Introducing Bayesian Argumentation Networks...Bayesian Argumentation Networks thentheprobabilityofe...

Introducing Bayesian ArgumentationNetworks

D M GabbayDepartment of Informatics Kingrsquos College London

Ashkelon Academic College IsraelBar Ilan University Ramat Gan IsraelUniversity of Luxembourg Luxembourg

http www inf kcl ac uk staff dgdovgabbaykclacuk

O RodriguesDepartment of Informatics Kingrsquos College London The Strand London WC2R

2LS UKhttp www inf kcl ac uk staff odinaldo

odinaldorodrigueskclacuk

Abstract

We give a faithful interpretation of Bayesian networks into a version of nu-merical argumentation networks based on Łukasiewicz infinite-valued logic withproduct conjunction The advantages of such a translation beyond the theoret-ical aspects of it are hopefully threefold 1) importing updating algorithms intoargumentation networks 2) importing the handling of loops into cyclic Bayesiannetworks and 3) importing logical proof theory into Bayesian networks

Keywords Bayesian Networks Argumentation Theory Numerical Networks

1 IntroductionIn this paper we compare probabilistic argumentation with Bayesian networks andmotivate the new definition of Bayesian Argumentation Networks We examine whatextra features are needed to extend traditional abstract argumentation frameworksto enable the extended frameworks to simulate Bayesian networks Once we identifysuch features then we can call the extended argumentation frameworks by the name

Vol 3 No 2 2016IFCoLog Journal of Logic and its Applications

D M Gabbay and O Rodrigues

Bayesian Argumentation Networks We shall see later that the extra features areall well-known features existing in the literature in various contexts

In order to illustrate these ideas consider the network 〈SR〉 of Fig 1 Inthis figure the arrows just indicate parenthood If the arrows are considered asattacks then 〈SR〉 is a traditional abstract argumentation framework (hencefortha ldquoDung networkrdquo) and there is only one complete extension E = XY UW(with A = ldquooutrdquo)

X Y

A

U W

Figure 1 A sample argumentation network

The operating assumptions (which are violated in Bayesian networks) are

1 Since there is no connection (ie attack) going into X and into Y then Xand Y are ldquoinrdquo (intuitively meaning X = Y = gt)

2 Since U and V have the same parent A we treat U and V in the same way

3 We do not mind having cycles ie R need not be acyclic

There are other assumptions in the case of argumentation networks but let usconcentrate only on the ones above

Bayesian networks do not allow for cycles (R must be acyclic) and they do notdetermine the values of nodes without parents such as X and Y Moreover theyare not committed to treating nodes with the same parents (such as U and W ) thesame way Such a view is not new to argumentation In fact such a view is sharedby Abstract Dialectical Frameworks (ADFs) [6] In an ADF each node α dependson its parents say β1 βn via a Boolean formula Ψα specific to α Thus wehave that α depends on Ψα(β1 βn) in the ADF case and we want that

αharr Ψα(β1 βn)

In Dungrsquos networks the same constraint Ψα is imposed on all nodes α namely

αharrnandi=1notβi

Bayesian Argumentation Networks

where ldquoandnotrdquo is the Peirce-Quine connective

darr Ei = andni=1notEi1

Still we do not have many options when we treat the network of Fig 1 as an ADFSince X and Y depend on the empty set then each can be either gt or perp A dependson X and Y and these being gt or perp allow for A to be either gt or perp and similarlyfor U and W So depending on the Boolean functions Ψα employed we can get allpossible distributions of perpgt among the nodes in Fig 12

If we allow source nodes such as X and Y to have arbitrary given values in[0 1] and are able to describe the desired dependencies between node values inan argumentation context then we can bridge the gap between the Bayesian andargumentation representations and hence analyse the properties of the former underthe perspective of the latter This can bring benefits to both areas which we willdiscuss later

We find that the probabilistic approach to argumentation is the nearest we canget to Bayesian networks We identify that what is missing in the probabilisticapproach is a representation of conditional probabilities a feature which is central inBayesian networks We further realise that if we define new argumentation networksbased on joint attacks defined numerically using Łukasiewicz infinite-valued logicswe will have what we need The integration of conditional probabilities and jointattacks is one of the objectives of this paper

