Probabilistic Argumentation and Decision Systemsarb1/publications/ps/AB03.pdfProbabilistic Argumentation and Decision Systems B. Anrig & D. Baziukaite Internal working paper no 03-03

Department of InformaticsUniversity of Fribourg

Probabilistic Argumentationand Decision Systems

B. Anrig & D. Baziukaite

Internal working paper no 03-03March 2003

Department of InformaticsUniversity

Rue Faucigny 2CH-1700 Fribourg

(Switzerland)

phone : +41 26 300 83 21fax : +41 26 300 97 [email protected]

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Probabilistic Argumentationand Decision System ∗

Bernhard Anrig Dalia Baziukaite

Department of Informatics Computer Science DepartmentUniversity of Fribourg Klaipeda University

CH–1700 Fribourg, Switzerland LT–5808 Klaipeda, [email protected] [email protected]

March 2003

The concept of probabilistic argumentation systems PAS is restricted to twotypes of variables: Assumptions, which model the uncertain part of the know-ledge, and propositions, which model the rest of the information. Here, weintroduce a third kind into PAS: So-called decision variables. This new kindallows to describe the decisions a user can make to react on some state ofthe system. Such a decision allows then to possibly reach a certain goal stateof the systems. Further, we present an algorithm which exploits the specialstructure of PAS with decision variables.

∗Some parts of this work have been published in (Anrig & Kohlas, 2002c).

2 Contents

Contents

1 Introduction 3

2 An Example 4

3 Argumentation Systems 6

3.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Propositional Argumentation Systems . . . . . . . . . . . . . . . . . . . . 8

3.3 Logical Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.4 Probabilistic Argumentation Systems . . . . . . . . . . . . . . . . . . . . . 11

4 Argumentation and Decision Systems 12

4.1 Propositional Argumentation and Decision Systems . . . . . . . . . . . . . 12

4.2 Decisions depending on System States . . . . . . . . . . . . . . . . . . . . 15

4.3 Logical Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.4 Probabilistic Argumentation and Decision Systems . . . . . . . . . . . . . 17

4.5 Choosing a Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Computation 18

5.1 Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.2 Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.3 Improvements in the Computation . . . . . . . . . . . . . . . . . . . . . . 22

5.4 Local Computations and Approximation Techniques . . . . . . . . . . . . 25

6 An Example 26

7 Conclusion 28

References 29

3

1 Introduction

The concept of probabilistic argumentation systems PAS has been used for dealing withproblems in different contexts. Different examples from a wide spectrum have beentreated (Haenni, 1996, Anrig et al., 1999, Haenni et al., 2000b, Kohlas et al., 2000).Lately, Anrig & Kohlas (2002a) have considered the relation between reliability (Kohlas,1987, Beichelt, 1993) and model-based diagnostics (Anrig, 2000, Kohlas et al., 1998,Anrig et al., 1996) on the basis of PAS. In this environment, the attention was broughtto a small but very interesting example. Although the theory of PAS was able to computethe required result for this example, this was only possible after a quite sophisticatedmodelization of the knowledge. Yet it turns out that there is a more elegant and naturalway.

Inspired from this example, we have extended the concept of PAS with decision variables.Consider the situation where we have modeled a system using a PAS (P,Σ) and theelements of the set A ⊆ P are the indicator variables (with values ok and not ok) ofthe components of the system. A requirement δ describes the expected behaviour of thesystem, typically its desired input-output relations. We are now interested in computingthose system states which allow to deduce the requirement δ. Anrig & Kohlas (2002a)have shown that those system states are just the supporting scenarios of δ. This is awell known problem in PAS.

Now consider that given a system state, the user can himself set the values of somedecision variables in order to guarantee the requirement. The interesting system statesare now those for which the user can find at least one settings of “his” variables underwhich the requirement imposed on the system can be fulfilled.

This can be seen as a game with two players, say nature against a user. Nature makesthe first move and the user the second. If the requirement is fulfilled, the user wins (asthe system is up), otherwise he looses.

For computing results for this generalized version of PAS, the original methods of a PAScan be applied and after a post-processing, they can be used for the generalized PAS.Yet this proves to be inefficient and a new approach is proposed.

The structure of the paper is as follows: In the next section an introductory exampleis presented. In Section 3 the general theory of argumentation systems is introduced.The generalization to argumentation and decision systems is then done in Section 4.This generalization is clearly connected to diagnostic and reliability (cf. Section 7).Section 5 discusses then the computational theory and introduces a new algorithm. Alarger example is treated in Section 6.

4 2 An Example

2 An Example

The following example — treated in (Anrig & Kohlas, 2002a) — has been the startingpoint of our ideas. We present it here to clarify these ideas.

Example 1: Availability of Energy Distribution Systems

The function of an electrical power distribution system as depicted in Fig. 1 is to supplya high-tension line L3 from one of two incoming high-tension lines L1 and L2 over thebus A1 (Frey & Reichert, 1973, Kohlas, 1987). The lines L1 , L2 , and L3 are protectedby corresponding high-tension switches S1 , S2 , and S3 . In case of a broken switch itis possible to redirect the electric current on a second bus A2 which is protected byanother high-tension switch S4 . The corresponding switches T1 , T2 , and T3 can onlybe operated under no tension.

Figure 1: Example of an energy distribution system.

The energy distribution system is considered to be operating if L3 is linked to L1 or L2over at least two protecting high-tension switches Si (i = 1, . . . , 4). The problem is todetermine the availability of an operating system.

The lines are modeled by the corresponding binary propositions with the same name,e.g. L1 means that the line L1 is under tension, whereas ¬L1 means that it is not. A

5

switch can be closed (e.g. S1 ) or open (e.g. ¬S1 ) and furthermore it can be in its correctworking mode (e.g. okS1 ) or faulty (e.g. ¬okS1 ).

For some of the variables we have probabilistic information available, namely for theworking modes of the components:

p(okSi) = 0.95, i = 1, 2, 3, 4,

p(okTi) = 0.97, i = 1, 2, 3.

p(okAi) = 0.99, i = 1, 2. (1)

The energy can pass through a switch only if the switch is closed. If a switch is notintact, then it cannot be closed. So every switch can be modeled by the two logicalformulas, where the second formula has always the form if switch x is closed, then thereis power on both lines or points specified on the right-hand side or on none of them:

¬okS1 → ¬S1 , S1 → (L1 ↔ P1 ),¬okT1 → ¬T1 , T1 → (L1 ↔ P5 ),¬okS2 → ¬S2 , S2 → (L2 ↔ P3 ),¬okT2 → ¬T2 , T2 → (L2 ↔ P7 ),¬okS3 → ¬S3 , S3 → (P2 ↔ L3 ),¬okT3 → ¬T3 , T3 → (P6 ↔ L3 ),¬okS4 → ¬S4 , S4 → (P4 ↔ P8 ).

(2)

The different connection points to the buses are modeled by the propositions P1 to P8 ; ifthe bus is working correctly, then either there is power on all of the respective connectionpoints or on none of them:

okA1 →((P1 ∧ P2 ∧ P3 ∧ P4 ) ∨ (¬P1 ∧ ¬P2 ∧ ¬P3 ∧ ¬P4 )

)(3)

okA2 →((P5 ∧ P6 ∧ P7 ∧ P8 ) ∨ (¬P5 ∧ ¬P6 ∧ ¬P7 ∧ ¬P8 )

)(4)

An important part of the model is the fact that only one of T1 , T2 and T3 is allowedto be closed at the same time, because otherwise there is no protection between thecorresponding lines:

¬(T1 ∧ T2 ), ¬(T1 ∧ T3 ), ¬(T2 ∧ T3 ). (5)

Until now, we have specified different things:

• propositions whose value is set by nature:

A = {okS1 , okS2 , okS3 , okS4 , okT1 , okT2 , okT3 , okA1 , okA2} (6)

• propositions which are (or may be) set by the user (under some restrictions):

D = {S1 ,S2 ,S3 ,S4 ,T1 ,T2 ,T3} (7)

6 3 Argumentation Systems

• other propositions: U = {P1 , . . . ,P8}

• knowledge about the system in form of logical sentences

Consider now the requirement that given power at the inputs lines L1 and L2 the line L3is really supplied. We will formulate this by the logical formula δ = L1 ∧ L2 → L3 .

