inference using compiled min-based possibilistic causal networks in the presence of interventions

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Fuzzy Sets and Systems 239 (2014) 104–136www.elsevier.com/locate/fss

Inference using compiled min-based possibilistic causal networksin the presence of interventions

Raouia Ayachia,b,∗, Nahla Ben Amora, Salem Benferhatb

a LARODEC, Institut Supérieur de Gestion Tunis, Le Bardo 2000, Tunisiab CRIL-CNRS, Université d’Artois, 62307, France

Available online 9 July 2013

Abstract

Qualitative possibilistic causal networks are important tools for handling uncertain information in the possibility theory framework.Contrary to possibilistic networks (Ayachi et al., 2011 [2]), the compilation principle has not been exploited to ensure causalreasoning in the possibility theory framework. This paper proposes mutilated-based inference approaches and augmented-basedinference approaches for qualitative possibilistic causal networks using two compilation methods. The first one is a possibilisticadaptation of the probabilistic inference approach (Darwiche, 2002 [13]) and the second is a purely possibilistic approach based onthe transformation of the graphical-based representation into a logic-based one (Benferhat et al., 2002 [3]). Each of the proposedmethods encodes the network or the possibilistic knowledge base into a propositional base and compiles this output in order toefficiently compute the effect of both observations and interventions. This paper also reports on a set of experimental resultsshowing cases in which augmentation outperforms mutilation under compilation and vice versa.© 2013 Elsevier B.V. All rights reserved.

Keywords: Possibilistic inference; Min-based possibilistic causal networks; Compilation techniques

1. Introduction

Knowledge compilation [8] is an important topic in many on-line applications that involve hard tasks. It transformsknowledge bases into new structures, with the intent being to improve the problem-solving efficiency. Assuming thatthe input knowledge base does not often change, so it can be turned into a compiled one during an off-line compilationphase which is then used to answer queries on-line. Answering such queries using the compiled base should becomputationally easier than answering them from the input base. One of the most prominent successful applicationsof knowledge compilation is in the context of Bayesian networks [9,10,13] referring to graphical models using theprobability theory. This framework is only appropriate when all numerical data are available, which is not alwayspossible. Moreover, there are some situations, like the case of total ignorance, which is not well handled and can makethe probabilistic reasoning unsound.

∗Corresponding author at: LARODEC, Institut Supérieur de Gestion Tunis, Le Bardo 2000, Tunisia. Tel.: +216 99076003.E-mail addresses: [email protected] (R. Ayachi), [email protected] (N. Ben Amor), [email protected] (S. Benferhat).

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R. Ayachi et al. / Fuzzy Sets and Systems 239 (2014) 104–136 105

Fig. 1. Proposed methods.

Several non-classical theories of uncertainty have been proposed in order to deal with uncertain and imprecise datasuch as Evidence theory [25,27,28], Spohn’s ordinal conditional functions [29,30] and Possibility theory [17,18] issuedfrom Fuzzy sets theory [31]. Possibility theory offers a natural and simple model to handle uncertain information. It isconsidered an appropriate framework for experts to express their opinions about uncertainty either numerically usingpossibility degrees or qualitatively using a total pre-order on the universe of discourse.

Recently, we have proposed two compilation-based inference approaches for min-based possibilistic networks [2].The main idea consists in compiling the Conjunctive Normal Form CNF encoding of the possibilistic network to havea polytime possibilistic inference in the presence of an evidence on a set of variables (i.e., observation). The firstmethod is an adaptation of the probabilistic compilation-based inference method [13] and the second one is a purelypossibilistic method based on compiling the possibilistic knowledge base associated with the possibilistic network [3].In our recent work, we have only studied the impact of evidences under compilation using possibilistic networks. Theso-called possibilistic causal networks [5], referring to possibilistic networks dealing with both observations which arethe results of testing some variables and interventions corresponding to external actions forcing some variables to havesome specific values, have not been explored under compilation. Interventions may have two different interpretationsdepending on whether we focus on the representational or on the reasoning issue. From a reasoning point of view,an intervention on a variable A is represented using the so-called mutilation, by ignoring relations between the intervenedvariable A and its direct causes. From a representational point of view, an intervention is depicted, using the so-calledaugmentation, by adding a new extra node as a parent-node to each intervened variable. Inference in causal networks,which focuses on determining the impact of either an observation or an intervention on the remaining variables, isknown as a hard problem [24].

In this paper, we will generalize the approaches proposed in [2] to deal with possibilistic causal networks. Our maincontributions are

• Handling interventions in min-based possibilistic causal networks using a compilation setting, which has not beenexplored yet in the possibility theory, neither on the probability theory (to the best of our knowledge). In fact, eachmethod studied in [2] will be extended and will deal with interventions in twofold: mutilation and augmentationas depicted by Fig. 1. More precisely, we will propose two mutilated-based methods that require the mutilation ofsymbolic compiled bases. This avoids re-compiling the network each time an intervention is occurred, which isintractable. We will also suggest two augmented-based methods that do not apply this constraint due to the new extranode. After compiling the network and handling interventions either by mutilation or augmentation, an efficientcomputation of the effect of both observations and interventions should be ensured using compiled bases. Fig. 1recapitulates the proposed methods in the present paper, namely two mutilated and two augmented approaches. Eachof them is either based on a possibilistic adaptation �-DNNF of a probabilistic method or based on a possibilisticlogic counterpart of a graphical model using possibilistic knowledge bases PKB.• Proposing a comparison study between mutilated-based approaches and augmented-based approaches under a

compilation framework. This study points out that the choice between mutilation and augmentation relies on thenumber of interventions. We will show that if the number of interventions is known in an instant t, augmented-basedapproaches are recommended, otherwise mutilated-based approaches are preferred.

106 R. Ayachi et al. / Fuzzy Sets and Systems 239 (2014) 104–136

The remaining paper 1 is organized as follows: Section 2 presents a brief refresher on possibility theory and knowledgecompilation. Section 3 presents two mutilated-based approaches. Section 4 describes two augmented-based approaches.Section 5 is dedicated to the comparative and experimental study. Section 6 concludes the paper.

2. Possibilistic causal networks

This section presents both possibilistic causal networks and knowledge compilation for possibilistic networks, whileproviding a brief refresher on possibility theory. In this paper, we will use the following notations:

Let V = {X1, X2, . . . , X N } be a set of variables. We denote by DXi = {x1, . . . , xn} the domain associated with thevariable Xi . By xi we denote any instance of Xi . The jth instance of Xi is denoted by xi j . When there is no confusionwe use xi instead of xi j to mean any instance of Xi . � denotes the universe of discourse, which is the Cartesian productof all variable domains in V. Each element � ∈ � is called a state of �. �[Xi ] = xi denotes an instantiation of Xi in �.� ⊆ � and � ⊆ � denote subclasses of � and are called events.

2.1. A refresher on possibility theory

Issued from fuzzy sets theory, possibility theory [18] is seen as a simple and natural model for handling uncertaindata. The basic building block in this theory is the concept of possibility distribution �, which is a mapping from theuniverse of discourse � to the unit interval [0, 1]. This later, called possibilistic scale, encodes our knowledge on the realworld. Contrary to the standard probability theory, the possibilistic scale could be interpreted in twofold: a numericalinterpretation when values have a real sense and an ordinal one when values only reflect a total pre-order between thedifferent states of the word. When �(�) = 1, this means that the realization of � is totally possible and when �(�) = 0,it means that � is an impossible state. When �(�) > �(�′), � is preferred to �′ for being the real state of the world.In the extreme case of total ignorance, �(�) = 1, ∀� ∈ �. It is generally assumed that there exists at least a state �which is totally possible. In this case, � is said to be normalized.

Example 1. Let us consider that we submit a scientific paper to an international conference. After reviewing, the papercan be rejected or accepted for either a conference or a poster. Then, the universe of discourse related to the submittedpaper after reviewing can be defined as follows: � = {accepted_con f erence, accepted_poster, rejected}.

Let us assume that a reviewer gives its judgment, denoted by Jd, in the form of a possibility distribution, then it canbe defined as follows:

�(Jd = accepted_conference) = 1,

�(Jd = accepted_poster) = 0.5,

�(Jd = rejected) = 0.1.

The possibility distribution given by the reviewer is normalized since max(1, 0.5, 0.1) = 1.

Given a possibility distribution �, we can define a mapping grading the possibility measure of any subset � ⊆ � by

�(�) = max�∈�

�(�) (1)

�(�) is called the possibility degree of �. It evaluates at which level � is consistent with our knowledge representedby �. For instance, �(�) = 1 makes this event possible but does not exclude ¬�. However, we conclude that only theevent � can be realized if �(¬�) = 0.

Example 2. Let us consider Example 1 such that the review process of the submitted paper is double blind. If we wantto know the possibility degree to have an accepted paper, we can say that such an event is fully possible. In other terms,its possibility degree � is equal to 1 since we have not yet received reviews and we have no information that contradictspaper’s acceptance.

1 This is an extended and revised version of the conference paper [1].


Table 1Joint possibility distribution after min-based conditioning.

Relevance Quality p(relevance, quality) p(relevance, quality|mw)

High Good 0.9 1High Bad 0.2 0.2Low Good 0.6 0Low Bad 0.7 0

The dual of the possibility measure is the necessity measure defined by ∀� ⊆ �:

N (�) = 1−�(¬�) = min�/∈�

(1− �(�)) (2)

N (�) is called the necessity degree of �. It corresponds to the certainty degree associated with �. In other terms,N evaluates at which level � is certainly implied by our knowledge represented by �. Let us illustrate the relationbetween N and �. If N (�) > 0, this implies that �(�) = 1. This means that � is completely possible before beingsomewhat certain. This property ensures the following inequality: N (�) � �(�).

Example 3. We assume that we received all reviews such that the relevance is equal to 9 of 10. If my friend asks me:Do you think the paper will be rejected? Then I will say No since it is nearly impossible to reject a paper having a veryhigh relevance. This is equivalent to say that it is almost necessary (certain) that the paper is accepted.

Conditioning is a crucial notion in possibility theory. It consists in revising our initial knowledge, encoded by apossibility distribution �, by the arrival of a new certain piece of information � ⊆ �. In a qualitative framework,min-based conditioning 2 is defined by the qualitative counterpart of the Bayesian rule defined by [17,22]

�(� ∧ �) = min(�(�|�), �(�)) (3)

The well-known definition of min-based conditioning using the minimum specificity principle is expressed by

�(�|m�) ={

�(� ∧ �) if �(� ∧ �) < �(�)

1 if �(� ∧ �) = �(�)(4)

When � ∧ � is inconsistent, then �(� ∧ �) = 0. If we want to compute the impact of � on each � ∈ �, thenmin-based conditioning is defined by

�(� |m �) =

⎧⎪⎨⎪⎩

�(�) if �(�) < �(�) and ��

1 if �(�) = �(�) and ��

0 otherwise

(5)

Example 4. Let us consider two variables relative to the quality of the paper and its relevance such that quali t y ={good, bad} and relevance = {high, low}. The joint distribution is given in Column 3 of Table 1. For instance, thepossibility degree of having an accepted paper given a high relevance and a good quality is equal to 0.9.