The rest of the paper is structured as follows We start with a descriptionof the probabilistic approach to argumentation in Section 2 The section is writ-ten in a Socratic manner leading the reader to our conclusions using examplesand semi-formal definitions Section 3 gives the formal definitions in a systematicmanner Section 4 contains a comprehensive example illustrating what we havedone Section 5 deals with complexity issues Section 6 discusses related literature[3 4 9 16 5 19 20 8 21 22 24 25 27] In Section 7 we conclude with a discussionand directions for future research

1Notice that the other boolean connectives can be defined in terms of darr for instance notP equiv P darrP

2As pointed out by one of the referees the reader might think that this is a shortcoming ofADFs in the sense that initial arguments being dependent on the empty set can only have a fixedvalue However there is also the possibility to consider all initial arguments A as self-looping withacceptance condition A In this case most semantics then yield a ldquoguessingrdquo value for A We shouldhowever be cautious in not allowing too many modifications It is known that enough modificationscan reduce ADFs to traditional argumentation systems (see [14])


2 Background Discussion

This section develops in a Socratic manner the features we need to reach a properrepresentation of a Bayesian network as an extension of an argumentation networkWe therefore turn to the probabilistic approach to argumentation it being the near-est to Bayesian networks We need some notation before we describe it As astarting point we consider the elements of S = XYAU V as classical proposi-tional atoms capable of getting the values α = gt (corresponding to α is ldquoinrdquo) andα = perp (corresponding to α is ldquooutrdquo) Let us understand the term full conjunction ofliterals to mean a conjunction containing for each atom α of the language (which isassumed to be finite) either α or notα Any full conjunction of literals of the form

e = andiαplusmni

can be considered a classical model m(e) We have

m(e) |= α iff e ` α

We can also associate with e a subset Se of S Se = α isin S | e ` α (rememberthat the elements of S are atoms without negation) So if we assign probabilitydistributions π on the modelsm(e) or on the set of full conjunctions of literals e weget a traditional probability function π on the space Ω = 22S = families of models =2e = the set of all propositional well-formed formulae (wffs) built-up from theatoms of S

In our example S = XYAU V 2S = all subsets of S = all models of thelanguage S Ω = 22S = all possible sets of models So π gives a value 0 le π(m) le 1for each model m of S We have that

summ π(m) = 1

According to [13] the probability π(m) for a typical conjunctive model m(e)where e = andαisinSαplusmn can be given in two main ways

i) The semantic way which gives values π(m(e)) directly for each e

ii) The syntactic way which gives values π(α) for each α isin S and then π(m(e))is defined as the product prod

α

πplusmn(α)

where π+(α) = π(α) and πminus(α) = 1minus π(α)

So for example in Fig 1 we either give probability directly to each model eg toe = X and Y and notA and U and notW or give probabilities to each of X Y A U and W and


then the probability of e can be calculated as

π(X) middot π(Y ) middot (1minus π(A)) middot π(U) middot (1minus π(W ))3

The version of the probabilistic approach to both Dung or ADF which can becompared with the Bayesian approach is the syntactical one ii) above the onewhich assigns probabilities to nodes (not the one that assigns probabilities to thesubnetworks) Thus each of the nodes X Y A U and V is assigned a probabilityvalue

The most general case of getting such probability is to regard the arguments asatoms in a space (ie S = XYAU V ) and assign probabilities to the subsets ofS This is a traditional probability distribution Note that the subsets E sube Ω = 22S

can also be identified with sets of models of formulas built-up using atoms from SWe now show a connection of probabilistic argumentation with Bayesian argu-

mentation Both Dung and ADFs would read Fig 1 as follows The figure suggeststhe probability space being the family Ω = 22S of all subsets of S = XYAU V and the connections (arrows) in the figure suggest restriction on the probability πon Ω We want to consider only those probabilities which satisfy for every α isin Swith parents β1 βn the following

π(α) = π(ψα(β1 βn))

Remember that each wff defines a set of models in which it holds and π is a prob-ability on sets of models Thus π gives a number 0 le π(m) le 1 to each model m ofthe language of S with

summ π(m) = 1

Bayesian argumentation looks at the elements of S as random variables capable ofgetting gt or perp and regards all probability functions P (α1 αn) where αi = S