There are two main problems: The first one is to compute a (logical) description of thosesystems states, where the user can react so that the requirement δ is fulfilled. Second,for every system state we want to have a description of possible decisions of the user(so that the requirement δ is fulfilled). In the rest of this paper we will show how thesolutions to both problems can be computed.

Clearly, in reliability theory this very example has been treated successfully: This meansthat for example in (Kohlas, 1987), the path sets have been determined by hand and onlythen based on this information, the reliability function has been computed. Here, wewill show how the structure function can be derived directly from the logical descriptionof the system.

3 Argumentation Systems

Probabilistic assumption-based argumentation systems have been developed as generalformalisms for expressing uncertain and partial knowledge and information in artificialintelligence. They combine in an original way logic and probability. Logic is used toderive arguments and probability serves to compute the reliability of these arguments.These systems can be used for example for model-based diagnostics as has been shownin (Anrig, 2000, Kohlas et al., 1998).

Argumentation systems can be based on different logics. For the sake of simplicity welimit ourselves here to the case of propositional logic. In this section we give a shortintroduction into propositional probabilistic argumentation systems. For a more detailedpresentation of the subject we refer to (Haenni et al., 2000b, Kohlas et al., 1998, Anriget al., 1999). We remark also that such systems have been implemented in a system calledABEL, Assumption-Based Evidential Language (Anrig et al., 1999), which is availableon the internet (cf. Haenni et al., 2000a), and which can be used to compute results ofthe examples of this paper as well as of other problems.

3.1 Basic Definitions

Propositional logic deals with declarative statements that can either be true or false.Such statements are called propositions. Let P = {p1, . . . , pn} be a finite set of proposi-tions. The symbols pi ∈ P together with > (tautology) and ⊥ (falsity), are called atomsor atomic formulas. Compound formulas are built by the following syntactic rules:

3.1 Basic Definitions 7

(1) atoms;

(2) if γ is a formula, then (¬γ) is a formula;

(3) if γ and δ are formulas, then (γ ∧ δ), (γ ∨ δ), (γ → δ), and (γ ↔ δ) are formulas.

Often, unnecessary parentheses can be omitted, e.g. γ∧δ instead of (γ∧δ). Furthermore,by assigning priority in decreasing order to ¬, ∧, ∨, →, some other parentheses can beeliminated, e.g. γ → δ ∧ λ instead of γ → (δ ∧ λ). The set LP of all formulas generatedby the above recursive rules is called propositional language over P . A formula γ ∈ LP

is also called a propositional sentence.

A literal is either an atom pi or the negation of an atom ¬pi. A term is a conjunctionof literals, and a clause is a disjunction of literals. Usually we will consider only properterms and clauses, where every atom occurs at most once in a term or a clause, eitherpositive of negative, and neither > nor ⊥ does occur. Additionally, we say that > isa (empty) proper term and ⊥ a (empty) proper clause. The set of all proper terms isdenoted by CP ⊆ LP , and the set of all proper clauses by DP ⊆ LP .

A clause (and similarly a term) is interpreted also as a set of its literals; this allows tosimplify the notations and algorithms. So for example for terms α, α′ ∈ LP , we haveα ⊆ α′ if and only if α′ = α∧β for some β ∈ LP . As a special case, ⊥ ⊆ α ⊆ > for everyclause α and ⊥ ⊇ β ⊇ > for every term β. If we want to emphasize the interpretationas a set, we write lit(β). Note that lit(¬β) for a term β is the set of literals of the clause¬β. For a set of formulas X we define ¬X = {¬ζ : ζ ∈ X}.

The meaning of a propositional sentence is obtained by assigning truth values true andfalse (represented by “1” and “0” respectively) to the propositions. The truth value ofcompound formulas can then recursively be obtained according to Table 1.

γ δ ⊥ > ¬γ γ ∧ δ γ ∨ δ γ → δ γ ↔ δ

0 0 0 1 1 0 0 1 10 1 0 1 1 0 1 1 01 0 0 1 0 0 1 0 01 1 0 1 0 1 1 1 1

Table 1: Truth values of compound formulas.

An assignment of truth values to the elements of a set P = {p1, . . . , pn} is called inter-pretation relative to P . NP = {0, 1}n denotes the set of all 2n different interpretations.Every interpretation x ∈ NP can be seen as a point or a vector x = (x1, . . . , xn) in then-dimensional binary product space NP . Each xi ∈ {0, 1} denotes a binary variable thatis associated with the corresponding proposition pi.


Let x be an arbitrary interpretation relative to P . If (according to Table 1) γ ∈ LP

evaluates to 1, then x is called a model of γ. Otherwise, x is a counter-model of γ.The set of all models of γ is denoted by NP (γ) ⊆ NP . If NP (γ) = ∅, then γ is calledunsatisfiable. Otherwise, it is called satisfiable.

The notions of models and counter-models links propositional logic to the Boolean al-gebra of subsets of interpretations:

(1) NP (⊥) = ∅,

(2) NP (>) = NP ,

(3) NP (¬γ) = NP −NP (γ),

(4) NP (γ ∧ δ) = NP (γ) ∩NP (δ),

(5) NP (γ ∨ δ) = NP (γ) ∪NP (δ).

A propositional sentence γ entails another sentence δ (denoted by γ |= δ) if and only ifNP (γ) ⊆ NP (δ). In that case, δ is also called a logical consequence of γ. For example,γ ∧ δ |= γ ∨ δ. Sometimes, it is convenient to write x |= γ instead of x ∈ NP (γ).Also we write γ |= ⊥ if γ is not satisfiable. Furthermore, two sentences γ and δ arelogically equivalent (denoted by γ ≡ δ), if and only if NP (γ) = NP (δ). For example,γ → δ ≡ ¬γ ∨ δ. Note that logically equivalent sentences represent exactly the sameinformation.

3.2 Propositional Argumentation Systems

Consider a finite set P = {p1, p2, . . . , pm} of propositions. We consider a fixed set offormulas Σ ⊆ LP called the knowledge base, which models the information available;sets of formulas are interpreted conjunctively, i.e. Σ =

∧{ξ ∈ Σ}. We assume that this

knowledge base is satisfiable.

Definition 1 A propositional argumentation system PAS is a tuple (P,Σ) whereP is a set of literals and Σ ⊆ LP a set of formulas.

Note that in the present formulation (and opposed to previous work on PAS, e.g. Haenniet al. 2000b, Anrig et al. 1999), we do not explicitly single out the assumptions, but allowthem to vary from one situation to another (cf. below). This does not contradict theinitial idea about argumentation system. Often, it is clear from the beginning whichpropositions are assumptions and which ones are not. Yet from the users perspective itis interesting to switch the type of variables from assumptions to propositions or viceversa in the development of a model. Further we will see in Section 4 that this can alsobe interesting for answering even more general questions.

3.3 Logical Representation 9

Definition 2 Consider a PAS (P,Σ) and a subset of assumptions A ⊆ P . For ahypothesis h ∈ LP we define

Inconsistent Scenarios: CSsA(Σ) = {s ∈ NA : s,Σ |= ⊥}

Quasi-Support Set: QSsA(h, Σ) = {s ∈ NA : s,Σ |= h}

Support Set: SPsA(h, Σ) = QSs

A(h, Σ)−QSsA(⊥,Σ)

Plausible Set: PLsA(h, Σ) = NA −QSs

A(¬h, Σ)

The elements of NA are called scenarios or system states. A scenario represents aspecification of all values of the assumptions in A.

Inconsistent scenarios are in contradiction with the knowledge base and therefore to beconsidered as excluded by the knowledge. CSs

A is also called conflict set. Supportingscenarios for a formula h are scenarios, which, together with the knowledge base imply hand are consistent with the knowledge. So, under a supporting scenario, the hypothesis his true. Possible scenarios for h are scenarios, which do not imply ¬h and thereby donot refute h. Quasi-supporting scenarios for h are the union of supporting scenarios andinconsistent scenarios. They are important especially for technical reasons.