Consider now that we receive a fully certain piece of information indicating that the relevance is high, then � ={high ∧ good, high ∧ bad} and �(�) = max(0.9, 0.2) = 0.9. Using Eq. (5), the qualitative possibility distribution ofTable 1 will be transformed into a new distribution presented in Column 4 of Table 1.

At the syntactic level, uncertain pieces of information can be represented by means of a possibilistic knowledgebase, which is a set of weighted formulae of the form � = {(i , ai ), i = 1, . . . , n}, where i is a propositional formula,

2 The numerical interpretation of possibility theory uses the product instead of the min, but this is beyond the scope of the present study.


and ai ∈]0, 1]. Each possibilistic logic formula (i , ai ) expresses that i is certain to at least the level ai , or moreformally by N (i ) � ai , where N is the necessity measure associated to i . Each possibilistic knowledge base inducesa unique possibility distribution such that ∀� ∈ �:

��(�) ={

1 if ∀(i , ai ) ∈ �, �� i

1−max{ai : (i , ai ) ∈ � and �� i } otherwise(6)

Example 5. Let us consider the following possibilistic knowledge base � = {(a, 0.5), (b, 0.3), (¬a ∨ ¬b, 0.2)}, thenwe can say that formulae (a), (b) and (¬a∨¬b) are quantified by necessity degrees equal to 0.5, 0.3 and 0.2, respectively.

2.2. Basics on possibilistic causal networks

This subsection presents basics on possibilistic causal networks, which represent another compact representation ofpossibility distributions.

2.2.1. DefinitionsA possibilistic causal network is a generalization of a possibilistic network such that its graphical component is

a Directed Acyclic Graph (DAG) where nodes represent variables and edges encode not only dependencies betweenvariables but also direct causal relationships [4]. Such a network is the possibilistic counterpart of a probabilistic causalnetwork [24]. In the literature [7,26], the two interpretations of the possibilistic scale lead to two kinds of possibilisticcausal networks, namely min-based ones in an ordinal setting and product-based ones in a numerical setting. So, in theremaining when we talk about possibilistic causal networks, it means min-based ones.

Formally, over a set of variables V = {X1, X2, . . . , X N }, a min-based possibilistic causal network, denoted by�Gmin , has two components:

• A graphical component that is a DAG where nodes represent variables and edges encode not only dependen-cies between variables but also direct causal relationships [4]. The parent set of any variable Xi , denoted byUi = {Ui1, Ui2, . . . , Uim} where m is the number of parents of Xi , represents all direct causes for Xi . In whatfollows, we use xi , ui , ui j to denote, respectively, possible instances of Xi , Ui and Ui j . For the particular case ofroot nodes, ui = ∅.• A numerical component that quantifies different links. Uncertainty of each node Xi is represented by a local normal-

ized conditional possibility table (denoted by C�Ti ) in the context of its parents (i.e., ∀ui , maxxi �(xi |ui ) = 1).The set of all C�Ti is denoted by C�T .

The set of a priori and conditional possibility degrees in a min-based possibilistic network �Gmin induces a uniquejoint possibility distribution defined by the following min-based chain rule:

�(X1, . . . , X N ) = mini=1..N

�(Xi |Ui ). (7)

Example 6. Let us consider the possibilistic network, depicted by Fig. 2, containing two binary variables A and B. Thejoint possibility distribution of �Gmin is represented in Table 2.

2.2.2. Inference in possibilistic causal networksAs probabilistic causal networks [24], possibilistic causal networks allow us to study two kinds of information [26]:

• A set of observations (also called evidences): consisting in observing some variables. They describe the spontaneousevolution of the system and allow us to reason diagnostically about causes of observed events.• A set of interventions: representing external events, coming from outside the system and forcing some variables to

take some specific values.

To study the impact of both observations and interventions, authors in [26] proposed an inference algorithm forpossibilistic causal networks using augmented junction trees. In such a work, interventions are only studied from arepresentational point of view, i.e., using augmentation. They also proposed a new representation format, called hybridpossibilistic causal networks, where local uncertainty is no longer represented by conditional possibility distributions but


Fig. 2. A possibilistic network.

Table 2Joint distribution of �Gmin .

A B P(A) P(B|A) p

a1 b1 1 1 1a1 b2 1 0.8 0.8a2 b1 0.4 0.8 0.4a2 b2 0.4 1 0.4

by possibilistic knowledge bases. The main advantage of this representation concerns space complexity. An adaptationof the junction tree inference algorithm was proposed for hybrid possibilistic causal networks. In our work, we considerthe two different interpretations of interventions, namely mutilation and augmentation depending on whether we focuson the reasoning or the representational issue. By do(xI ), we denote an intervention on a variable X I ∈ V .

(A) Mutilation: From a reasoning point of view, an intervention is handled by the so-called mutilation operation [24],which refers to altering the network structure by excluding all direct causes related to the variable of interest andmaintaining the remaining variables unchanged [24]. The intuition behind such mutilation is that interventions areresults of external actions, and hence beliefs on direct causes of the intervened variable should not change. Thepossibility distribution associated with the mutilated network �Gmut is denoted by �mut . In possibility theory,the effect of do(xI ) is to transform �(�) into �mut (�|xI ), which gives us [4]:

∀�; �mut (�|xI ) = �(�|do(xI )). (8)

By mutilating the network, parents of X I become independent of X I . Moreover, the event that attributes the value xI

to X I becomes sure after performing intervention do(xI ). More formally, �mut (xI ) = 1 and ∀xi , xi � xI , �mut (xi ) = 0.The effect of do(xI ) on � is given as follows, ∀�:

�(�|do(xI )) ={

mini � I �(xi |ui ) if �[Xi ] = xI

0 otherwise.(9)

Example 7. Let us consider the possibilistic network �Gmin of Fig. 3, which will be used throughout the whole paper.It is composed of two variables A and B s.t. DA = {a1, a2} and DB = {b1, b2}. We can see that nodes A and B arequantified by possibility distributions �(A) and �(B|A) in the context of its parent(s). Note that the normalizationconstraint is satisfied, for instance for �(b1|a1) and �(b2|a1), we have max(�(b1|a1), �(b2|a1)) = 1.

Let B be the variable in �Gmin forced to take the value b1 by the intervention do(b1). Such intervention is reflectedgraphically by deleting the edge between A and B since the parent A is no longer responsible for the state of B afterintervention. The resulting mutilated network is depicted by Fig. 4.

To compute the effect of do(b1) on a2, b1, i.e., �(a2, b1|do(b1)), we use Eq. (9) as follows: �(a2, b1|do(b1)) =min(�(a2), �(b1|a2)) = min(0.4, 0.8) = 0.4. This value corresponds exactly to �mut (a2, b1|b1) = min(�mut (a2),�mut (b1)) = min(0.4, 1) = 0.4, according to Eq. (8).


Fig. 3. A possibilistic causal network �Gmin .

Fig. 4. The mutilated network �Gmut .

(B) Augmentation: From a representational point of view, an intervention can be depicted by augmentation,an alternative but equivalent approach for handling interventions. This allows us to represent interventions as observa-tions on special new variables. In fact, the augmenting scheme creates a new node DOI for each variable X I having anintervention and a link DOI → X I [21,23]. The new extra node takes k values {do(xI ) : ∀xI ∈ DX I } ∪ {doI−NoAct }such that k is the number of instances xI of X I and the new instance I − NoAct . The value doI−NoAct means thatthere is no intervention performed for X I , while values do(xI ) intend that the system forces the value xI for X I .We denote by doI any value of DOI .

The possibility distribution associated with the augmented network �Gaug is denoted by �aug . The new parent setof X I is represented by U ′I = UI ∪ DOI . The new possibility distribution of X I after performing do(xI ) is givenby [4]

�(xi |u′i ) =

⎧⎪⎨⎪⎩

�(xi |ui ) if DOI = doI−NoAct

1 if xi = xI

0 if xi � xI

(10)

However, possibility distributions associated with added nodes DOI are not a priori stated. In [5], it has beenproposed to define �aug(doI−NoAct ) = 1, while ∀xI ∈ DX I , �aug(do(xI )) = s.t., is a very small positive numberclose to 0. Note that should satisfy � min�∈� ��Gmin (�) where ��Gmin is the possibility distribution associated to�Gmin .

In fact, by affecting the degree 1 to values doI−NoAct of variables DOI , events {DOI = doI−NoAct } are by defaultaccepted as an expected truth. By affecting to �(DOI = do(xI )), ∀xI ∈ DX I , events {DOI = doxI } are consideredthe least preferred without being totally excluded. This possibility distribution allows us to express that by default thereis no intervention without excluding future interventions [4].

Example 8. Let us re-consider the �Gmin of Fig. 3, then the augmented network resulting of the intervention do(b1)on the variable B is represented by Fig. 5. The new possibility distribution of B is obtained using Eq. (10). For thedistribution of DOB , we should just find the smallest possibility degree in the distributions of the initial network ofFig. 3, namely 0.4 and assign it to both of �(do(b1)) and �(do(b2)).


Fig. 5. The augmented network �Gaug of �Gmin of Fig. 3.

The two ways of handling interventions are equivalent in the possibility theory framework. More formally, [4,26]:

Definition 1. Let �Gmin be a min-based possibilistic causal network. Let do(xI ) be an intervention forcing X I to takethe value xI . Let �Gmut (resp. �Gaug) be the mutilated (resp. augmented) network obtained after mutilation (resp.augmentation). Then, two situations are considered:

1. No intervention: ∀�, ∀xI ∈ DX I , �(�) = �aug(�|DOI = doI−NoAct ).2. An intervention do(xI ) occurs: �(�|do(xI )) = �mut (�|X I = xI ) = �aug(�|DOI = do(xI )).

Example 9. Let us re-consider the �Gmin of Fig. 3. Let B be the variable in �Gmin forced to take the value b1 by theintervention do(b1). This latter implies:

• Mutilation: �mut (b1) = 1 and �mut (b2) = 0.• Augmentation: �aug(do(b1)) = 1 and �aug(do(B − NoAct)) = �aug(do(b2)) = 0.

We can check if �mut (a2, b1) computed from �Gmut is equal to �aug(a2, b1) computed from �Gaug . Effectively,�mut (a2, b1) = min(�mut (a2), �mut (b1)) = �aug(a2, b1) = max(�aug(a2, b1, do(B − NoAct)), �aug(a2, b1, do(b1)),�aug(a2, b1, do(b2))) = 0.4, which is in accordance with Definition 1.

The problem of handling interventions and observations in possibilistic causal networks is stated as an NP-hardproblem [26]. In a previous work [2], we have addressed such a strategy in possibilistic networks, so that we tend togeneralize it for the particular case of possibilistic causal networks. Basics of knowledge compilation for possibilisticnetworks are given in the next subsection.