In our case we have probability functions P (XYAU V ) Thus for each com-bination of values of gt or perp to the variables in S P will give a probability

Such a combination can also be viewed as a model m for the language of S andso P gives probability to models This is the same as π but the restrictions onP and the manipulation of P are different in this case We have according to theBayesian view that Fig 1 gives the dependencies of the variables on each other Letβ1 βn be all the parents of α and P (α|Z) denote the conditional probabilityof α given Z We have the following equations

P (α) = P (α | β1 βn) middot P (β1 βn)

3See [13]


If α has no parents then P (α) must be given If α does have parents β1 βnthen P (α | β1 βn) must be given P (α | β1 βn) can be given as a functiongiving a value in [0 1] for every choice of gtperp to each βi

Thus the Bayesian approach specifies a syntactical type probability on the atomsα isin S by using the graph of the network and giving conditional probabilities forthe dependencies of the graph

So for the network of Fig 1 we need the following values to specify a specificBayesian distribution P

bull Values of the probabilities of the source nodes X and Y ie P (X) and P (Y )

bull A table of values v describing the coefficients of the function P (A|XY ) Weuse the notation FA|11 for the case A|X andY FA|10 for the case A|X andnotY etcWe denote the transmission coefficient for each case as eFA|11 eFA|10 and so on

X Y v

gt gt eFA|11

gt perp eFA|10

perp gt eFA|01

perp perp eFA|00

bull A table for P (U |A)

A v

gt eFU|1

perp eFU|0

bull A table for P (W |A)

A v

gt eFW |1

perp eFW |0

Note that the network of Fig 1 actually depicts Pearlrsquos famous Earthquakeexample [20] as described in the book ldquoBayesian Artificial Intelligencerdquo by Korb andNicholson [17]


ldquoYou have a new burglar alarm installed It reliably detects burglarybut also responds to minor earthquakes Two neighbors John and Marypromise to call the police when they hear the alarm John always callswhen he hears the alarm but sometimes confuses the alarm with thephone ringing and calls then also On the other hand Mary likes loudmusic and sometimes doesnrsquot hear the alarm Given evidence aboutwho has and hasnrsquot called yoursquod like to estimate the probability of aburglaryrdquo

Replacing X with ldquoBurglaryrdquo Y with ldquoEarthquakerdquo A with ldquoAlarmrdquo U withldquoJohn callsrdquo and W with ldquoMary callsrdquo gives the Bayesian network

Burglary Earthquake

Alarm

John calls Mary calls

For simplicity we will continue to use the letters X Y A U and W Let us assume the following values P (X) = 001 giving P (X = gt) = 001 and

P (X = perp) = 099 P (Y = gt) = 002 giving P (Y = perp) = 098 and the valuesgiven by the tables below

X Y v

gt gt eFA|11 = 095gt perp eFA|10 = 094perp gt eFA|01 = 029perp perp eFA|00 = 0001

A v

gt eFU|1 = 09perp eFU|0 = 005

A v

gt eFW |1 = 070perp eFW |0 = 001


We can now compute the probabilities P (A) P (U) and P (W )

P (A) = eFA|11 times P (X)times P (Y ) +eFA|10 times P (X)times (1minus P (Y )) +eFA|01 times (1minus P (X))times P (Y ) +eFA|00 times (1minus P (X))times (1minus P (Y ))

P (W ) = eFW |1 times P (A) + eFW |0 times (1minus P (A))

P (U) = eFU|1 times P (A) + eFU|0 times (1minus P (A))

So

P (A) = 095times 001times 002 +094times 001times 098 +029times 099times 002 +0001times 099times 098

= 000019 + 0009212 + 0005742 + 00009702= 00161142 asymp 0016

So P (notA) asymp 0984 Now for U and W we get

P (W ) = (07times 0016) + (001times 0984) = 00112 + 000984 asymp 0021

P (U) = (09times 0016) + (005times 0984) = 0014 + 00492 asymp 0063

Thus we can see that we have a syntactical probability distribution on S

21 A Common Ground for Bayesian Argumentation and AbstractDialectical Frameworks

To compare Bayesian networks with say ADFs or with traditional Dung networks weneed to go to a common ground First we note that with any formal system whetherit be a logic such as classic or intuitionistic logic or whether it be a Bayesian ADFor traditional argumentation network there are always two components The firstone is the intended meaning of the system The second is the formal mathematicalrepresentation of the system and the mathematical machinery of handling it Whenwe compare two such systems we can compare them in regard to their formal ma-chinery or we can compare them in their intended meaning It may be that twosystems have the same formal machineries but completely different meanings This