3.3 Logical Representation

Scenarios are the basic concepts of assumption-based reasoning. However, sets of in-consistent, quasi-supporting, supporting and possible scenarios may become very large.Therefore, more economical, logical representations of these sets are needed. For thispurpose, the following concepts are defined:

Definition 3 Consider a PAS (P,Σ), a set of assumptions A ⊆ P and a hypothesish ∈ LP , then we call

Conflicts: α ∈ CA such that NA(α) ⊆ CSsA(Σ)

Quasi-Supporting Argument for h: α ∈ CA such that NA(α) ⊆ QSsA(h, Σ)

Supporting Argument for h: α ∈ CA such that NA(α) ⊆ SPsA(h, Σ)

Possible Argument for h: α ∈ CA such that NA(α) ⊆ PLsA(h, Σ)

Conflicts (or inconsistent terms) represent sets of literals of assumptions, which arein contradiction to the knowledge base Σ. Supporting arguments for a hypothesis hrepresent sets of literals of assumptions which, together with the knowledge base Σ,are sufficient to guarantee the truth of h, whereas possible arguments represent sets ofliterals of assumptions, which are sufficient to exclude the guarantee of the falsity of h.


We define QSA(h, Σ), SPA(h, Σ), and PLA(h, Σ) = SPAc(¬h, Σ) to be the sets of quasi-

supporting, supporting and possible arguments for h, respectively. QS(⊥,Σ) denotesthen the set of conflicts. These sets are all upward closed, e.g. if α ∈ SPA(h, Σ) thenevery α′ ⊃ α is also in SPA(h, Σ).

Theorem 4 (Haenni et al., 2000b)

QSA(h, Σ) = {α ∈ CA : α, Σ |= h}

SPA(h, Σ) =

{α ∈ CA :

α, Σ |= hα′,Σ 6|= ⊥ for every α′ ∈ CA, with α′ ⊇ α,

}PLA(h, Σ) = {α ∈ CA : α′,Σ 6|= ¬h for every α′ ∈ CA, with α′ ⊇ α} (8)

A conjunction α is a minimal element of a set of conjunctions if for every conjunction α′

of this set satisfying α′ ⊆ α we have α = α′. According to the theorem above the sets ofarguments are already determined by their minimal elements. We denote by µQSA(h, Σ),µSPA(h, Σ) and µPLA(h, Σ) the respective sets of minimal quasi-supporting, supportingand possible arguments. In general for an upward closed set S of conjunctions (orclauses), µS denotes the set of minimal elements of S.

Definition 5 Consider a PAS (P,Σ), a set of assumptions A ⊆ P and a hypothesish ∈ LP , then we define

Conflict: csA(Σ) =∨

α∈µQSA(⊥,Σ)

α

Quasi-Support of h: qsA(h, Σ) =∨

α∈µQSA(h,Σ)

α

Support of h: spA(h, Σ) =∨

α∈µSPA(h,Σ)

α

Possibility of h: plA(h, Σ) =∨

α∈µPLA(h,Σ)

α

Using these definition, we have NA(csA(Σ)) = CSsA(h, Σ), NA(qsA(h, Σ)) = QSs

A(h, Σ),NA(spA(h, Σ)) = SPs

A(h, Σ) and NA(plA(h, Σ)) = PLsA(h, Σ).

Clearly, these logical formulas are only representants for the corresponding sets of scen-arios, hence for example any formula ζ ∈ LA which is logically equivalent to spA(h, Σ)is as interesting as spA(h, Σ) itself. Clearly, a compact representant of the equivalenceclass is of interest.

3.4 Probabilistic Argumentation Systems 11

3.4 Probabilistic Argumentation Systems

On top of the structure of a propositional argumentation systems, we may easily adda probability structure. We assume that there is a probability p(ai) = pi for everyassumption ai ∈ A given. Assuming stochastic independence between assumptions, ascenario s = (s1, . . . , sn) gets the probability

p(s) =n∏

i=1

psii (1− pi)1−si . (9)

This induces a probability measure p on the language LA,

p(f) =∑

s∈NA(f)

p(s) (10)

for f ∈ LA. A tuple (Σ, A, P,Π) with Π = (p1, . . . , pn) is then called a probabilistic(propositional) argumentation system PAS.

More general, any probability measure p on LA (or more rigorously on NA) can be used.The local structure described above is just a very convenient but also frequent specialcase, yet there are other local structures which can be used.

Once we have such a probability structure on top of a propositional argumentationsystem, we can exploit it to compute likelihoods (or in fact, reliabilities) of supportingand possible arguments for hypotheses h. First, we note, that the knowledge base Σimposes that we eliminate the inconsistent scenarios and condition the probability onthe consistent ones. In other words, Σ is an event that restricts the possible scenariosto the set NA − CSs

A(Σ), hence their probability has to be conditioned on the event Σ.This conditional probability is defined by

p′(s) =p(s)

1− p(qsA(⊥,Σ)).

for consistent scenarios s. p(qsA(h, Σ)) = dqsA(h) is the so-called degree of quasi-supportfor h. Now, the degree of support dspA for hypotheses h is defined by

dspA(h) = p′(spA(h, Σ)) =p(spA(h, Σ))

1− p(qsA(⊥,Σ))

=dqsA(h, Σ)− dqsA(⊥,Σ)

1− dqsA(⊥,Σ).

This result explains the technical importance of quasi-support. It is sufficient to computedegrees of quasi-supports. Further, we obtain the degree of plausibility of h,

dplA(h) = p′(plA(h, Σ)) =p(plA(h, Σ))

1− dqsA(⊥,Σ)=

1− dqsA(¬h, Σ)1− dqsA(⊥,Σ)

= 1− dspA(¬h).

12 4 Argumentation and Decision Systems

For some information about computing these probabilities see Section 5.4.

We remark, that the degree of quasi-support dqsA(h) of h corresponds in fact to unnor-malized belief, the degree of support to normalized belief in the Dempster-Shafer theoryof evidence (Shafer, 1976, Kohlas & Monney, 1995, Haenni et al., 2000b).

4 Argumentation and Decision Systems

The concept of a PAS allows that the user can compute symbolical and numerical argu-ments for and against any hypotheses he likes, but it does not allow to include actionsof the user. The introductory Example 1 is a typical situation where in some cases addi-tional user interaction allows to fulfill a specified requirement. Hence we will incorporatea limited range of user actions (or better: possible user actions) into our system. In thesequel, we will especially look at the situation when the user makes a reaction withrespect to some previous action of “nature”, in the sense that given some componentsare functioning, he selects which ones of these components should do the job.

4.1 Propositional Argumentation and Decision Systems

Consider a PAS (P,Σ) and a set of assumptions A ⊆ P . A subset of variables D ⊆ P issingled out. This set D contains all those variables which can be set by the user. Theidea is that given the knowledge base Σ, the variables specified by A are set withoutany possible intervention of the user, i.e. these are the variables set by nature or anenemy. The user cannot influence them. Yet after these variables are set, the usercan set the values of the variables specified by D and try to deduce therewith somehypothesis h ∈ LP . More precisely: Given the setting of the variables in D, is therealways a possibility for the user to select a setting of the variables in D so that togetherwith the knowledge Σ these settings are not contradictory and imply the hypothesis h?

Some definitions will help us to formalize these ideas:

Definition 6 A propositional argumentation and decision system PADS is atuple (P,D,Σ) with D ⊆ P , and Σ ⊆ LP .

Definition 7 A scenario d ∈ ND is called a user decision.

We assume that the propositions in D can directly be influenced by the user, whereas theother propositions can not. Especially the assumptions, i.e. a specified set A ⊆ P −D,cannot be influenced by the user but nature does influence them. Here we will considerthe situation where A ∩ D = ∅, i.e. there are no propositions which can be influenceddirectly by nature as well as by the user.