2.3. Knowledge compilation for possibilistic networks

Propositional logic is a knowledge representation and reasoning formalism. It is within the simplest logics butexpressive enough for many applications. It is used to express statements to which we assign the so-called truth values:true or false and no other possible value. We start with introducing basic concepts used in propositional logic.

From a syntactic point of view, the propositional variable represents the central element in propositional logic. It isa boolean variable subject to take two truth values, namely True (�) or False (⊥). A literal l is either a propositionalvariable, called a positive literal, or its negation, called a negative literal.

Let P be a finite set of propositional variables. The language of a propositional logic, denoted by L, is constructedover the set of propositional variables P, boolean constants True (�) and False (⊥), logical connectors (negation (¬),conjunction (∧), disjunction (∨), implication (→), equivalence (↔) 3 ) and parentheses. Elements ofL are propositional

3 Connectors ∧, ∨,→ and↔ have the same priority, while the negation connector (i.e., ¬) has the higher priority over all connectors.


formulae, also called well-formed formulae. They are formed using a set of propositional variables and can be definedin the following way:

1. Each propositional variable as well as � and ⊥ is propositional formulae.2. If and � are propositional formulae, then so are (¬), ( ∨ �), ( ∧ �), (→ � 4 ), (↔ �).3. Propositional formulae are only obtained using (1) and (2).

Relying on literals, a clause is a finite disjunction of literals (in particular the constant ⊥, when the set of literalsis empty) and a term is a finite conjunction of literals (in particular the constant �, when the set of literals is empty).A propositional formula is said to be in a Conjunctive Normal Form (CNF) iff is a conjunction of clauses.

Knowledge compilation is a common technique for propositional logic knowledge bases. It is a mapping from agiven knowledge base (from which the inference process is hard) into a special form of propositional bases, from whichqueries can be answered efficiently. Such mapping, which is ensured using the so-called target compilation languages,consists in splitting query answering of a particular problem into two phases [8]. In the first phase, knowledge basesare preprocessed in order to obtain the data structures the most appropriate for the given application (such a phasecalled the off-line reasoning is very expensive but it has to be performed only once). In the second phase, queries areanswered efficiently using the output of the first phase (such a phase is the on-line reasoning).

A logical form is qualified to be a target compilation language if it supports some set of nontrivial queries, usuallyincluding clausal entailment, in a polynomial time. There are several target compilation languages as it has been studiedin the knowledge map of [16]. We are in particular interested in Decomposable Negation Normal Form (DNNF) [11]and �-Decomposable Negation Normal Form (�-DNNF) [2]. We start with the DNNF language, which is the set ofall NNF (Negation Normal Form) sentences that satisfy the decomposability property stating that conjuncts of anyconjunction should share no variables. It supports a rich set of polynomial-time operations which can be performedsimply and efficiently. We cite in particular:

• Conditioning: Let be a propositional formula. Let � be a consistent term. 5 Then conditioning on �, denoted by|�, generates a new formula where each variable Pi of is set to:

Pi ={� if Pi is consistent6 with �

⊥ otherwise.(11)

For instance, if we have = (A∨ B)∧ (¬A∨C), then |A = (�∨ B)∧ (⊥ ∨C) which is equivalent to C. In whatfollows, the function condition(,Pi ) will be used to condition on Pi .• Forgetting: Let be a propositional formula, let Pi be a propositional variable, then forgetting Pi from , denoted

by ∃Pi · comes down to remove any reference of Pi in . Formally, it can be inductively defined as follows:

∃Pi · = |Pi ∨ |¬Pi (12)

where |Pi (resp. |¬Pi ) is the result of conditioning of on Pi (resp. ¬Pi ). For instance, forgetting the variable Afrom the following DNNF ≡ (A ∧ ¬B) ∨ C gives us: ∃A. ≡ (|A) ∨ (|¬A) ≡ ¬B ∨ C .In [19], authors have generalized the set of NNF languages by the Valued Negation Normal Form (VNNF) which offers

an enriched representation of functions ranging over an ordered scale. Within VNNF’s operations, we cite max-variableelimination which consists in forgetting variables using the max operator. As a special case of VNNFs, �-DNNFs,which are the possibilistic counterpart of DNNFs using minimum and maximum operators, have been explored in[2]. Note that formulae can be represented, in a compact manner, as Directed Acyclic graphs (DAGs) using circuits(e.g., boolean circuits, arithmetic circuits, etc.).

Recently, we proposed two alternative compilation-based inference methods for min-based possibilistic networksto efficiently compute the effect of an observation on remaining variables [2]. The first method is a possibilistic

4 Using the disjunction rule, we obtain ¬ ∨ �.5 A term that does not contain a contradiction.6 Pi is consistent with � if there exists an interpretation that satisfies both Pi and �.


adaptation of the so-called arithmetic circuit method [13]. It requires the use of the �-DNNF language [2] insteadof the propositional DNNF language [11]. The main idea is based on encoding the possibilistic network using theConjunctive Normal Form (CNF) propositional language, then compiling it to ensure inference in polytime. The CNFencoding used takes advantage of the structure exhibited by network parameters, known as local structure. Such astructure, consisting in incorporating particular values of parameters in the encoding phase, for instance degree 1, equalparameters, etc., induces a reduction of the time and the size of compiled bases [15].

The second method is a purely possibilistic inference method which is not grounded on encoding probabilistic works.The idea consists in transforming the possibilistic network into a possibilistic knowledge base [3], encoding it intoa CNF encoding by incorporating a set of additional variables corresponding exactly to the different weights of thebase [6]. After that a compilation step is required from which the inference query should be ensured in a polyno-mial time. The second method is qualified to be flexible since it allows the use of all existing target compilationlanguages.

Note that we used Darwiche’s work aiming to compile a CNF formula into DNNF using the compiler c2d [12].Interestingly enough, c2d imposes decomposability on formulae by disconnecting the underlying formula into sub-formulae that do no share a variable and instantiating enough variables using conditioning. This process, called caseanalysis, is then applied recursively until each sub-formula becomes decomposable.

3. Mutilated-based approaches

In this section, we will extend compilation-based inference methods proposed in [2] to deal with both observationsand interventions using mutilation which gives rise to two mutilated-based approaches, namely Mut-�-DNNF andMut-PKB.

3.1. Mutilated �-DNNF (Mut-�-DNNF)

One immediate way for handling sets of interventions consists first in mutilating �Gmin , encoding the mutilatednetwork using the CNF propositional theory and then compiling it to offer a polytime handling of queries. However,such a way is not efficient since it needs a re-compilation of the network each time an intervention occurs. TheMutilated �-DNNF (denoted by Mut-�-DNNF) proposed here avoids this problem by handling both observations andinterventions while avoiding re-compiling the initial network. We detail in what follows the three phases mentionedin Fig. 6.

3.1.1. Encoding and compilation phaseThe starting point of Mut-�-DNNF is the CNF encoding of the possibilistic network �Gmin using two types of

propositional variables, namely instance indicators xi for recording variables instances and network parameters �xi |ui

for recording possibility degrees. Formally, the CNF encoding handling n-ary variables and denoted by Cmin , requires

Fig. 6. Principle of Mut-�-DNNF method.


two types of propositional variables, namely:

• ∀Xi ∈ V , ∀xi j ∈ DXi , we associate an instance indicator xi j .• ∀Xi ∈ V , ∀xi j ∈ DXi , ∀ui ∈ DUi s.t. ui = {ui1, ui2, . . . , uim}, we associate a parameter variable �

xi |ui1,ui2,. . .,uim

for each network parameter �(xi |ui1, ui2, . . . , uim). Note that for any root node Xi , this parameter correspondsto �xi .

When there is no ambiguity, we use xi (resp. �xi |ui ) instead of xi j (resp. �xi j |ui1,ui2,. . .,uim

). The CNF encoding Cmin

is defined in the same way as in [2]:

Definition 2. Let �Gmin be a possibilistic network composed of N variables, having each n instances, xi j

(i = 1, . . . , N ), ( j = {1, . . . , n}) be the set of instance indicators and �xi |ui1,ui2,. . .,uim

be the set of parameter variables,then ∀Xi ∈ V , Cmin contains the following clauses:

• Mutually exclusive clauses:

xi1 ∨ xi2 ∨ · · · ∨ xin (13)

¬ xi j ∨ ¬ xik ( j = {1, . . . , n}) � (k = {1, . . . , n}) (14)

• Network parameter clauses: ∀�xi |ui1,ui2,. . .,uim

, we have

xi ∧ ui1 ∧ · · · ∧ uim → �xi |ui1,ui2,. . .,uim

(15)

�xi |ui1,ui2,. . .,uim

→ xi (16)

�xi |ui1,ui2,. . .,uim

→ ui1, . . . , �xi |ui1,ui2,. . .,uim

→ uim (17)

The encoding Cmin given in Definition 2 is in a CNF form, i.e., a conjunction of all clauses induced by (13)–(17)where clauses (13) and (14) state that indicator variables are exclusive and clauses (15)–(17) simply encode the factthat the possibility degree of xi |ui1, ui2, . . . , uim (represented by the propositional formula xi ∧ ui1 ∧ · · · ∧ uim )is equal (represented in the logical setting by logical equivalence ⇔) to �(xi |ui1, ui2, . . . , uim) (represented by thepropositional variable �

xi |ui1,ui2,. . .,uim). In what follows, the function encode (�Gmin) will be used to encode �Gmin

into Cmin .The CNF encoding of �Gmin , resulting from Definition 2, is then compiled into the common target compilation

language DNNF [14]. The resulting compiled base, denoted by CDN N F , is symbolic since we associate a propositionalvariable �xi |ui for each parameter value �(xi |ui ) without taking into consideration any numerical value. The use of thestrategy one variable per parameter represents the key of handling interventions by mutilation under a compilationframework, as we will vindicate next. In what follows, the function compile(Cmin) will be used to compile Cmin intoCDN N F .

Example 10. Considering the network �Gmin of Fig. 3, then its CNF encoding using Definition 2 contains the clausesof Table 3.

We can notice that the degree 0.8 which appears twice in the C�Ti of B is encoded by two different propositionalvariables, namely �b1|a2 and �b2|a1 . The CNF encoding Cmin is then compiled into CDN N F as represented by Fig. 7.At this stage, we ignore � and ⊥ assigned to instances indicators in Fig. 7.

3.1.2. Mutilation phaseThe mutilation phase proceeds with three inputs:

(i) the compiled base resulting from the previous step (CDN N F ),(ii) the intervention do(xI ) forcing the variable X I to take the value xI ,

(iii) the initial possibility degrees of �Gmin .


Table 3The CNF encoding Cmin of �Gmin of Fig. 3.

