score in our multi-attribute settings, where preferences are

defined by the utility weights. Given ω = ω 1 , . . . , ω n , the Borda score s(x, ω)

of an alternative x is


s(x, ω) =

si (x, ω i )


where si (x, ω i ) = |{y ∈ X | x i y}| counts the number of alternatives that are

strictly beaten by x according to the preference relation induced from ω i , where

i is the asymmetric part of

i: x

i y iff

i and ¬(y

i x). Our definition

allows for ties in each ranking. When using only linear orders (i.e. the ω i s are

such that there are no ties) we get the usual Borda count.

When the weights of the agents are not known to the system with certainty,

we need to reason about partially specified preferences. This is done by assuming

a vector Ω = Ω 1 , . . . , Ω n where each Ω i is the set of feasible ω i that are

consistent with the available preference information on agent i. Later, we will

use Ω (that represents our uncertainty about the weights associated with the

agents) in order to provide a recommendation based on minimax regret. At the

level of a single agent i, we can check whether pairs of alternatives are in a

necessary preference relation given Ω i .

Definition 1. Alternative x is necessarily weakly preferred to y for agent i,





written x N

i y, iff ∀ω ∈ Ω ,

k=1 ωk (xk − yk ) ≥ 0. Similarly, x is necessarily





strictly preferred to y for agent i, written x N

i y, iff ∀ω ∈ Ω ,

k=1 ωk (xk −

yk ) > 0.

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3 Soundness/Completeness of Multiple-Agent Possibilistic Logic

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