4.1 Propositional Argumentation and Decision Systems 13

Formalizing the ideas from above, we are now interested in the following scenarios givena hypothesis h ∈ LP and a set of assumptions A ⊆ P −D:

{s ∈ NA : There is a d ∈ ND s.t. s,d,Σ |= h and s,d,Σ 6|= ⊥} (11)

In the special case when there are no decidable variables, D = ∅, then we make theconvention that the set of scenarios N∅ consists of one element �, for which �, ζ ≡ ζ forany formula ζ. Then the problem above is just the problem of computing the supportSPs

A(h, Σ) of the hypothesis h (considered as an ordinary hypothesis). Therefore were-use the same notation for the set (11):

Definition 8 The set of supporting scenarios SPsA(h, Σ; D) of a hypothesis h with

respect to the decision variables in D is defined as

SPsA(h, Σ; D) = {s ∈ NA : There is a d ∈ ND s.t. s,d,Σ |= h and s,d,Σ 6|= ⊥}

For every supporting scenario, i.e. for every setting of the variables in A chosen bynature, the user can find at least one setting of the variables in D, i.e. those which canbe changed by him, so that the knowledge based together with these settings allow todeduce the hypothesis h but are at the same time not contradictory.

Lemma 9 SPsA(h, Σ; ∅) = SPs

A(h, Σ).

Proof SPsA(h, Σ; ∅) = {s ∈ NA : There is a d ∈ N∅ s.t. s,d,Σ |= h and s,d,Σ 6|= ⊥} =

{s ∈ NA : s,Σ |= h and s,Σ 6|= ⊥} = SPsA(h, Σ) as N∅ = {�} and �, s,Σ ≡ s,Σ. ut

Example 2: (Cont. of Example 1)

The information modeled in Example 1 is already in the form of a PADS, namely wehave defined the assumptions

A = {okS1 , okS2 , okS3 , okS4 , okT1 , okT2 , okT3 , okA1 , okA2} (12)

and the decision variables

D = {S1 ,S2 ,S3 ,S4 ,T1 ,T2 ,T3 ,A1 ,A2} (13)

There are some further variables U = {P1 , . . . ,P8}. The knowledge about the systemhas been modeled using several logical formulas, which, together, specify Σ:

Σ =

¬okS1 → ¬S1 , S1 → (L1 ↔ P1 ),¬okT1 → ¬T1 , T1 → (L1 ↔ P5 ),¬okS2 → ¬S2 , S2 → (L2 ↔ P3 ),¬okT2 → ¬T2 , T2 → (L2 ↔ P7 ),¬okS3 → ¬S3 , S3 → (P2 ↔ L3 ),¬okT3 → ¬T3 , T3 → (P6 ↔ L3 ),¬okS4 → ¬S4 , S4 → (P4 ↔ P8 ),okA1 → ((P1 ∧ P2 ∧ P3 ∧ P4 ) ∨ (¬P1 ∧ ¬P2 ∧ ¬P3 ∧ ¬P4 )),okA2 → ((P5 ∧ P6 ∧ P7 ∧ P8 ) ∨ (¬P5 ∧ ¬P6 ∧ ¬P7 ∧ ¬P8 )),¬(T1 ∧ T2 ), ¬(T1 ∧ T3 ), ¬(T2 ∧ T3 ).


These ingredients form the PADS (P,D,Σ) with P = D ∪ U ∪A.

For the hypothesis δ = L1 ∧ L2 → L3 the following supporting scenarios can be com-puted:

SPsA(δ,Σ; D) =

(1, ∗, 1, ∗, ∗, ∗, ∗, 1, ∗),(∗, 1, 1, ∗, ∗, ∗, ∗, 1, ∗),(1, ∗, ∗, 1, ∗, ∗, 1, 1, 1),(∗, 1, ∗, 1, ∗, ∗, 1, 1, 1),(∗, ∗, 1, 1, 1, ∗, ∗, 1, 1),(∗, ∗, 1, 1, ∗, 1, ∗, 1, 1)

(14)

where the assumptions are ordered as in (12) and a star “∗” signifies that either a 0 ora 1 can be inserted in the place. This gives finally a total of 170 supporting scenarios.For example, the scenario s = (1, 0, 1, 0, 0, 0, 0, 1, 0) ∈ SPs

A(δ,Σ; D) states that only theswitches S1 and S3 are working correctly together with the bus A1 and indeed thisscenario allows the user to take a specific action (namely close the switches which workcorrectly) and the requirement δ can be fulfilled as the power can go securely throughS1 on A1 and from there through S3 to L3 .

Lemma 10 SPsA(h, Σ; D) is monotone in D, i.e. for D′ ⊆ D we have

SPsA(h, Σ; D′) ⊆ SPs

A(h, Σ; D). (15)

Proof Let s ∈ SPsA(h, Σ; D′), i.e. there is a d′ ∈ ND′ with s,d′,Σ |= h and s,d′,Σ 6|= ⊥.

Now consider D′′ = D −D′. If D′′ = ∅ the lemma holds trivially. Otherwise, considerN ′ = NP (s,d′,Σ). As s,d′,Σ 6|= ⊥, the set N ′ is not empty and also N ′↓D′′ 6= ∅. (SeeSection 5.1 for the definition of X↓Y .) So choose a d′′ from N ′↓D′′

, and set d = (d′, d′′).Then we have s,d,Σ |= h and s,d,Σ 6|= ⊥, hence s ∈ SPs

A(h, Σ; D). ut

Note that Lemma 9 is just a “boundary case” of this result. Several results from supportfunctions (cf. Theorem 2.2 in Haenni et al., 2000b) can be restated for the present case.However, note the set inclusion instead of equality in point (3) below.

Lemma 11 If h1, h2 are sentences in LP then

(1) SPsA(⊥,Σ; D) = ∅.

(2) SPsA(>,Σ; D) = CSs

A(Σ).

(3) SPsA(h1 ∧ h2,Σ; D) ⊆ SPs

A(h1,Σ; D) ∩ SPsA(h2,Σ; D).

(4) SPsA(h1 ∨ h2,Σ; D) ⊇ SPs

A(h1,Σ; D) ∪ SPsA(h2,Σ; D).

(5) h1 |= h2 implies SPsA(h1,Σ; D) ⊆ SPs

A(h2,Σ; D).

4.2 Decisions depending on System States 15

Proof (1) SPsA(⊥,Σ; D) = {s ∈ NA : There is a d ∈ ND s.t. s,d,Σ |= ⊥ and s,d,Σ 6|=

⊥} = ∅.

(2) SPsA(⊥,Σ; D) = {s ∈ NA : There is a d ∈ ND s.t. s,d,Σ |= > and s,d,Σ 6|= ⊥} =

{s ∈ NA : s,Σ 6|= ⊥} = CSsA(Σ).

(3) Let s ∈ SPsA(h1 ∧ h2,Σ; D), then there is a d ∈ ND with s,d,Σ |= h1 ∧ h2 and

s,d,Σ 6|= ⊥. For i = 1, 2, as h1 ∧ h2 |= hi, we have for this d that s,d,Σ |= hi, hences ∈ SPs

A(hi,Σ, D), which proves the statement.

(4) Follows from (5) using hi |= h1 ∨ h2 for i = 1, 2.

(5) Let h1 |= h2. Then there is a h′ ∈ LP s.t. h1 = h2 ∧ h′. Using (3), we getSPs

A(h1,Σ; D) = SPsA(h2 ∧ h′,Σ; D) ⊆ SPs

A(h2,Σ; D) ∩ SPsA(h′,Σ; D) ⊆ SPs

A(h2,Σ; D).ut

4.2 Decisions depending on System States

The second question which has been raised in the introductory example was: given asystem state which decision can be taken by the user? Given a PADS, this can easily beanswered using the following concept:

Definition 12 The set of supporting decisions SDsA(h, Σ; D) of a hypothesis h with

respect to the decision variables in D is defined as

SDsA(h, Σ; D) = {(s,d) ∈ NA ×ND : s,d,Σ |= h and s,d,Σ 6|= ⊥}

A supporting decision (s,d) ∈ SDsA(h, Σ; D) specifies first a system state s and a corres-

ponding decision d which can be taken by the user in order to fulfill the hypothesis h.For a system state, there may be several possible decision, hence this is not an optimalrepresentation.