Mutually exclusive clausesA ( a1 ∨ a2 ) ∧ (¬ a1 ∨ ¬ a2 )

B ( b1 ∨ b2 ) ∧ (¬ b1 ∨ ¬ b2 )

Parameter clauses of A�(a1) = 1 ( a1 → �a1 ) ∧ (�a1 → a1 )

�(a2) = 0.4 ( a2 → �a2 ) ∧ (�a2 → a2 )

Parameter clauses of B

�(b1|a1) = 1 ( a1 ∧ b1 → �b1|a1 ) ∧ (�b1|a1 → b1 ) ∧ (�b1|a1 → a1 )

�(b1|a2) = 0.8 ( a2 ∧ b1 → �b1|a2 ) ∧ (�b1|a2 → b1 ) ∧ (�b1|a2 → a2 )

�(b2|a1) = 0.8 ( a1 ∧ b2 → �b2 |a1 ) ∧ (�b2 |a1 → b2 ) ∧ (�b2 |a1 → a1 )

�(b2|a2) = 1 ( a2 ∧ b2 → �b2 |a2 ) ∧ (�b2 |a2 → b2 ) ∧ (�b2 |a2 → a2 )

Fig. 7. The mutilated compiled base CmDN N F .

Given these inputs, we need to express that after intervention, the value of X I is xI for sure using CDN N F . Technicallyspeaking, this is established by conditioning the compiled base CDN N F on xI . Formally, ∀ �xi |ui relative to X I ,we have

�xi |ui ={� if xi = xI

⊥ otherwise (i.e., xi � xI )(18)

The resulting mutilated compiled base is denoted by CmDN N F . By conditioning, as if we assign �mut (xI ) = 1 and

�mut (xi ) = 0, ∀xi � xI , i.e., we alter X I ’s parameters to 1 (�) and 0 (⊥). Hence, given interventions, new possibilitydegrees are affected to propositional variables �xi |ui corresponding to the intervened variable X I which results in a newcompiled base as illustrated by Fig. 6. The function mutilate (CDN N F ) will be used to mutilate CDN N F .

It is worth to point out that the factor that makes the first step achievable is the strategy one variable per parameterused while encoding the causal network and providing a symbolic compiled base that do not take into account parametersvalues. Note also that each time an intervention occurs, we do not need to re-compile the network but we just need to‘logically’ mutilate the current compiled base by applying conditioning.

Example 11. Let us consider the compiled base resulting from the previous phase. Let do(b1) be an interventionforcing the variable B to take the value b1. Then, applying Eq. (18) to CDN N F consists in assigning �b1|a1 and �b1|a2

(resp. �b2|a1 and �b2|a2 ) to � (resp. ⊥). The resulted mutilated compiled base is depicted by Fig. 7.


3.1.3. Inference phaseGiven the mutilated compiled base Cm

DN N F resulting from the previous phase, an instance of interest y of somevariables X ⊆ V , an observation e and an intervention do(xI ), we should be able to efficiently compute the effect of eand do(xI ) on y, namely �c(y|e, do(xI )). Using Eq. (4), it is clear that we should first compute �c(y, e, do(xI )) and�c(e, do(xI )) following these three steps:

Step 1: Updating instance indicators. This step serves to record the instance of interest y, the observation e and theintervention do(xI ) into instance indicators xi . It corresponds to conditioning the mutilated compiled base resultingfrom the previous phase using instance indicators. Formally:

• CmDN N F is conditioned on e, i.e., ∀ Xi � X I , we have

xi ={� if xi ∼ e

⊥ otherwise(19)

where ∼ denotes the compatibility relation, i.e., xi and y agree on values.• Cm

DN N F is conditioned on y, i.e., ∀Xi � X I , we have

xi ={� if xi ∼ y

⊥ otherwise(20)

• CmDN N F is conditioned on xI , i.e., ∀X I , we have

xi ={� if xi = xI

⊥ otherwise(21)

The resulting compiled base is denoted by [CmDN N F |e, y, xI ].

Step 2: Mapping from logical to numerical representation. In this step, we transform the logical compiled baseresulting from the previous step into a numerical representation, denoted by Cm

�−DN N F , from which we will be ableto ensure efficient possibilistic computation. By mapping, we mean

• replacing ∨ and ∧ by max and min, respectively,• substituting each � (resp. ⊥) by 1 (resp. 0),• associating �(xi |ui ) to each �xi |ui related to ∀Xi � X I .

It is obvious that the mapping from logical to numerical representation is established in a polynomial time sinceit corresponds to a set of trivial substitution operations. The function map ([Cm

DN N F |e, y, xI ]) will be used to map[Cm

DN N F |e, y, xI ] into Cm�−DN N F .

Step 3: Computation. The last step corresponds to evaluating Cm�−DN N F in order to compute �c(y, e, do(xI )) and

�c(e, do(xI )). In fact, evaluation consists in applying min and max operators in a bottom-up way. The final resultcan be found at the root. Thus, inference is guaranteed to be established in polytime since it corresponds to a simplepropagation from leaves to root. Note that such computation corresponds also to a max-variable elimination operation.In what follows, we will use the function evaluate (Cm

�−DN N F ) to evaluate Cm�−DN N F .

Example 12. Let us consider the mutilated compiled base of Fig. 7. Let do(b1) be an intervention forcing the vari-able B in �Gmin to take the value b1 and a2 be the instance of interest of A, then to compute �c(a2, do(b1))we should:

1. Record a2 and do(b1) into instance indicators using Eqs. (20) and (21). This consists in setting a2 and b1 (resp. a1 and b2 ) to � (resp. ⊥). The resulting compiled base Cm

DN N F |a2, do(b1) is shown in Fig. 8(a).2. Transform the logical compiled base Cm

DN N F |a2, do(b1) into a numerical representation Cm�−DN N F (see Fig. 8(b)).

3. Evaluate Cm�−DN N F leading to �c(a2, do(b1)) = 0.4 as shown in Fig. 8(c). Thus, �c(a2|do(b1)) = 0.4 since

�c(a2, do(b1)) = 0.4 < �c(do(b1)) = 1.


Fig. 8. Inference steps: (a) The conditioned compiled base CmDN N F |a2, b1), (b) The numerical representation Cm

�−DN N F , and (c) Computing�c(a2, do(b1)).

3.1.4. Mut-�-DNNF algorithmThe whole Mut-�-DNNF approach, outlined by Algorithm 1, guarantees the equivalence between possibility degrees

computed using Mut-�-DNNF and the joint distribution as shown by the following proposition:

Algorithm 1. Mut-�-DNNF.

Data: �Gmin , instance of interest y, evidence e, intervention do(xI )

Result: �c(y|e, do(xI ))

begin∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

% Encoding and compilation phaseCmin ← encode(�Gmin)

CDN N F ← compile(Cmin)

% Mutilation phaseCm

DN N F ← mutilate(CDN N F )

% Inference phaseI nt ← {y, e, do(xI )}�c(y, e, do(xI ))← Computing(Cm

DN N F , I nt, do(xI ))

I nt ← {e, do(xI )}�c(e, do(xI ))← Computing(Cm

DN N F , I nt, do(xI ))

if �c(y, e, do(xI )) < �c(e, do(xI )) then �c(y|e, do(xI ))← �c(y, e, do(xI ))

else �c(y|e, do(xI ))← 1

return �c(y|e, do(xI ))end


Algorithm 2. Computing.

Data: CmDN N F , instance of interest Int, intervention do(xI )

Result: �c(I nt, do(xI ))begin∣∣∣∣∣∣∣∣

CmDN N F |I nt, xI ← condition(Cm

DN N F , I nt, xI )Cm

�−DN N F ← map(CmDN N F |I nt, do(xI ))

�c(I nt, do(xI ))← evaluate(Cm�−DN N F )

return �c(I nt, do(xI ))end

Proposition 1. Let CDN N F be the compiled base of a possibilistic network �Gmin . Let do(xI ) be an intervention thatforces the variable X I to take the value xI .

(i) Let � ∈ �, we have

�c(�|do(xI )) = �(�|do(xI )) (22)

where �c(�|do(xI )) (resp. �(�|do(xI ))) is computed using Algorithm 2 (resp. Eq. (9)).(ii) Let y be an instantiation of any variables Y ∈ V and e be an instantiation of any variables E ∈ V . Then

�c(y|e, do(xI )) = �(y|e, do(xI )) (23)

where �c(y|e, do(xI )) (resp. �(y|e, do(xI ))) is computed using Algorithm 1 (resp. Definition 1).

Proof. We have CDN N F ≡∨

v

∧v�xi∧ui

xi �xi |ui

Mutilating CDN N F gives us: CDN N F ≡∨

v

∧v�xi∧ui

xi �xi |ui s.t. �xI |ui (�¬xI |ui ) encodes the degree 1 (resp. 0)(Eq. (19)). The compiled base resulting from updating each instance indicator xi and the mapping from logical tonumerical representation is the following:

⇒ C�−DN N F = min(xi ,ui )∈�

�xi |ui if �[Xi ] = xI (Eqs. (20) and (21)),

⇒ �c(�|do(xI )) = min(xi ,ui )∈�

�(xi |ui ) if �[Xi ] = xI (since �xi |ui encodes �(xi |ui )),

⇒ �c(�|do(xI )) = �(�|do(xI )) (Eq. (9)).

For an observation e, we obtain �c(�|e, do(xI )) = �(�|e, do(xI )) From this step: ∀y an instantiation of Y ∈ V ,we obtain

max��y

�c(�|e, do(xI )) = max��y

�(�|e, do(xI ))

⇒ �c(y|e, do(xI )) = �(y|e, do(xI )) (Eq. (19)). �

It is important to note that our approach takes advantage from the fact that the compiled base considers a set ofsymbols (propositional variables) without regard to numerical values. In particular, for any new intervention, we canre-use the same compiled base with a simple updating of parameters values. Hence, we can conclude that our methoddo not depend on interventions, i.e., even if we grow the number of interventions, the complexity is not altered sincethe mutilation process, which is a conditioning operation, is ensured in a polynomial time and the computation of theeffect of interventions is also polynomial with respect to the size of the compiled base.

As a negative side, the constraint of one variable per parameter should be applied even for equal parameters sincewhen handling interventions, the environment is qualified as dynamic. So, the so-called local structure enhance-ment related to equal parameters, used to reduce the set of added variables cannot be explored in mut-�-DNNF.


More precisely, we cannot attribute the same propositional variable even for equal parameters within C�Ti . This isnot the case in [2] if only observations occur because parameters are constant. For instance, assuming that we have�b2|a1 = �b1|a2 = 0.8. Then, after performing intervention do(b1), we should set �b2|a1 (resp. �b1|a2 ) to 0 (resp. 1). Thisis infeasible when we use the same propositional variable � for both of �b2|a1 and �b1|a2 .

3.2. Mutilated compiled possibilistic knowledge bases (Mut-PKB)

The mutilated compiled possibilistic knowledge bases (denoted by Mut-PKB) is a purely possibilistic inferencemethod based on the transformation of the initial network into a possibilistic knowledge base [3] and then using thissecondary structure as input of the encoding and compilation phase. The three main phases of Mut-PKB presented inFig. 9 are detailed in what follows.