happens a lot in modal logics with possible world semantics In the case of a network〈SR〉 the intended meaning may impose some restrictions on the graph In an ar-gumentation network the arrows mean attack the variables get values ldquoinrdquo ldquooutrdquoand ldquoundecidedrdquo and unattacked nodes must get value ldquoinrdquo In Bayesian networksthe arrows represent dependencies and there is the requirement of the network beingacyclic In addition nodes with the same parents in Bayesian networks can behavedifferently which is not the case in the traditional argumentation but is the case inADFs In ADFs the arrows represent dependencies and there is no requirement ofbeing acyclic Having said all that let us now compare the systems on the basis oftheir mathematical machinery which can be captured by the two points below

a Since Bayesian networks allow for points without parents to have an arbi-trary probability assigned to them Bayesian networks can agree to limit suchassignment for the sake of common grounds with argumentation and assignprobability 1 we can assume a similar sacrifice and the same property forADFs If on the other hand we want to leave Bayesian networks as they are(not ask them to make any limitations) but we still want to have commonground with respect to this property with ordinary Dung networks we canmodify Dungrsquos networks and add for each node α a new node called notα withα and notα attacking each other This will allow any node α which was origi-nally unattacked to get any value in the modified network because it will bepart of the cycle αnotαWe can also assume for Bayesian networks the sacrifice limitation that nodeswith the same parents behave the same (we can call these Bayesian networksBNA nets4 We can also add this requirement (that nodes with the sameparents behave the same way) as an additional assumption on ADFs to makethem more in line with traditional Dung networks

b Bayesian networks are acyclic and since ADFs and traditional Dung networkscan also be acyclic let us accept this additional restriction on them

So we compare acyclic probabilistic ADFs with BNA nets and see what else we needto add to argumentation networks to be able to implement Bayesian networks inthem

The above discussion outlined several possibilities for finding commongrounds between Bayesian networks and Dungrsquos networks We made what we thinkis the best choiceapproach which we now proceed to explain

4The letters ldquoNArdquo stand for ldquonice-to-argumentationrdquo


Looking again at Fig 1 we see that what is missing in order to do a propercomparison is the fact that Bayesian networks have the conditional probabilitiesLet us look at Fig 2 which is the top part of Fig 1

X Y

A

Figure 2 The top part of the network in Fig 1

What the Bayesian approach does is to give arbitrary values P (X) P (Y ) (so wehave syntactical probabilities for X and Y ) but to get P (A) it uses transmissionvalues as given in Fig 3 eFA|ij

are transmission coefficients in the sense of [1 2] andF ij is an attack formation in the sense of [12] to be explained below and formallydefined in Section 3

F 11 X and Y F 10 X and notY F 01 notX and Y F 00 notX and notY

A eFA|00

eFA|10

eFA|11

eFA|01

We have P (A) =sumeijP (F ij)

Figure 3 An argumentation network with attack formations transmission coeffi-cients and joint attacks

So what we need to accommodate Bayesian networks are argumentation net-works with attack formations transmission coefficients and joint attacks obeyingthe attack formula of Fig 3 The semantics of such networks is best given using theequational approach [15]5

5There are several different interpretations of basic argumentation notions In traditional Dungnetworks arcs represent attacks while in ADFs they represent dependencies It is natural then tothink that ADFs are closer in meaning to Bayesian networks because certainly arcs in Bayesiannetworks are not attacks but dependencies Our reader may therefore be puzzled at our translationof Bayesian networks into argumentation where arcs represent attacks We even use joint attacksWe remind the reader that ADFs can be translated into traditional argumentation networks usingjoint attacks and additional nodes The additional nodes are used to help simulate the booleandependencies (see [14]) It may be possible to translate Bayesian networks into ADFs but wewould need additional points and some kind of fuzzy propagation It therefore makes more sense totranslate directly into traditional networks with attacks Note also that numerical values associatedwith nodes can have several interpretations 1) a fuzzy truth-value 2) a probability value express-ing uncertainty about argument acceptance 3) a value obtained in the context of the equational