Consider again the PADS as modeled in Example 2. The hypothesis δ = L1 ∧L2 → L3has a lot of supporting decision. An example is the scenario

(s,d) = ((1, 0, 1, 0, 0, 0, 0, 1, 0), (1, 0, 1, 0, 0, 0, 0)) ∈ SDA(δ,Σ)

which signifies that the system state s, which signifies that only the switches S1 and S3together with the bus A1 are working correctly, allows the user to take the action d inorder to fulfill the requirement δ, i.e. he has to close only the switches S1 and S3 and— as they are working correctly — the power from L2 can pass to L3 .


4.3 Logical Representation

Analog to the case of ordinary PAS, we define the notion of argument for a PADS.

Definition 13 Consider a PADS (P,D,Σ), a set of assumptions A ⊆ P − D and ahypothesis h ∈ LP , then we call α ∈ CA a supporting argument for h iff NA(α) ⊆SPs

A(h, Σ; D) The set of supporting arguments for h is denoted by SPA(h, Σ; D).As in the case of PAS, sets of supporting arguments are upward closed, i.e. if α ∈SPA(h, Σ; D) then every α′ ⊃ α is also in SPA(h, Σ; D). Hence µSPA(h, Σ; D) denotesthe corresponding set of minimal arguments. The support is

spA(δ,Σ; D) =∨

α∈µSPA(h,Σ;D)

α. (16)

Definition 14 Consider a PADS (P,D,Σ), a set of assumptions A ⊆ P − D and ahypothesis h ∈ LP , then we call α ∈ CA∪D a supporting decision (argument) for hiff NA∪D(α) ⊆ SDs

A(h, Σ; D) The set of supporting decision (arguments) for h isdenoted by SDA(h, Σ; D); this set is upward closed. Hence µSDA(h, Σ; D) is the corres-ponding set of minimal ones. The support of decision is

sdA(δ,Σ; D) =∨

α∈µSDA(h,Σ;D)

α. (17)

As before, we are interested in a compact representation of the supporting arguments.Hence we are interested in computing a formula which is logically equivalent but morecompact. We will face this problem in Section 5.


A logical version of the supporting arguments is more readable:

µSPA(h, Σ; D) =

okA1 ∧ okS1 ∧ okS3 ,okA1 ∧ okS2 ∧ okS3 ,okA1 ∧ okA2 ∧ okS1 ∧ okS4 ∧ okT3 ,okA1 ∧ okA2 ∧ okS2 ∧ okS4 ∧ okT3 ,okA1 ∧ okA2 ∧ okS3 ∧ okS4 ∧ okT1 ,okA1 ∧ okA2 ∧ okS3 ∧ okS4 ∧ okT2

(18)

These six formulas represent indeed the result expected from reliability theory, i.e. theydetermine the structure function of the example. The formulas corresponds to the sixminimal paths given in (Kohlas, 1987).


A corresponding logical formulation of the supporting decisions of Example 3 is morereadable. Below the twelve shortest arguments are listed, in total there are 48.

4.4 Probabilistic Argumentation and Decision Systems 17

µSDA(δ,Σ; D) = (19)

okA1 ∧ okS1 ∧ okS3 ∧ S1 ∧ S3 ∧ ¬S2 ∧ ¬S4 ∧ ¬T1 ∧ ¬T2 ∧ ¬T3 ,okA1 ∧ okS2 ∧ okS3 ∧ S2 ∧ S3 ∧ ¬S1 ∧ ¬S4 ∧ ¬T1 ∧ ¬T2 ∧ ¬T3 ,okA1 ∧ okS1 ∧ okS3 ∧ okT3 ∧ S1 ∧ S3 ∧ ¬S2 ∧ ¬S4 ∧ ¬T1 ∧ ¬T2 ,okA1 ∧ okS1 ∧ okS3 ∧ okT2 ∧ S1 ∧ S3 ∧ ¬S2 ∧ ¬S4 ∧ ¬T1 ∧ ¬T3 ,okA1 ∧ okS2 ∧ okS3 ∧ okT3 ∧ S2 ∧ S3 ∧ ¬S1 ∧ ¬S4 ∧ ¬T1 ∧ ¬T2 ,okA1 ∧ okS2 ∧ okS3 ∧ okT2 ∧ S2 ∧ S3 ∧ ¬S1 ∧ ¬S4 ∧ ¬T1 ∧ ¬T3 ,okA1 ∧ okS2 ∧ okS3 ∧ okT1 ∧ S2 ∧ S3 ∧ ¬S1 ∧ ¬S4 ∧ ¬T2 ∧ ¬T3 ,okA1 ∧ okS1 ∧ okS3 ∧ okT1 ∧ S1 ∧ S3 ∧ ¬S2 ∧ ¬S4 ∧ ¬T2 ∧ ¬T3 ,okA1 ∧ okS2 ∧ okS3 ∧ okS4 ∧ S2 ∧ S3 ∧ ¬S1 ∧ ¬T1 ∧ ¬T2 ∧ ¬T3 ,okA1 ∧ okS1 ∧ okS3 ∧ okS4 ∧ S1 ∧ S3 ∧ ¬S2 ∧ ¬T1 ∧ ¬T2 ∧ ¬T3 ,okA1 ∧ okS1 ∧ okS2 ∧ okS3 ∧ S1 ∧ S3 ∧ ¬S4 ∧ ¬T1 ∧ ¬T2 ∧ ¬T3 ,okA1 ∧ okS1 ∧ okS2 ∧ okS3 ∧ S2 ∧ S3 ∧ ¬S4 ∧ ¬T1 ∧ ¬T2 ∧ ¬T3 ,...

This example shows the problem with results for general PADS: While they are under-standable, their generality makes the results quite complex.

A possible solution for this problem is to consider a policy for the decision maker. Forexample, consider the policy that only those components, which are explicitly specifiedin the supporting decision, are turned on. This policy makes sense in the present context,yet in other situations it will be problematic. For the previous example, this means thatall negated literals can be omitted and only six supporting decisions remain.

4.4 Probabilistic Argumentation and Decision Systems

So far we have only been concerned with symbolical results. However, given the nu-merical information we can weigh the arguments and compute — for example — thereliability. Along similar lines as in the case of probabilistic argumentation systems (cf.Section 3.4), we introduce probabilities into the framework of PADS. Using the samenotation, we define the degree of support dspA for hypotheses h by

dspA(h, Σ; D) = p′(spA(h, Σ; D)) =p(spA(h, Σ; D))1− p(csA(Σ))

. (20)


The numerical reliability is pδ = dspA(δ,Σ; D) = 0.985.

In the case of PAS, such a probability can be computed directly using the concept ofDempster-Shafer belief functions (Lehmann, 2001, Haenni & Lehmann, 2003) with localcomputations (cf. Section 5.4). It is not yet clear if and how this approach can beextended to PADS.

18 5 Computation

4.5 Choosing a Decision

If for a given scenario several decision are possible to be taken, we have a problem ofchoice. We will not develop further this problem here but only mention several possiblestrategies for this problem.

• Minimize changes: Take the decision which needs a minimal number of change-ments w.r.t. the actual situation

• Robustness: Select the decision which can also handle most “neighbor” states

• Reliable: Select the decision which leads to the maximum reliability

Clearly, the same type of problem arises when we are considering arguments.

5 Computation

Consider a PADS (P,D,Σ) and a set of assumptions A ⊆ P − D. In this section wefocus on the problem of computing the set of supporting scenarios and the set of minimalarguments for a hypothesis h ∈ LP .