3.2.1. Encoding and compilation phaseThe principle of the transformation of the initial �Gmin into �min is to associate to each Xi a local possibilistic

knowledge base �Xi and to combine them into a global possibilistic knowledge base �min . Formally, this transformationis described by Definition 3.

Definition 3. Let �Gmin be a possibilistic causal network. Then, its possibilistic knowledge base, denoted by �min ,is expressed by

�min =⋃

Xi∈V

�Xi (24)

where �Xi = {(¬xi ∨ ¬ui , ai ) : ai = 1−�(xi |ui ) � 0}.

The function transform(�Gmin) will transform �Gmin into �min .The min-based possibilistic knowledge base �min is then encoded into a CNF base by affecting a new propositional

variable, denoted by Ai , for each N (i ), where i ∈ �min . The set of propositional variables, denoted by {A1, . . . , An},encodes {N (1), . . . , N (n)}. The propositional encoding of �min , denoted by K�, is expressed by

K� = {i ∨ Ai : (i , ai ) ∈ �min} (25)

where a propositional formula i ∨ Ai will encode each possibilistic formula (i , ai ). In what follows, the functionencode-PKB (�min) will be used to encode �min into K�.

It is important to note that each degree ai in �min is encoded using a propositional variable Ai (one variable perparameter) without taking into account any parameter value, which maintains the network structure. Consequently,even if we transform the network into a logic-based representation, causal links are held. This strategy is required inMut-PKB since some parameters values are not stable and will be updated depending on interventions. This is thefundamental difference between handling only observations and handling both observations and interventions.

The CNF encoding K� of �min is then compiled into any target compilation language supporting both of con-ditioning and clausal entailment, which are the required operations to compute the effect of both observations andinterventions, thanks to these operations which make Mut-PKB flexible. In what follows, we pick on the most succincttarget compilation language, i.e., DNNF and the resulting compiled base is denoted by Kc.

Example 13. Let us consider the �Gmin of Fig. 3. The CNF encoding of the possibilistic knowledge base �min of�Gmin is shown in Table 4.

The CNF encoding K� is then compiled into DNNF. The resulting compiled base is as follows: Kc = {[(a2 ∧ A1)∧(b2 ∨ (b1 ∧ A3))] ∨ [a1 ∧ (b1 ∨ (b2 ∧ A2))]}.

3.2.2. Mutilation phaseGiven an intervention do(xI ) performed on X I , the compiled base resulting from the previous phase Kc should be

mutilated as depicted by Fig. 9. This phase makes the connection between mutilating �Gmin and mutilating Kc. More


Table 4The CNF encoding K� of �min of Fig. 3.

P(ai) Clauses of A�(a2) = 0.4 (a1, 0.6) (a1 ∨ A1)

P(bj|ai) Clauses of B�(b2|a1) = 0.8 (a2 ∨ b1, 0.2) (a2 ∨ b1 ∨ A2)

�(b1|a2) = 0.8 (a1 ∨ b2, 0.2) (a1 ∨ b2 ∨ A3)

Fig. 9. Principle of Mut-PKB method.

explicitly, it consists in updating the necessity degrees of variables Ai related to the variable of interest X I . In fact, thenecessity degree 1 should be assigned for formulae of ¬xi s.t. xi � xI (since �mut (¬xi ) = 0) and 0 for formulae of ¬xI

(since �mut (xI ) = 1). The mutilated compiled base is denoted by K mc .

It is worth pointing out that before the mutilation step we cannot attribute the same propositional variable Ai even forequal degrees in �min . For instance, assuming that we have the following formulae (a2∨b1, A1) and (a1∨b2, A1) suchthat A1 encodes the necessity degree 0.2. Let do(b1) be an intervention that forces B to take the value b1. By mutilation,we mean setting the degree 0 (resp. 1) to the Ai corresponding to (a1 ∨ b2, 0.2) (resp. (a2 ∨ b1, 0.2)). However, this isinfeasible since the degree 0.2 is encoded twice using the same propositional variable A1.

Example 14. We continue with Example 13. Let B be the variable forced to take the value b1 by the intervention do(b1).Then, mutilating the compiled base Kc consists in updating the degree of A2 (resp. A3) corresponding to (a2 ∨ b1, 0.2)(resp. (a1 ∨ b2, 0.2)) from 0.2 to 1 (resp. 0.2 to 0).

3.2.3. Inference phaseAfter the mutilation phase, we should sort variables Ai by associating a new variable B j to variables Ai encoding

equal degrees in K mc . To this end, let us consider g the number of different degrees in Kc after mutilation. Then, we will

associate the set of propositional variables B = {B1, . . . , Bg} for all different degrees pertaining to Kc after mutilation.The function sort(A) will be used to sort A and the new set of variables will be B.

Once we sort propositional variables Ai , we should compute �c(y|e, do(xI )) using the compiled base K mc given

an instance of interest y, an observation e and an intervention do(xI ) performed on X I . The computation processis established in an iterative manner by applying entailment and conditioning as outlined by Algorithm 3. In fact,we should first test if K m

c �B1 ∨ ¬e ∨ ¬xI . If this deduction is not satisfied, we condition K mc on ¬B1 and then


test if K mc entails ¬y. If this is the case, we compute �c(y|e, do(xI )) by subtracting 1 from the necessity degree

associated to the current propositional variable, else we move to the next variable B j and we re-iterate the sametreatment. In the worst case, this computation is performed g − 1 times since the last variable Bg encodes thedegree 0.

It is crucial to note that our method do not depend on the number of interventions, thanks to the symbolic compiledbase that allows us to update parameters values linearly regardless of the number of interventions.

Example 15. Let us continue Example 14 to compute the effect of do(b1) on the observation a2. First, we should sortpropositional variables after mutilation. The new set of variables is the following: B = {B1(1), B2(0.6), B3(0)}. Then,we should compute �c(a2|do(b1)) as follows:

1. Iteration 1: K mc �b2 ∨ B1 ⇒ (K m

c |¬B1)�a1 ⇒ i ← i + 1,

2. Iteration 2: K mc �b2 ∨ B2 ⇒ (K m

c |¬B2)�a1 ⇒ StopCompute← true.

This means that �c(a2|do(b1)) = 1− degree(2) = 1− 0.6 = 0.4, where degree(2) designates the weight associatedto B2, i.e., 0.6.

3.2.4. Mut-PKB algorithmThe Mut-PKB method, outlined by Algorithm 3, guarantees the equivalence between possibility degrees computed

using Mut-PKB and the joint distribution. Formally, if we consider a possibilistic network �Gmin and an interventiondo(xI ) that forces the variable X I to take the value xI . Then, for any y ∈ DY and e ∈ DE , we have �c(y|e, do(xI )) =�(y|e, do(xI )), where �c(y|e, do(xI )) (resp. �(y|e, do(xI ))) is computed using Algorithm 3 (resp. Proposition 1).In fact, encoding �Gmin using the strategy one variable per parameter, then mutilating the compiled base dependingon do(xI ) enables the computation of the effect of do(xI ) and e on y by applying conditioning and entailment in aniterative manner.

Proposition 2. Let Kc be the compiled base of a possibilistic network �Gmin . Let do(xI ) be an intervention thatforces the variable X I to take the value xI .

(i) ∀� ∈ �, we have

�c(�|do(xI )) = ��(�|do(xI )) (26)

where �c(�|do(xI )) (resp. ��(�|do(xI ))) is computed using Algorithm 3 (resp. Eq. (9)).(ii) Let y be an instantiation of a variable Y ∈ V and e be an instantiation of any variables E ⊆ V . Then

�c(y|e, do(xI )) = ��(y|e, do(xI )) (27)

where �c(y|e, do(xI )) (resp. ��(y|e, do(xI ))) is computed using Algorithm 3 (resp. Eq. (9)).

Proof. From Eq. (6), we have �(�) = ��(�). By mutilating �Gmin , its means setting 1 to �(xI ) and 0 to �(xi ),∀xi � xI . Using CNF encodings of possibilistic knowledge bases, mutilation associates 1 or 0 to propositional variablesAi encoding necessity degrees.We obtain �c(�|do(xI )) = ��(�|do(xI )). Thus, Eq. (26) is established.

This result is relative to an interpretation �, to generalize it to any instantiation y of a variable Y ∈ V , we obtain

max��y

�c(�|do(xI )) = max��y

�c(�|do(xI ))

Thus, �c(y|do(xI )) = ��(y|do(xI )).When we deal with an observation e, we obtain

�c(y|e, do(xI )) = ��(y|e, do(xI )). �


Algorithm 3. Mut-PKB.

Data: �Gmin , instance of interest y, evidence e, intervention do(xI )Result: �c(y|e, do(xI ))begin∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

% Encoding and compilation phase�min ← transform(�Gmin)K� ← encode-PKB(�min)Kc ← compile(K�)%Mutilation phaseK m

c ← mutilate(Kc)%Inference phaseLet B = {B1, . . . , Bg} ← sort(A)i ← 1, StopCompute← false, �c(y|e, do(xI ))← 1while (K m

c �Bi ∨ ¬e ∨ ¬xI ) and (i < g) and (StopCompute = f alse) do⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

K mc |¬Bi ← condition(K m

c ,¬Bi )if (K m

c |¬Bi )�¬y then⎢⎢⎢⎣ StopCompute← trueLet degree(i) be the weight associated to Bi

�c(y|e, do(xI ))← 1− degree(i)else i ← i + 1


4. Augmented-based approaches

In this section, we will explore the augmentation-based approach for handling interventions. The focus will be onthe new extra node.

4.1. Augmented �-DNNF (Aug-�-DNNF)

The Augmented �-DNNF, denoted by Aug-�-DNNF, deals with interventions from the first phase (i.e., encoding andcompilation phase) by adding a new extra node to the possibilistic network. Contrarily to mut-�-DNNF, Aug-�-DNNFbenefits from local structure aiming to exploit parameters values while encoding the network. Mainly based on thelocal structure, Aug-�-DNNF is composed of two phases as shown in Fig. 10 and detailed below.

4.1.1. Encoding and compilation phaseAug-�-DNNF proceeds by encoding the augmented possibilistic network arisen from performing intervention on the

initial network. The question that may arise is: Can we handle interventions before the encoding and compilation phase?We have seen that the answer is NO when using mutilated-based approaches, which is not the case in augmented-basedapproaches, more precisely in Aug-�-DNNF since the new extra node DOI enables to focus on both observations andinterventions by using the augmented network.