We now explain the components needed

a Attack formations

Consider Fig 4 (L) and Fig 4 (R) In Fig 4 (L) we have that α attacks βWe have

(a) If α = ldquoinrdquo then β = ldquooutrdquo(b) If α = ldquooutrdquo then β = ldquoinrdquo (unless β is attacked by something else that

is not ldquooutrdquo)(c) If α = ldquoundecidedrdquo then β = ldquoundecidedrdquo (unless it is attacked by

something else that is ldquoinrdquo in which case β has to be ldquooutrdquo)

α

β

α

Xαβ

Yαβ

β

(L) (R)

Figure 4

In Fig 4 (R) we have two intermediary points unique to the pair (α β) Weview this as an attack formation It does the job of Fig 4 (L) (a) (b) and (c)still hold for Fig 4 (R) Fig 4 (R) is a general replacement for Fig 4 (L) usedby Gabbay in [14] It allows for the implementation of higher level attacks6

approach as the result of some calculation 4) a Bayesian probability value The conditions whentwo of such values coincide need to be investigated For example we do know that for the Eqinv

equational approach the numerical values can be viewed as probabilistic values where the argumentsare mutually independent [13]

6Higher level attacks are attacks on attacks So for example an academic professional argumentβ put forward in favour of promoting three members of staff to the rank of full professor by virtue of


For example the attack γ rarr (α rarr β) can be implemented as γ rarr Yαβ Thetwo additional dummy pointsXαβ and Yαβ are just two ldquotransmitting pointsrdquowhich allow the node γ to attack the transmission by attacking the point YαβWe do not need higher level attacks in this paper but it is useful to knowhow useful attack formations are in translationsimplementations We use thenotation F [α β] as in Fig 5

α

F [α β] =

auxiliary pointsxαβ yαβ

appearing only inthis diamond

β

Input α

Output β

Figure 5

Attack formations can be single arguments as in Fig 6 The argument A isboth the input and the output point of the formationAttack formations can attack each other as in Fig 7 The output point offormation F 1 attacks the input point of formation F 2

b Transmission coefficients

their brilliant performance may be attacked by an argument α which says that there is not enoughmoney to pay their higher salaries and benefits if indeed promoted If the claims of both argumentsare true (ie the individuals did perform well and indeed there is not enough money to pay forthe extra expenditure caused by the promotion there is nothing to say except to put forward anargument γ which says that budgetary considerations should not be arguments against promotionHere γ is a higher level attack on the very attack arrow αrarr β We write this as γ rarr (αrarr β)


inputoutput

A

Figure 6

inputoutput F 1

inputoutput F 2

Input 1

Output 1

Input 2

Output 2

Figure 7 An attack formation attacking another attack formation


β1 β2 βk

α

Figure 8 A typical attack configuration in an argumentation network

Consider Fig 8 under the equational approach of [15] Let β1 βk be allof the attackers of node α Each node has a numerical value f(α) f(β1) f(βk) The equational approach (under the Eqinv Equational semantics)requires that

f(α) =kprodi=1

(1minus f(βi)) (1)

The equational approach obtains f as a solution to the equations of type (1)for every α isin S and at least all the preferred extensions (in Dungrsquos sense) areobtained via the correspondence

(a) α = ldquoinrdquo if f(α) = 1(b) α = ldquooutrdquo if f(α) = 0(c) α = ldquoundecidedrdquo if 0 lt f(α) lt 1

Such solutions also can be seen as syntactical probability distributions for thenodes as shown in [13] So if we implement Bayesian networks in argumen-tation networks with the Eqinv interpretation we hope to get the Bayesianprobabilities as preferred extensionsWhen we have a transmission coefficient ei between the attacker βi and αthe strength of the attack from βi is adjusted by the coefficient ei and henceits value is only ei times βi Thus we get for Fig 9 of attacks with transmissioncoefficients that

f(α) =kprodi=1

(1minus ei times f(βi)) (2)

It is worth noting that the transmission coefficient does not change the expres-sive power of Eqinv since the effects of the coefficients can be implementedthrough additional nodes in Eqinv

c Joint attacks


β1 β2 βk

α

e1 e2 ek

Figure 9 A typical attack configuration in an argumentation network with trans-mission coefficients