5.1 Scenarios

First, we focus on the computation of the scenarios SPsA(h, Σ, D). We need the concept

of projection. Consider a set of variables B, a scenario s ∈ NB and a subset C ⊆ B. Thens↓C denotes the projection of s to the subset C of variables. That is, s↓C contains onlythe components corresponding to variables in the set C. We have already considered thespecial case when C = ∅ in Section 4.1 by defining N∅ = {�}. More general, we defines↓∅ = � for any s ∈ B.

If S is a subset of NB and C ⊆ B, then the projection is defined on the items of the set:

S↓C = {s↓C : s ∈ S} (21)

For the element � we have (s, �) = s for any scenario s, i.e. it contains no informationabout any variable. Hence we also have �, ζ ≡ ζ.

For the computation of SPsA(h, Σ; D), we have to vary the set of assumptions and consider

a PAS with respect to the set of assumptions A ∪D:

Lemma 15 SPsA(h, Σ; D) = (SPs

A∪D(h, Σ))↓A

5.2 Arguments 19

Proof SPsA(h, Σ; D) = {s ∈ NA : There is a d ∈ ND s.t. s,d,Σ |= h and s,d,Σ 6|= ⊥} =

{(s,d) ∈ NA ×ND : s,d,Σ |= h, s,d,Σ 6|= ⊥}↓A = (SPsA∪D(h, Σ))↓A. ut

Second, we have to compute the supporting decision. Yet, this follows mainly from thelemma above:

Lemma 16 SDsA(h, Σ; D) = SPs

A∪D(h, Σ))

Proof SDsA(h, Σ; D) = {(s,d) ∈ NA×ND : s,d,Σ |= h and s,d,Σ 6|= ⊥} = SPs

A∪D(h, Σ).ut

According to the Lemmas 15 and 16, computing results requires there the computation ofscenarios in a PAS. Usually, this is not done directly, but only arguments are computed,cf. for example Haenni et al. (2000b).

5.2 Arguments

Clearly, the computation of a compact representation of the supporting scenarios andthe supporting decisions is a key point. We re-use here the well-developed concepts fromPAS.

First, we have to introduce the concept of projection w.r.t. conjunctions. Consider aconjunction α ∈ CP . The projection α↓P ′

of α to the subset P ′ ⊆ P means to eliminateall literals outside P ′ from the conjunction. The projection of a set of formulas S ⊆ CP

to P ′ is computed using the projection of every element, S↓P ′= {α↓P ′

: α ∈ S}.1

Example 7:

Let A = {a1, a2} and P = A ∪ {p}. Then (a1 ∧ p)↓A = a1 and {a1 ∧ a2, a1 ∧ p}↓A ={a1 ∧ a2, a1}.

For a discussion of the computation of minimal arguments in PAS see Section 5.4 and(Haenni et al., 2000b, Kohlas et al., 1999). For the computation of arguments in a PADS,we can use the techniques of computing arguments in PAS by changing temporarily thetype of some variables as done above for the scenarios:

Lemma 17 NA(SPA(h, Σ; D)) = NA((SPA∪D(h, Σ))↓A).

Proof By definition and Lemma 15, we have NA(SPA(h, Σ; D)) = SPsA(h, Σ, D) =

(SPsA∪D(h, Σ))↓A = NA((SPA∪D(h, Σ))↓A). ut

The lemma above is stated on the level of scenarios. If we try to lift it up to the level ofarguments, we can only prove the following result:

1Note that this operation denotes the projection; this is different from variable elimination (Haenni et al.,2000b) also known as variable forgetting!

20 5 Computation

Lemma 18 SPA(h, Σ; D) ⊇ (SPA∪D(h, Σ))↓A.

Proof Let α ∈ CA and β ∈ CD s.t. α∧β ∈ SPA∪D(h, Σ). Then, for every a ∈ NA(α) thereis at least one b s.t. (a,b) ∈ SPs

A∪D(h, Σ; D), hence by definition α ∈ SPA(h, Σ; D). ut

The lemma above states that the corresponding PAS allows to compute only a subset ofthe supporting arguments for the PADS, yet the “missing ones” are included in largerarguments:

Corollary 19 For every α ∈ SPA(h, Σ; D) there is a α′ ∈ (SPA∪D(h, Σ))↓A so thatα ∧ β = α′ for some β ∈ CA.

Proof Follows from Lemmas 17 and 18 using the fact that every argument is a conjunctionin CA. ut

So this means that from PAS, essentially a correct logical representation of the supportin PADS can be computed; however, some arguments might be missing.

Example 8:

Consider the situation of a PADS with A = {a1, a2}, D = {d}, P = A ∪D ∪ {h}, andΣ = {a1∧a2∧d → h, ¬a1∧a2∧¬d → h}. The correct result of the supporting argumentsfor h are

SPA(h, Σ; D) = {a1 ∧ a2, ¬a1 ∧ a2, a2}

Now consider the supporting arguments in the PAS:

SPA∪D(h, Σ) = {a1 ∧ a2 ∧ d, ¬a1 ∧ a2 ∧ ¬d}

and if we project this on A we get

(SPA∪D(h, Σ))↓A = {a1 ∧ a2, ¬a1 ∧ a2},

hence the argument a2 is missing, yet the logical representation is correct as (a1 ∧ a2)∨(¬a1 ∧ a2) ≡ (a1 ∧ a2) ∨ (¬a1 ∧ a2) ∨ a2.

The same situation arises in the computation of arguments in PAS, and the use of theoperator ConsA for computing all arguments is discussed in (Haenni et al., 2000b). Theseresults can be applied to our situation. Hence this yields a stronger result, which allows— if really needed — to explicitly compute all the arguments of the PADS from theresults of the PAS in logical form using the idea of resolution.

Definition 20 The operator ConsA is defined on a set of conjunctions C as follows,with A = {a1, . . . , an}:

ConsA(C) = ¬(Consa1(Consa2(· · · (Consan(¬C)) · · ·))) (22)

5.2 Arguments 21

and Consa is the set of all resolvents with respect to the assumption a, i.e. for a set ofclauses Ξ,

Consa(Ξ) = Ξ ∪ {ρ(ξ, ξ′) : ξ, ξ′ ∈ Ξ} (23)

and ρ is defined as in (Haenni et al., 2000b).

Note that the definition is unambiguous since the order of the assumptions a1 to an in(22) does not matter because Haenni et al. (2000b) have shown that

Consai(Consaj (X)) = Consaj (Consai(X)),Consa(Consa(X)) = Consa(X).

The order however is crucial for efficiency of computations.

Lemma 21 SPA(h, Σ; D) = ConsA((SPA∪D(h, Σ))↓A).2

Proof “⊇”: Follows from Lemma 18.

“⊆”: Let α ∈ SPA(h, Σ; D). Using Lemma 17, we get α |= (SPA∪D(h, Σ))↓A, or —switching to clauses — ¬(SPA∪D(h, Σ))↓A |= ¬α. So we are back in the problem ofcomputing implicates (clauses) of a set of clauses, and the operator above is known tocompute all implicates, cf. the procedure for computing prime implicates in (Haenniet al., 2000b). ut

Example 9:

Consider the example above where (SPA∪D(h, Σ))↓A = {a1 ∧ a2, ¬a1 ∧ a2}. Lemma 18implies that a1 ∧ a2 and ¬a1 ∧ a2 are contained in SPA(h, Σ; D). Lemma 21 tells us howto compute the remaining arguments: If we apply Cons{a1,a2} on the set of arguments{a1 ∧ a2, ¬a1 ∧ a2}, we get one additional arguments a2, hence SPA(h, Σ; D) = {a1 ∧a2, ¬a1 ∧ a2, a2}.

Corollary 22 µSPA(h, Σ; D) = µConsA((µSPA∪D(h, Σ))↓A).

The set of supporting arguments µSPA∪D(h, Σ) can be computed from µQSA∪D(h, Σ)and µQSA∪D(⊥,Σ) (Haenni et al., 2000b) as follows:

Algorithm Compute Supporting ArgumentsInput: µQSA∪D(h, Σ), µQSA∪D(⊥,Σ)Output: µSPA∪D(h, Σ)

let R = ∅for every α ∈ µQSA∪D(δ,Σ)

transform α ∧∧¬µQSA∪D(⊥,Σ) into a DNF

and add the conjunctions to Rreturn µConsA(µR)

2Note that the negation applied to a set of conjunctions means to apply the negation to every conjunctionin the set, which results in a set of clauses, and vice versa.