Once we obtain the augmented network �Gaug , we should encode it using the CNF encoding given in Definition 2,but we will focus on �Gaug instead of �Gmin . In other words, we should just consider the new extra node DOI asa variable of V (i.e., V = V ∪ DOI ). The key point that should be highlighted is the strategy of encoding, whichdepends on the handled node, i.e., an extra node DOI it is or a node Xi ∈ V . In fact, for any node Xi ∈ V except DOI ,parameter values are constant and do not depend on interventions. For this reason, the so-called local structure can beaddressed. More precisely, each network parameter �xi |ui equal to 0 can be dropped from the encoding by replacingclauses (15)–(17) by a shorter clause involving only indicator variables, namely:

¬ xi ∨ ¬ ui1 ∨ · · · ∨ ¬ uim (28)


Fig. 10. Principle of Aug-�-DNNF method.

Moreover, each set of equal parameters per CPT should be encoded by the same propositional variable �i whichreduces the number of network parameters. Such simplification can involve in an inconsistent theory if we keep clauses(16) and (17). For instance, if we replace two equal parameters �b1|a1 and �b2|a2 by a single one (i.e., �), then clauses(16) and (17) generate �→ b1 and �→ b2 which is inconsistent since b1 and b2 cannot be both true in the samemodel. This inconsistency can be avoided by just dropping these additional clauses from the encoding.

Regarding the new extra node DOI , the local structure cannot be exploited due to the variability of DOI ’s parameters.For this reason, DOI should be encoded using symbolic propositional variables without taking into consideration anynumerical value, which results in a local symbolic encoding associated with DOI . More explicitly, the strategy onevariable per parameter per DOI ’s C�Ti will be adopted regardless of whether there are some equal or zero parametersin DOI ’s C�Ti . This is the key point that allows us to express that by default there is no intervention and does notexclude future interventions by just setting the appropriate values to DOI ’s network parameters.

Once we have obtained the CNF encoding of �Gaug , denoted by Caugmin , we should then compile it into DNNF. The

resulting compiled base is denoted by CaugDN N F .

Example 16. Considering the network of Fig. 5, then at first, we should define the set of instance indicators andnetwork parameters given respectively, in Tables 5 and 6. Using these propositional variables, the CNF encoding of�Gaug contains clauses of Table 7.

It is obvious that we are using the strategy one variable per parameter per DOB’s C�Ti , without taking into accountany numerical value, i.e., 1, 0.4 and 0.4. The compiled base of Caug

min is depicted by Fig. 11.

4.1.2. Inference phaseThe inference phase consists in computing efficiently the effect of both observations and interventions using the

compiled base CaugDN N F resulting from the previous phase. Hence, given Caug

DN N F , an instance of interest y of somevariables X ⊆ V , an observation e and an intervention do(xI ), we should compute both of �c(y, e, do(xI )) and�c(e, do(xI )), by applying the following steps, to have �c(y|e, do(xI )).

Step 1: Updating instance indicators. The first step consists in updating instances indicators according to observatione, instance of interest y and intervention do(xI ). We should first condition Caug

DN N F on e and y using Eqs. (19) and (20)


Table 5Set of instance indicators used in Caug .

Instances Caug

a1 a1

a2 a2

b1 b1

b2 b2

do(B − NoAct) do(B−NoAct)

do(b1) do(b1)

do(b2) do(b2)

Table 6Set of network parameters used in Caug .

Variables Possibility degrees Caug

A 1 �a1

0.4 �a2

B �(b1|a1, do(B − NoAct) = 1 �1

�(b1|a1, do(b1)) = 1 �1

�(b1|a1, do(b2)) = 0 –�(b1|a2, do(B − NoAct)) = 0.8 �2

�(b1|a2, do(b1)) = 1 �1

�(b1|a2, do(b2)) = 0 –�(b2|a1, do(B − NoAct)) = 0.8 �2

�(b2|a1, do(b1)) = 0 –�(b2|a1, do(b2)) = 1 �1

�(b2|a2, do(B − NoAct)) = 1 �1

�(b2|a2, do(b1)) = 0 –�(b2|a2, do(b2)) = 1 �1

DOB �(do(B − NoAct)) (not initialized) �do(B−NoAct)

�(do(b1)) (not initialized) �do(b1)

�(do(b2)) (not initialized) �do(b2)

as in mut-�-DNNF. A slight modification should be performed in both of these equations by replacing ∀Xi � X I by∀Xi � DOI . Then, we should set each doI of DOI to � or ⊥ depending on do(xI ).

• Intervention do(xI ): ∀ doI of DOI , we have

doI ={ � if doI = do(xI )⊥ otherwise (i.e., doI � do(xI ))

(29)

• No interventions: ∀ doI of DOI , we have

doI ={ � if doI = doI−NoAct

⊥ otherwise (i.e., doI � doI−NoAct )(30)

Step 2: Mapping from logical to numerical representation. This step transforms the logical representation arisenfrom Step 1 into a numerical one, denoted by Caug

�−DN N F . As in mut-�-DNNF, we should replace each ∨, ∧, � and⊥ by max, min, 1 and 0, respectively. Also, each �xi |ui of Xi � DOI should be replaced by the appropriate possibilitydegree �(xi |ui ). An additional substitution should be performed for DOI ’s network parameters. In fact, if we onlyfocus on observations, we should set initial possibility values (i.e., 1 and which is a very small positive number close


Table 7The CNF encoding Caug of �Gaug of Fig. 5.

Mutually exclusive clausesA ( a1 ∨ a2 ) ∧ (¬ a1 ∨ ¬ a2 )B ( b1 ∨ b2 ) ∧ (¬ b1 ∨ ¬ b2 )DOB ( do(B−NoAct) ∨ do(b1) ∨ do(b2))

∧(¬ do(B−NoAct) ∨ ¬ do(b1))∧(¬ do(B−NoAct) ∨ ¬ do(b2))∧(¬ do(b1) ∨ ¬ do(b2))

Parameter clauses of A�(a1) = 1 ( a1 → �a1 ) ∧ (�a1 → a1 )�(a2) = 0.4 ( a2 → �2) ∧ (�a2 → a2 )

Parameter clauses of B�(b1|a1, do(B − NoAct)) = 1 ( a1 ∧ b1 ) ∧ do(B−NoAct) → �1

�(b1|a1, do(b1)) = 1 a1 ∧ b1 ∧ do(b1) → �1

�(b1|a1, do(b2)) = 0 ¬ a1 ∨ ¬ b1 ∨ ¬ do(b2)

�(b1|a2, do(B − NoAct)) = 0.8 a2 ∧ b1 ∧ do(B−NoAct) → �2

�(b1|a2, do(b1)) = 1 a2 ∧ b1 ∧ do(b1) → �1

�(b1|a2, do(b2)) = 0 ¬ a2 ∨ ¬ b2 ∨ ¬ do(b2)

�(b2|a1, do(B − NoAct)) = 0.8 a1 ∧ b2 ∧ do(B−NoAct) → �2

�(b2|a1, do(b1)) = 0 ¬ a1 ∨ ¬ b2 ∨ ¬ do(b1)

�(b2|a1, do(b2)) = 1 a1 ∧ b2 ∧ do(b2) → �1

�(b2|a2, do(B − NoAct)) = 1 a2 ∧ b2 ∧ do(B−NoAct) → �1

�(b2|a2, do(b1)) = 0 ¬ a2 ∨ ¬ b2 ∨ ¬ do(b1)

�(b2|a2, do(b2)) = 1 a2 ∧ b2 ∧ do(b2) → �1

Parameter clauses of DOB

�(do(B − NoAct)) ( do(B−NoAct) → �do(B−NoAct))∧(�do(B−NoAct) → do(B−NoAct))

�(do(b1)) ( do(b1) → �do(b1)) ∧ (�do(b1) → do(b1))�(do(b2)) ( do(b2) → �do(b2)) ∧ (�do(b2) → do(b2))

to 0) to DOI ’s network parameters. However, if an intervention occurs, the degree 1 or 0 is assigned to each DOI ’snetwork parameter depending on do(xI ). The new compiled base, denoted by Caug

�−DN N F , corresponds to the compiledbase associated with interventions.

Step 3: Computation. The last step corresponds to computing �c(y, e, do(xI )) and �c(e, do(xI )) using Caug�−DN N F

by applying operators min and max in a bottom-up way as in mut-�-DNNF.

Example 17. Considering the compiled base CaugDN N F of Fig. 11, a2 an instance of interest and do(b1) an intervention

forcing B to take the value b1, then �c(a2|do(b1)) is obtained by comparing both of �c(a2, do(b1)) and �c(do(b1)).To compute �c(a2, do(b1)):

1. We should first set a2 , do(b1), b1 to � and do(B−NoAct), a1 , b2 , do(b2) to ⊥.2. We should replace each ∧ and ∨ by min and max, respectively.3. Also, we should set the possibility degree 1 to �do(b1) and 0 for both of �do(b2) and �do(B−NoAct).4. Finally, we compute �c(a2, do(b1)) in a bottom-up way as shown in Fig. 12.

Hence, �c(a2|do(b1)) = 0.4 since �c(a2|do(b1)) = 0.4 < �c(do(b1)) = 1.

4.1.3. Aug-�-DNNF algorithmAug-�-DNNF method, outlined by Algorithm 4, guarantees the equivalence between possibility degrees computed

using Aug-�-DNNF and the joint distribution as shown by the following proposition:


Fig. 11. The CaugDN N F .

Fig. 12. Computing �c(a2, do(b1)).

Proposition 3. Let CaugDN N F be the compiled base of a possibilistic network �Gaug . Let do(xI ) be an intervention that

forces the variable X I to take the value xI .

(i) Let � ∈ �, we have

�c(�|do(xI )) = �(�|do(xI )) (31)

where �c(�|do(xI )) (resp. �(�|do(xI ))) is computed using Algorithm 5 (resp. Eq. (9)).(ii) Let y be an instantiation of any variables Y ∈ V and e be an instantiation of any variables E ∈ V . Then

�c(y|e, do(xI )) = �(y|e, do(xI )) (32)

where �c(y|e, do(xI )) (resp. �(y|e, do(xI ))) is computed using Algorithm 4 (resp. Definition 1).


Proof. We have CaugDN N F ≡

∨v

∧v�xi∧ui

xi �xi |ui

Updating each instance indicator xi and transforming the resulting logical compiled base into a numerical baseresults in:

⇒ Caug�−DN N F = min

(xi ,ui )∈��xi |ui if �[DOI ] = do(xI ) (Eq. (29))

⇒ �c(�|do(xI )) = min(xi ,ui )∈�

�(xi |ui ) if �[DOI ] = do(xI ) (since �xi |ui encodes �(xi |ui )),

⇒ �c(�|do(xI )) = �(�|do(xI )) (Eq. (10)).

For an observation e, we obtain �c(�|e, do(xI )) = �(�|e, do(xI ))From this step: ∀y an instantiation of Y ∈ V , we obtain

max��y

�c(�|e, do(xI )) = max��y

�(�|e, do(xI ))

⇒ �c(y|e, do(xI )) = �(y|e, do(xI )) (Eq. (19)). �

Algorithm 4. Aug-�-DNNF.