In [14] Gabbay et al used the notation of Fig 10 in the context of Fibringnetworks joint attacks and disjunctive attacks The idea of joint attacks onits own was earlier introduced in [18] The intended meaning of a joint attackis that α is ldquooutrdquo if all βi are ldquoinrdquo In a numerical context (ie under anequational approach) we can write the equation

f(α) = 1minusprodi

f(βi)

Clearly α is ldquooutrdquo exactly when all of βi are ldquoinrdquo

β1 βk

α

Figure 10 A joint attack from β1 βk to α

Remark 1 Note that if we have such joint attacks (as given in [18]) underthe equational approach we can implement products (π-attacks) with the helpof auxiliary points For example the value of α as the product of β1 and β2in Fig 11 can be implemented as Fig 12 and vice-versa In Fig 12 we havethat f(x) = 1minus β1 middot β2 and f(α) = 1minus x = β1 middot β2Fig 13 can be implemented as Fig 14 In Fig 13 we have that f(α) =1minus β1 middot β2 In Fig 14 f(y) = β1 middot β2 and f(α) = 1minus y = 1minus β1 middot β2

For implementing Bayesian networks we need a different understanding forjoint attacks yielding a different equation What we need is the followingequation

f(α) = min(1sumj(1minus f(βj))) (3)


β1 β2

α= β1 middot β2

π product

Figure 11

β1 β2

x

α

and joint

Figure 12

β1 β2

α

and joint

Figure 13

β1 β2

y

α

π product

Figure 14

From (3) one can see that only if all f(βj) = 1 do we get f(α) = 0


So we are using the equation below for the joint attacks of Fig 10

α = min(1sumj 1minus βj)

We make some comments about this equation

(a) First note that this equational truth-table is definable in Łukasiewiczlogic [23] In this logic propositions get value in [0 1] 1 represents truthand 0 represents falsity The truth-tables for not and rarr are

notX = 1minusXX rarr Y = min(1 1minusX + Y )

ThereforeX rarr notY = min(1 1minusX + 1minus Y )

Let ϕ be a new connective operating on a non-empty list [X1 Xn]7defined as follows

ϕ([X1]) = notX1

ϕ([X1 X2]) = X2 rarr ϕ([X1])

By induction assume ϕ(X1 Xn) is definable and satisfies

ϕ(X1 Xn) = min(1sumi

(1minusXi))

Then

Xn+1 rarr ϕ(X1 Xn) = min(1 1minusXn+1 + ϕ(X1 Xn))= min(1

sumn+1i=1 (1minusXi))

= ϕ(X1 Xn+1)

Note that ϕ(X1) = min(1 1minusX1) = notX1 So we have

ϕ(X1) = notX1

ϕ(X1 Xn Xn+1) = Xn+1 rarr ϕ(X1 Xn)

(b) Also note that argumentation networks with a formula of the type of ϕjust defined for joint attacks are not definable through the traditionalDung semantics This gives new semantics Consider Fig 15


a b

c

Figure 15

a b x d e

c

c = min(1 1minus a+ 1minus b)timesmin(1 1minus x)timesmin(1 1minus d+ 1minus e)= min(1 1minus a+ 1minus b)times (1minus x)timesmin(1 1minus d+ 1minus e)

Figure 16

Use only ϕ We get a = 12 b = 1

2 c = min(1 12 + 1

2) = 1 So c = ldquoinrdquoThe rationale behind this semantics is as followsWe reject c if there are joint attacks on c where all the attackers areldquoinrdquo If some attackers are ldquoundecidedrdquo then so is c However if toomany attackers are ldquoundecidedrdquo (and remember this is a consortium jointattack where too many members of the consortium are undecided) thenwe disregard the attack and let c = ldquoinrdquoTo get a better idea of how Eqinv with Łukasiewicz joint attacks workscompare Figures 16 and 17The equation in Fig 17 is given by Eqinv for β1 being the joint attack ofa b β2 being the attack of x and β3 being the joint attack of d eThe joint attacks of β1 and β3 are calculated each according to Equation 3(see page 15) So going back to Fig 16 under the above considerationswe get

c = min(1 1minus a+ 1minus b)times (1minus x)timesmin(1 1minus d+ 1minus e)