22 5 Computation

The supporting decision arguments can be computed similarly using the following results:

Lemma 23 SDA(h, Σ; D) = SPA∪D(h, Σ).

Proof Using Lemma 16, we have SDA(h, Σ; D) = {α ∈ CA∪D : NA∪D(α) ⊆ SDsA(h, Σ; D)}

= {α ∈ CA∪D : NA∪D(α) ⊆ SPsA∪D(h, Σ)} = SPA∪D(h, Σ). ut

Corollary 24 µSDA(h, Σ; D) = µSPA∪D(h, Σ).

5.3 Improvements in the Computation

The computations described in the previous chapter are correct, yet in applications, onebecomes aware that some of them require a lot of time. Where is the problem? Thecomputation of a typical support µSPA(h, Σ; D) as defined above requires according toCorollary 22 essentially that we determine µSPA∪P (h, Σ). This is done by computing thetwo minimal sets of arguments µQSA∪D(h, Σ) and µQSA∪D(⊥,Σ), which both can becomputed usually in short time. The computationally hard part is then the combinationof them according to the algorithm on Page 21. There, especially the conversion ofa negated DNF into a DNF is time consuming, i.e. a translation from a CNF to aDNF. This is a problem whose complexity is known to be non-polynomial (without theintroduction of additional literals).

In the general situation this is a hard problem, but here, we are in the situation where weare not interested in the supporting arguments µSPA∪D(h, Σ) but only in µSPA(h, Σ; D).Hence we essentially have to combine the computation of µSPA∪D(h, Σ) with the sub-sequent projection to A.

The following algorithm exploits the special situation:

AlgorithmInput: µQSA∪D(δ,Σ), µQSA∪D(⊥,Σ)Output: µSPA(δ,Σ; D)

let R = ∅for every α ∈ µQSA∪D(δ,Σ)

let Ξα =

β↓(var(α))c:

β ∈ µQSA∪D(⊥,Σ),var(β) ⊆ A ∪ var(α)lit(β) ∩ lit(¬α) = ∅

transform α ∧

∧¬Ξα into a DNF δ = δ1 ∨ · · · ∨ δq(α)

and add δ↓Ai to R for every i = 1, . . . , q(α)return µConsA(µR)

Remember that for a conjunction α = α1 ∧ · · · ∧αq, the set of literals lit(¬α) consists ofthe literals contained in the clause ¬α1 ∨ · · · ∨ ¬αq.

5.3 Improvements in the Computation 23

Lemma 25 The result of the algorithm is equal to µSPA(δ,Σ, D).

Proof We denote the result of transforming a set of formulas X into a DNF by DNF(X).Note that in general there are several possible DNF’s for a formula, but different DNF’s ofthe same formulas are always logically equivalent and so are their respective projectionsto A. Let α ∈ µQSA∪D(δ,Σ) and Ξ = µQSA∪D(⊥,Σ). The proof is divided into fourparts.

Part 1: Assume that there is a β ∈ Ξ which does not satisfy the first condition in theconstruction of Ξa, i.e. var(β) 6⊆ A ∪ var(α), or in other words, there is a b ∈ var(β)with b /∈ A ∪ var(α). Then β = b ∧ β′. Let Ξ′ = Ξ− {β}, then

α ∧ ¬Ξ = α ∧ ¬(β ∨ Ξ′) = α ∧ ¬β ∧ ¬Ξ′

= (α ∧ ¬b ∧ ¬Ξ′) ∨ (α ∧ ¬β′ ∧ ¬Ξ′) . (24)

Considering the projection to A, the literal ¬b in the first conjunction of (24) is elim-inated (because b /∈ A) and the second conjunction is subsumed by the first one, i.e.

(DNF(α ∧ ¬Ξ))↓A ≡ (DNF(α ∧ ¬Ξ′))↓A.

This justifies the first condition in the definition of Ξα.

Part 2: Assume that there is a β ∈ Ξ which does not satisfy the second condition in theconstruction of Ξa, i.e. lit(β) ∩ lit(¬α) 6= ∅. This implies that α ∧ ¬β ≡ α, hence forΞ′ = Ξ− {β},

α ∧ ¬Ξ = α ∧ ¬(β ∨ Ξ′) ≡ α ∧ ¬Ξ′ (25)

and this implies

(DNF(α ∧ ¬Ξ))↓A ≡ (DNF(α ∧ ¬Ξ′))↓A.

This justifies the second condition in the definition of Ξα.

Part 3: Assume that there is a β ∈ Ξ which satisfies both conditions in the definitionof Ξα, i.e.

var(β) ⊆ A ∪ var(α) (26)lit(β) ∩ lit(¬α) = ∅. (27)

Let now Ξ′ = Ξ− {β} and β = β1 ∧ · · · ∧ βr. Then,

α ∧ ¬Ξ = α ∧ ¬(β ∨ Ξ′) = α ∧ ¬β ∧ ¬Ξ′ =r∨

i=1

α ∧ ¬βi ∧ ¬Ξ′.

24 5 Computation

Considering the projection to A we get

(DNF(α ∧ ¬Ξ))↓A ≡r∨

i=1

(DNF(α ∧ ¬βi ∧ ¬Ξ′))↓A

≡∨

i=1,...,rβi /∈var(α)

(DNF(α ∧ ¬βi ∧ ¬Ξ′))↓A

because (27) implies that ¬βi /∈ lit(α) and (26) implies that βi ∈ A or βi ∈ var(α), butin the second case α∧¬βi |= ⊥, hence only those βi with βi /∈ var(α) have to be takeninto consideration.

So finally

(DNF(α ∧ ¬Ξ))↓A ≡ (DNF(α ∧ ¬β↓(var(α))c ∧ ¬Ξ′))↓A.

Part 4: Consider the set of formulas R as defined in the algorithms. At the end of thealgorithm, R is a set of terms, a DNF. R↓A is logically equivalent to µConsA(µR↓A).µSPA(δ,Σ; D) is the set of all prime implicates of the support spA(δ,Σ; D). In Parts 1to 3 of the proof we have shown that R is logically equivalent to this set. The operatorsConsA and µ assure that the result of the algorithm is as well the set of all primeimplicates. Hence these two sets are equal, and this proves the lemma. ut

Example 10:

Consider the set of assumptions A = {a1, a2, a3} and the decision variables D = {d1, d2}.Assume that we have computed

µQSA∪D(δ,Σ) = {a1 ∧ d1, d2, a2}µQSA∪D(⊥,Σ) = {a3 ∧ d2, ¬d1 ∧ a2, a1 ∧ ¬a2 ∧ d1}

and we have to compute the set of supporting arguments µSPA(δ,Σ; D).

We use the algorithm of Page 22. First, we set R = 0. Consider the first element a1∧d1.According to the algorithm, we have to look for elements in µQSA∪D(⊥,Σ) which respectthe two conditions of the definition of Ξα. The first argument, a3 ∧ d2 does not respectthe first condition because var(a3 ∧ d2) = {a3, d2} 6⊆ A ∪ var(a1 ∧ d1) = {a1, a2, a3, d1}.Hence this argument can be omitted. The second one, ¬d1 ∧ a2, does not respect thesecond condition because lit(¬d1 ∧ a2) ∩ lit(¬(a1 ∧ d1)) = {¬d1, a2} ∩ {¬a1,¬d1} ={¬d1} 6= ∅ and can therefore be omitted too. The third one, a1 ∧ ¬a2 ∧ d1, doesrespect both conditions, hence we have to project it to (var(a1 ∧ d1))c = {a2, a3, d2},i.e. (a1 ∧ ¬a2 ∧ d1)↓{a2,a3,d2} = ¬a2. Then, the resulting formula a1 ∧ d1 ∧ ¬(a2) has tobe transformed into a DNF (which it already is in this case) and its projection to A isadded to R, hence

R = {a1 ∧ ¬a2}

5.4 Local Computations and Approximation Techniques 25

The same procedure is now done with the second element d2 of µQSA∪D(δ,Σ). Thisyields

R = {a1 ∧ ¬a2, ¬a3}

and, using the third element a2, we get

R = {a1 ∧ ¬a2, ¬a3, a2}.