Data: �Gaug , instance of interest y, evidence e, intervention do(xI )Result: �c(y|e, do(xI ))begin∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

% Encoding and compilation phaseCaug

min ← encode(�Gaug)Caug

DN N F ← compile(Caugmin )

% Inference phaseI nt ← {y, e, do(xI )}�c(y, e, do(xI ))← Computing-Aug (Caug

DN N F , I nt, do(xI ))I nt ← {e, do(xI )}�c(y, do(xI ))← Computing-Aug (Caug

DN N F , I nt, do(xI ))if �c(y, e, do(xI )) < �c(e, do(xI )) then �c(y|e, do(xI ))← �c(y, e, do(xI ))else �c(y|e, do(xI ))← 1return �c(y|e, do(xI ))

end

Algorithm 5. Computing-Aug.

Data: CaugDN N F , instance of interest Int, intervention do(xI )

Result: �c(I nt, do(xI ))begin∣∣∣∣∣∣∣∣∣

CaugDN N F |I nt, do(xI )← Condition(Caug

DN N F , I nt, do(xI ))Caug

�−DN N F ← map(CaugDN N F |I nt, do(xI ))

�c(I nt, do(xI ))← evaluate(Caug�−DN N F )

return �c(I nt, do(xI ))end

4.2. Augmented compiled possibilistic knowledge bases (Aug-PKB)

The starting point of the Augmented compiled possibilistic knowledge bases approach (denoted by Aug-PKB) is atransformation of the augmented possibilistic network into a possibilistic knowledge base which will be used to ensurethe phases depicted by Fig. 13.


Fig. 13. Principle of Aug-PKB method.

4.2.1. Encoding and compilation phaseThe transformation of �Gaug into an augmented possibilistic knowledge base �aug consists in associating a local

knowledge base for both DOI and each Xi ∈ V . Formally, this transformation is described by Definition 4.

Definition 4. Let �Gmin be a possibilistic causal network. Let do(xI ) be an intervention that forces the variable X I

to take the value xI . Let �Gaug be the augmented network and �aug be its possibilistic knowledge base expressed by

�aug =⋃

Xi∈V

�Xi ∪ �DOI (33)

where �Xi = {(¬xi ∨ ¬ui , ai ) : ai = 1−�(xi |ui ) � 0} and �DOI = {(¬doI , bi )}.

Proposition 4. Let �Gaug be an augmented network. Let �aug be its augmented possibilistic knowledge base usingDefinition 4. Let �� : �→ [0, 1] be the possibility distribution associated with �aug . Then

∀� ∈ �, �(�) = ��(�). (34)

where � is the possibility distribution associated with �Gaug using Eq. (7).

The proof follows immediately results given in [3]. Clearly, in Definition 4, �DOI and �Xi , ∀Xi ∈ V are handledseparately since these latters do not require the same strategy of encoding. Note also that we should incorporate themutually exclusive clauses with a necessity degree of 1 for each non-binary variable.

After the transition from a graphical to a logic-based representation, an encoding phase is established as inMut-PKB. However, the strategy of encoding do not remain the same since in Aug-PKB, it depends on the node


Table 8Set of propositional variables used in K�aug .

Variables Necessity degrees K�aug

A 0.6 A2

B 1 A1

0.3 A3

DOB (Not initialized) B1

(Not initialized) B2

(Not initialized) B3

Table 9The CNF encoding K�aug of �aug of �Gaug of Fig. 5.

Mutually exclusive clausesA (a1 ∨ a2 ∨ A1) ∧ (¬a1 ∨ ¬a2 ∨ A1)

B (b1 ∨ b2 ∨ A1) ∧ (¬b1 ∨ ¬b2 ∨ A1)

DOB (do(B − NoAct) ∨ do(b1) ∨ do(b2) ∨ A1)

∧(¬do(B − NoAct) ∨ ¬do(b1) ∨ A1)

∧(¬do(B − NoAct) ∨ ¬do(b2) ∨ A1)

∧(¬do(b1) ∨ ¬do(b2) ∨ A1)

Clauses of A(¬a2, 0.6) (¬a2 ∨ A2)

Clauses of B(¬a1 ∨ ¬b1 ∨ ¬do(b2), 1) (¬a1 ∨ ¬b1 ∨ ¬do(b2) ∨ A1)

(¬a2 ∨ ¬b1 ∨ ¬do(B − NoAct), 0.3) (¬a2 ∨ ¬b1 ∨ ¬do(B − NoAct) ∨ A3)

(¬a2 ∨ ¬b1 ∨ ¬do(b2), 1) (¬a2 ∨ ¬b1 ∨ ¬do(b2) ∨ A1)

(¬a1 ∨ ¬b2 ∨ ¬do(B − NoAct), 0.3) (¬a1 ∨ ¬b2 ∨ ¬do(B − NoAct) ∨ A3)

(¬a1 ∨ ¬b2 ∨ ¬do(b1), 1) (¬a1 ∨ ¬b2 ∨ ¬do(b1) ∨ A1)

(¬a2 ∨ ¬b2 ∨ ¬do(b1), 1) (¬a2 ∨ ¬b2 ∨ ¬do(b1) ∨ A1)

Clauses of DOB

¬do(B − NoAct) (¬do(B − NoAct) ∨ B1)

¬do(b1) (¬do(b1) ∨ B2)

¬do(b2) (¬do(b2) ∨ B3)

related to each possibilistic formula. In fact, possibilistic formulae i related to each Xi ∈ V are encoded usingpropositional variables Ai such that Ai may encode a set of equal parameters ai pertaining to any �Xi , ∀Xi ∈ V .However, each possibilistic formula related to DOI is encoded using a propositional variable Bi without consideringany necessity value. Interestingly enough, local structure is only exploited for non-extra nodes (i.e., ∀Xi ∈ V exceptDOI ) since necessity degrees associated for their possibilistic formulae are constant and do not depend on do(xI ).This not the case for parameters of the new extra node DOI which are unstable and depend on do(xI ). More formally,the CNF encoding of �aug is expressed by

K�aug = {(i ∨ Ai : (i , ai ) ∈ �Xi , ∀Xi ∈ V )} ∪ {(i ∨ Bi : (i , bi ) ∈ �DOI )} (35)

It is worth pointing out that we should use the strategy one variable per parameter per DOI ’s formula without takinginto account any numerical value bi , which is not the case for each Xi ∈ V . This is the key point that allows us tohandle both the non-intervention and the intervention cases in a representational framework.

Once we have encoded the possibilistic knowledge base, we should then compile it into DNNF. The augmentedcompiled base is denoted by K aug

c .


Fig. 14. The augmented compiled base K augc .

Example 18. Let us re-consider the network �Gaug of Fig. 5. The set of propositional variables used to encodenecessity degrees of the possibilistic knowledge base �aug associated with �Gaug is presented in Table 8.

The CNF encoding K�aug of the possibilistic base of �Gaug is presented in Table 9 s.t. B1, B2 and B3 encode DOB’sdegrees. Compiling K�aug into DNNF results in an augmented compiled base represented by Fig. 14.

4.2.2. Inference phaseOnce we have compiled the augmented possibilistic knowledge base, we should first update the necessity degree of

each propositional variable Bi associated with �DOI depending on do(xI ). In fact, we should assign the necessity degree1 to variables Bi corresponding to formulae of ¬do(I − NoAct) and ¬do(xi ), i � I (since �aug(do(I − NoAct)) = 0and �aug(do(xi )) = 0) and 0 to variables Bi corresponding to formula of ¬do(xI ) (since �aug(do(xI )) = 1). In whatfollows, the function update(B) will update necessity degrees of each Bi ∈ B depending on do(xI ).

Then, propositional variables Ai and Bi should be unified into one set of variables A = {A1, . . . , Ag}, where gis the new number of variables. This guarantees that the same necessity degree is not encoded using two differentpropositional variables. The function unify(A, B) will be used in order to ensure this condition. Finally, we shouldcompute efficiently the effect of the intervention do(xI ) and the observation e on the instance of interest y by applyingentailment and conditioning in an iterative manner as shown by Algorithm 6.

Example 19. Let us continue Example 18. First, we should update the necessity degree of DOB’s parameters. In fact,B1 and B3 will encode the degree 1 and B2 will encode the degree 0. After that we unify variables Ai and Bi such thateach B1 and B3 (resp. B2) appearing in K�aug is substituted by A1 (resp. A4). The new set of variables is the following:


A = {A1(1), A2(0.6), A3(0.3), A4(0)}. Finally, the computation process is ensured as follows:

1. Iteration 1: K augc �¬do(b1) ∨ A1 ⇒ (K aug

c |¬A1)�a1 ⇒ i ← i + 1.2. Iteration 2: K aug

c �¬do(b1) ∨ A2 ⇒ (K augc |¬A2)�a1 ⇒ StopCompute← true.

Hence, �c(a2|do(b1)) = 1−degree(2) = 1−0.6 = 0.4 where degree(2) designates the necessity degree associatedto A2.

4.2.3. Aug-PKB algorithmThe Aug-PKB method, outlined by Algorithm 6, guarantees the equivalence between possibility degrees computed

using Aug-PKB and the joint distribution. Formally, if we consider a possibilistic network �Gmin and an interventiondo(xI ) that force the variable X I to take the value xI , then we can show that for any y ∈ DX and e ∈ DE , wehave �c(y|e, do(xI )) = �(y|e, do(xI )) where �c(y|e, do(xI )) (resp. �(y|e, do(xI ))) is computed using Algorithm 6(resp. Proposition 1). In fact, encoding DOI ’s parameters using the strategy one variable per parameter and updatingthem after compilation depending on do(xI ) enable the computation of the effect of do(xI ) and e on y by applyingconditioning and entailment in an iterative manner.

Algorithm 6. Aug-PKB.

Data: �Gaug , instance of interest x, observation e, intervention do(xI )Result: �c(y|e, do(xI ))begin∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

% Encoding and compilation phase�aug ← transform(�Gaug)K�aug ← encode-PKB(�aug)K aug

c ← compile(K�aug )%Inference phase% A : the set o f proposi tional variables of �Xi ,∀Xi ∈ V% B : the set o f proposi tional variables of �DOI

B ← update(B)A = {A1, . . . , Ag} ← unify(A, B)i ← 1, StopCompute← false,�c(y|e, do(xI ))← 1 while (K aug

c �Ai∨¬e ∨ ¬do(xI )) and (i < g) and (StopCompute = f alse) do⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

K augc |¬Ai ← condition(K aug

c ,¬Ai )if K aug

c |¬Ai �¬y then⎢⎢⎢⎣ StopCompute ← trueLet degree(i) be the weight associated to Ai

�c(y|e, do(xI ))← 1− degree(i)else i ← i + 1


Proposition 5. Let K augc be the compiled base of an augmented network �Gaug . Let do(xI ) be an intervention that

forces the variable X I to take the value xI .

(i) ∀� ∈ �, we have

�c(�|do(xI )) = �a(�|do(xI )) (36)

where �c(�|do(xI )) (resp. �a(�|do(xI ))) is computed using Algorithm 6 (resp. Definition (1)).(ii) Let y be an instantiation of a variable Y ∈ V and e be an instantiation of a set of variables E ⊆ V . Then

�c(y|e, do(xI )) = �a(y|e, do(xI )) (37)

where �c(y|e, do(xI )) (resp. �a(y|e, do(xI ))) is computed using Algorithm 6 (resp. Definition (1)).


Proof. For any min-based possibilistic network �Gmin , from Eq. (6), we have �min(�) = ��(�).We obtain �a(�|do(xI )) = ��aug (�|do(xI )) since �Gaug= �Gmin ∪ DOI . Thus, Eq. (36) is established.

This result is relative to an interpretation �, to generalize it to any instantiation y of a variable Y ∈ V , we obtain

max��y

�c(�|do(xI )) = max��y

�c(�|do(xI )).

Thus, �c(y|do(xI )) = �a(y|do(xI )). When we deal with an observation e, we obtain

�c(y|e, do(xI )) = �a(y|e, do(xI )). �

5. Comparative and experimental study

We propose now to compare theoretically and experimentally the already developed methods for compiling min-based possibilistic causal networks (i.e., Mut-�-DNNF, Mut-PKB, Aug-�-DNNF and Aug-PKB). The objective behindthis study is to compare the reasoning and the representational interpretations of interventions under a compilationframework in order to highlight cases in which mutilation outperforms augmentation and vice versa.

5.1. Theoretical study

Both of Mut-�-DNNF and Mut-PKB exploit symbolic bases regardless of parameters. The main difference residesin the set of clauses associated with each parameter (whether a possibility or a necessity degree it is). Aug-�-DNNFand Aug-PKB handle an additional extra node while dealing with numerical values for all nodes except for the newextra node DOI . Let us consider the two extreme cases regarding the number of variables and clauses.

• Best case: each C�Ti having p values contains only two different degrees, namely 1 and another parameter (e.g.,the C�Ti relative to B in the network of Fig. 3). The behavior of our methods is as follows:◦ Mut-�-DNNF: generates p propositional variables.◦ Mut-PKB: produces p propositional variables and uses less clauses than Mut-�-DNNF.◦ Aug-�-DNNF: uses two propositional variables and k propositional variables for DOI .◦ Aug-PKB: generates one propositional variable and k propositional variables for DOI .We recall that k corresponds to the number of instances of DOI . This case is qualified to be the best one in termsof propositional variables and clauses.• Worst case: each C�Ti having p values contains p/2 degrees equal to 1 and p/2 different degrees (e.g., assigning

this C�Ti [1,0.8; 1,0.4;1,0.7;1,0.9] to the node B of Fig. 3 illustrates the worst case). Our methods acquit as follows:◦ Mut-�-DNNF: generates p propositional variables.◦ Mut-PKB: yields p propositional variables and uses less clauses than Mut-�-DNNF.◦ Aug-�-DNNF: uses p/2+ 1 propositional variables and k propositional variables for DOI .◦ Aug-PKB: generates p/2 propositional variables and k propositional variables for DOI .

These two extreme cases allow us to deduce a key theoretical result regarding the number of variables and clausesin Aug-PKB and Aug-�-DNNF described in the following proposition:

Proposition 6. Aug-PKB is more compact (uses less propositional variables and clauses) than Aug-�-DNNF.

Proof. The proof is immediate:

1. Propositional variables:• Aug-PKB uses the strategy one variable per equal parameters per possibilistic knowledge base.• Aug-�-DNNF uses the strategy one variable per equal parameters per �Gaug’s C�Ti .⇒ Aug-PKB uses less propositional variables than Aug-�-DNNF.

2. CNF clauses:• Aug-PKB associates one clause (Eq. (35)) per Ai ,


• Aug-�-DNNF associates a set of clauses (Eqs. (15)–(17)) per �xi |ui .⇒ Aug-PKB uses less clauses than Aug-�-DNNF. �

5.2. Empirical study

In this section, we study mutilated-based approaches and augmented-based approaches dealing with interventionsunder a compilation framework from an experimental point of view. As comparison criteria, we use the number ofpropositional variables of CNF encodings, the number of clauses of CNF encodings and the number of edges ofcompiled bases.

As possibilistic networks have a graphical component and a numerical one, then it is judicious to specify which kindof networks to use during the experimental process. We have outlined in Section 3 that mutilated-based approachesare not sensitive to the number of interventions since we mutilate the compiled base instead of the initial possibilisticnetwork, this is not the case of augmented-based approaches. Our idea consists in generating a random possibilisticnetwork by setting the number of nodes to 50, variables cardinality to 2 and 3 and the maximum number of parentsper node to 3 in the case of mutilated-based approaches. For augmented-based approaches that depend strongly on thenumber of interventions, we generate augmented possibilistic networks by varying the number of interventions from 1to 50 and the redundancy percentage of equal parameters per conditional possibility table. This percentage is denotedby E PC�Ti ranging from 0 to 100 as follows: E PC�Ti = {0, 10, 30, 50, 70, 100} s.t. E PC�Ti = 50% means thata possibility degree appears in 50% of the possibility degrees different from 1 in the current C�Ti . Such parameterrepresents the amount of local structure affected per table.

Table 10 depicts experimental results by considering one intervention, then 10, 20, 30, 40 and 50 interventions.When the number of occurred interventions is not well-known in an instant t, the price to be paid is to consider 50 extranodes and compile a network composed of 100 nodes (50 network variables and 50 extra nodes). We call this particularsituation the extreme case. Let us now study in depth the results of Table 10 via the following pairwise comparison:

• Mut-�-DNNF vs Mut-PKB: It is well attested that both of Mut-�-DNNF and Mut-PKB do not exploit parametersvalues. This means that a propositional variable is associated for each parameter, regardless of its numerical value.As shown in rows 1 and 2 of Table 10, Mut-�-DNNF, which has a number of variables and clauses higher than thoseof Mut-PKB, gives rise to more compact compiled bases having less edges. This proves that dealing with symbolicbases using only right-side clauses makes the resulting knowledge base harder to compile. Consequently, we canconclude that both of left-side clauses and right-side clauses play a primary role in the compilation process sinceeach parameter variable is encoded using a logical equivalence and therefore cannot imply instance indicators otherthan those compatible with variables instantiation.• Mut-�-DNNF vs Aug-�-DNNF: From Fig. 15, it is clear that Aug-�-DNNF outperforms Mut-�-DNNF even in

the extreme case in which an extra node DOi is associated for each network variable. This means that even ifAug-�-DNNF deals with additional extra nodes DOI , it produces more compact compiled bases having less edgesthan those of Mut-�-DNNF. In the extreme case, a key result that should be highlighted is that augmenting a networkby 50 extra nodes and compiling 100 nodes with local structure gives better results than compiling 50 nodes withouttaking into account any numerical value. This means that increasing the size of possibility distributions of involvedvariables using parameters 1 and 0 and adding nodes DOI which are encoded without taking into considerationany numerical value do not degrade the quality of compiled bases, in comparison with Mut-�-DNNF. Interestinglyenough, local structure plays a leading role in augmentation.By paying more attention on edges of Aug-�-DNNF, we can see that the number of edges varies from 11,892 whenwe deal with a unique intervention to 19,995 in the case of 50 interventions. This means that given the compiledbase of a network augmented by a unique intervention, it is easy to obtain the compiled base of the extreme case.To strengthen this result, we have established another experiment for networks composed of 100 nodes. We haveobtained compiled bases composed of 104,339 (resp. 127,821) edges when the number of interventions is equal to1 (resp. 100). This experiment serves to demonstrate that augmentation is linear with respect to the compiled basesize of a network augmented by a unique intervention.Note that we only deal with Aug-�-DNNF since it turns out that Mut-�-DNNF is the most efficient and compactmethod.Ten further experiments on different DAG structures have been established and the same behavior and conclusions

have been observed.


Table 10Mut-�-DNNF vs Mut-PKB vs Aug-�-DNNF (better values are in bold).

Method Nb interventions EPCPTi Variables Clauses Edges

Mut-�-DNNF – 0 · · · 100 734 2573 29,943

Mut-PKB – 0 · · · 100 478 502 196,179

Aug-�-DNNF 1 0 507 1687 11,89210 453 1383 927230 357 1065 624650 297 942 426670 285 896 3986100 238 856 1571

Aug-�-DNNF 10 0 571 2414 14,08010 560 2326 13,26530 484 2032 10,62750 445 1953 995970 437 1922 9795100 399 1886 2743

Aug-�-DNNF 20 0 631 2718 15,05810 620 2630 14,24330 556 2392 11,78350 523 2331 11,13770 519 2316 10,975100 489 2296 3826

Aug-�-DNNF 30 0 691 3178 16,56910 660 3040 15,18730 654 3015 15,01450 628 2975 13,85770 611 2745 13,184100 591 2737 4707

Aug-�-DNNF 40 0 785 3637 18,74210 774 3549 18,00630 759 3521 17,31350 751 3504 16,80370 748 3492 16,625100 738 3476 11,521

Aug-�-DNNF 50 0 849 4054 19,99510 838 3970 19,18030 824 3952 18,43950 819 3938 18,12670 801 3872 18,006100 771 3764 12,682

6. Conclusion

The concept of intervention is of major importance to have a complete and coherent causal analysis in a dynamicframework. It is an external event that forces some variables to have some particular values [24]. There are two waysto handle interventions using possibilistic causal networks [5]. The most intuitive one, called mutilation, consists inignoring relations between the intervened variable and its direct causes. The rest of the network remains intact. Hence,


Fig. 15. Mut-�-DNNF vs Aug-�-DNNF.

causal inference resides in applying the inference algorithm to the mutilated possibilistic network. A different butequivalent approach to represent intervention in possibilistic causal networks, called augmentation, is to consider it asan additional variable into the system.

Our contribution in this paper is to extend our inference approaches proposed for min-based possibilistic net-works and propose mutilated-based approaches and augmented-based approaches aiming to compute the effect of bothobservations and interventions in an efficient manner in possibilistic causal networks. In fact, mutilated-based meth-ods require the mutilation of symbolic compiled bases, while augmented-based methods involve encoding augmentednetworks while ignoring numerical values of new extra nodes. Mutilated-based approaches are not sensitive to thenumber of interventions since the compiled base is mutilated instead of the initial possibilistic network, which enablesthe handling of a set of interventions without the need for re-compiling the network each time an intervention occurs.This is not the case of augmented-based approaches since the augmented network is compiled after performing the setof interventions. Our results show that augmented-based approaches outperform mutilated-based approaches even inthe extreme case in which an extra node is associated for each network variable.

An interesting future work is to explore the issue of fault detection from an applicative point of view. In fact,in complex systems, fast detection of faults and accurate diagnosis of root cause of failure is a crucial aspect ofoperational safety and considered as a critical factor in reducing system down-time. We are thinking about the useof compilation in qualitative models which have proved their efficiency by providing a realistic representation andaccurate diagnosis in [20].

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