Finally, we have to apply the minimality operator µ which yields µR = {¬a3, a2}and then apply the operator ConsA to µR which, in this special case, does not changeanything, hence the result is µSPA(δ,Σ) = {¬a3, a2}.

5.4 Local Computations and Approximation Techniques

For the computation of supporting scenarios, we can re-use techniques from PAS (Haenniet al., 2000b). These techniques are based on the concept of variable elimination. Oneof the main problems of variable elimination (Haenni et al., 2000b) is the order in whichthese variables are actually eliminated (the results are equivalent for all orderings, butthe computation time depends very much on the ordering). An ordering correspondsto a computation on a certain hypertree structure, a local computation technique basedon ideas from Lauritzen & Shenoy (1995) which allows to distribute the computationson different virtual or real processors (Shenoy & Kohlas, 2000, Kohlas, 2003). We referalso to (Anrig & Kohlas, 2002a) for applications of these computation techniques toreliability and diagnostic.

Based on these symbolical results, the numerical results can then be computed using or-thogonalization techniques in reliability theory (Abraham, 1979, Heidtmann, 1989, 1997,Bertschy & Monney, 1996, Anrig & Beichelt, 2001). Besides, new promising approchesfocus on more general decomposition techniques (Darwiche & Marquis, 2001, Darwiche,2002).

If we are only interested in numerical results, then there is a second, often more efficientway to compute them: The knowledge base Σ can be transformed into a set of belieffunctions which are combined by local computation techniques, see (Lehmann, 2001,Haenni & Lehmann, 2001, 2003) for more information.

When applying the framework of argumentation systems to larger problems, there is needfor approximation techniques due to the complexity of the computations. We will notgo into details here but ask the interested reader to consider (Haenni, 2001b), where ananytime algorithm is presented and (Haenni, 2001a), where approximation is discussedon the abstract level of valuation algebras in the sense of (Kohlas, 2003). Its applicationto PADS is subject to further research.

26 6 An Example

6 An Example

Example 11: Availability of Energy Distribution Systems

In Example 1, an energy distribution system has been considered. For the same purposeFrey & Reichert (1973) proposed a second system using more redundancy, cf. Fig. 2.

Figure 2: Example of an energy distribution system with more redundancy.

The modelization of a PADS follows the lines of Example 1. So first, we consider thedifferent types of variables:

• Assumptions:

A = {okSi : i = 1, . . . , 5} ∪ {okTj : j = 1, . . . , 11}∪ {okA1 , okA2 , okA3}

• Decision Variables:

D = {Si : i = 1, . . . , 5} ∪ {Tj : j = 1, . . . , 11} ∪ {L1 ,L2 ,L3}

In this example decision variables Si : i = 1, . . . , 5} ∪ {Tj : j = 1, . . . , 11} aremodeled as assumptions.

• Other variables:

U = {Pi : i = 1, . . . , 14}

27

The knowledge about the system can be divided in several parts. First of all we modelthe possibilities for supplying the line L3 with a tension trough the bus A1 . There aretwo possibilities for that: From the line L1 coming through the switches S1 and T1 , orfrom the line L2 , coming through the switches S2 and T4 . The point P3 located on thebus A1 plays the role of the collecting point, which is under the tension if and only if oneof the points P1 , P2 , P4 , or P5 is under the tension and the bus A1 is not damaged.If P3 has tension, then if switches T7 and S3 are switched on and not damaged, theline L3 is supplied with tension. The proper work of the switches is modeled using ruleslike ¬okTj → ¬Tj and ¬okSi → ¬Si .

The knowledge base to model the power supply from the lines L1 and L2 to the line L3trough the bus A1 is defined as follows:

Σ1 =

L1 ∧ S1 ∧ T1 → P1L2 ∧ S2 ∧ T4 → P2

(P1 ∨ P2 ∨ P4 ∨ P5 ) ∧ okA1 → P3P3 ∧ T7 ∧ S3 → L3

Secondly, we model the alternatives for the bus A2 . The collecting point on the bus A2is point P8, which is under the tension if and only if one of the points P6, P7, P9,or P10 is under the tension and the bus A2 is working correctly. Of course, we considerhere that corresponding switches, cf. Fig. 2, are switched on and are working correctly.Applying similar investigations as for a bus A1 , we construct the knowledge base formodeling the power supply to the line L3 through the bus A2 :

Σ2 =

L1 ∧ S1 ∧ T2 → P6L2 ∧ S2 ∧ T5 → P7

(P6 ∨ P7 ∨ P9 ∨ P10 ) ∧ okA2 → P8P8 ∧ T8 ∧ S3 → L3

The bus A2 is an alternative bus for A1 and by a definition is used only if a power supplythrough the bus A1 cannot be reached. The bus A3 is the alternative for the previousboth and is used only if there is no possibility to supply line L3 with a tension usingonly the buses A1 and A2 . The point P13 is a collecting point for the bus A3 . Startingfrom the similar considerations as before the knowledge base for modeling connectionsbetween the buses A1 , A2 , and A3 is as follows:

Σ3 =

(P1 ∨ P2 ) ∧ okA1 ∧ S5 → P10(P6 ∨ P7 ) ∧ okA2 ∧ S5 → P5

L1 ∧ T3 → P11L2 ∧ T6 → P12

(P11 ∨ P12 ) ∧ okA3 ∧ S4 ∧ T10 → P4(P11 ∨ P12 ) ∧ okA3 ∧ S4 ∧ T11 → P9

(P1 ∨ P2 ) ∧ okA1 ∧ T10 ∧ S4 → P14(P6 ∨ P7 ) ∧ okA2 ∧ T11 ∧ S4 → P14

P14 ∧ okA3 → P13P13 ∧ T9 → L3

28 7 Conclusion

The PADS is then specified by (P,A, D, Σ,Π) where P = A∪D∪U and Σ = Σ1∪Σ2∪Σ3.The probabilities Π are defined as in (1).

The requirement is the same as before, namely to supply L3 from the inputs L1 andL2 , i.e. δ = L1 ∧ L2 → L3 . The logical version of the supporting arguments consists ofsixteen elements:

µSPA(δ,Σ; D) =

okA2 ∧ okS2 ∧ okS3 ∧ okT5 ∧ okT8




okA2 ∧ okA3 ∧ okS2 ∧ okS4 ∧ okT11 ∧ okT5 ∧ okT9








okA1 ∧ okA2 ∧ okS1 ∧ okS3 ∧ okS5 ∧ okT1 ∧ okT8




7 Conclusion

We have shown how the well-known formalism of PAS can be enriched with decisionvariables. A specialized algorithm for computing the respective arguments has beenpresented. This allows now to use this framework for example in model-based reliabilitytheory (Anrig & Kohlas, 2002a,b,c) for computing structure functions.

Several open questions arose during the research: First, the numerical counterpart ofPAS is — in some sense — the computation with belief functions (Haenni & Lehmann,2001, Lehmann, 2001). But is it possible to apply the same concepts for PADS? Second,several approximation techniques are known for PAS and also for belief functions. Is itpossible to use the same techniques for PADS? Using these approximation technique wewould get then (symbolical) upper and lower bounds of structure functions in reliabilitytheory.

References 29

Acknowledgments

Most of this work was done while the second author was visiting researcher at theDepartment of Informatics, University of Fribourg, Switzerland. The visit was supportedby Swiss Baltic Net (Gebert Ruef Foundation).

The Research of the first author is supported by grant No. 2000-061454.00 of the SwissNational Foundation for Research.

We thank Jurg Kohlas and Norbert Lehmann for valuable comments on earlier drafts.

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