7For example conjunction is an operator that can be seen to be operating on a list whereand([X]) = X and

and([X1 Xn]) = Xn

and([X1 Xnminus1])


β1 β2 β3

c

c = min(1 1minus β1)timesmin(1 1minus β2)timesmin(1 1minus β3)= (1minus β1)times (1minus β2)times (1minus β3)=

prodi

(1minus βi)

Figure 17

ϕ11 ϕ10 ϕ01 ϕ00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1

Figure 18 A complex configuration of joint attacks with transmission factors

22 Combining It All

Consider Fig 18The equational approach will give a solution f to this configuration as

f(xij) = 1minus eij times f(ϕij)

and

f(A) = min (1sum

(1minus f(xij)))f(A) = min (1

sum(1minus (1minus eij times f(ϕij))))

f(A) = min (1sumeij times f(ϕij))

Now look again at Fig 3 If we can instantiate ϕij by an appropriate attack


notX X Y notY

c00

notX X Y notY

c01

notX X Y notY

c10

notX X Y notY

c11

1 1

1 1

1 1

1 1

F 00[notXnotY ] =

c00 = (1minus P (X)) middot (1minus P (Y ))

F 01[notXY ] =

c01 = (1minus P (X)) middot (1minus 1 + P (Y )) = (1minus P (X)) middot P (Y )

F 10[XnotY ] =

c10 = (1minus 1 + P (X)) middot (1minus P (Y )) = P (X) middot (1minus P (Y ))

F 11[XY ] =

c11 = (1minus 1 + P (X)) middot (1minus 1 + P (Y )) = P (X) middot P (Y )

Figure 19

formation F ij such that

f(ϕij) = f(F ij) = Pplusmn(X)times Pplusmn(Y )

then we have implemented that figure as Fig 18This is easy to do Look at Fig 19 and remember that since X notX Y and

notY are end points attacking each other respectively we can give them arbitraryprobabilities In Fig 19 cij is the output point (realising f(ϕij)) and X andY are given probabilities P (X) and P (Y ) respectively This gives probabilitiesP (notX) = 1minus P (X) and P (notY ) = 1minus P (Y )


Remark 2 Note that the above implementation required two types of attacks Theproduct attack of Fig 11 and the new joint attack of Fig 10 namely

α = min(1sumj(1minus βj))

We know that the new joint attack can be expressed in Łukasiewicz infinite-valuedlogic Therefore in the extension of this logic with product conjunctions namely withthe additional connective lowast8

x lowast y = x middot y

we can implement Bayesian networks

3 Formal DefinitionsThis section will describe the formal machinery of Bayesian Argumentation Networks(BANs) and the mechanism to translate a Bayesian network into a BAN

Definition 1 (Bayesian Argumentation Network (BAN))

a A BAN has the form B = 〈SR e〉 where S is a non-empty set of argumentsR sube (2S minus empty) times S is the attack relation between non-empty subsets of S andan element of S For each pair (Hx) isin R such that H sube S and x isin S e is atransmission function giving each h in the pair (Hx) a real value e(Hx h) isin[0 1]We can describe this situation in Fig 20 where (Hx) isin R H = h1 hkand ei = e(Hx hi)The general attack configuration of a node is depicted in Fig 21 where H1 =h1

1 h1k1 H i Hm = hm1 hmkm

are all attackers of the nodex In such configuration eij = e(H i x hij)

b Let f be a function from S into [0 1] The equation associated with f and theconfiguration of Fig 21 is

f(x) =mprodj=1

min(1kjsumi=1

(1minus eij middot f(hji ))) (4)

c A solution to equation (4) of item b is called an extension to B

8See [10] for details


h1 hk

x

e1 ek

H = h1 hk(Hx h1) = e1

(Hx h2) = e2

(Hx hk) = ek

Figure 20

H1 H i Hm

h11 h1

k1hm1 hmkm

x

e11 e1k1 em1 emkm

Figure 21

Example 1 Consider Fig 22 In this figure we have

S = a b c d e xR = (a b c) (x c) (d e c)

e(a b c a) = e1

e(a b c b) = e2

e(x c x) = e3

e(d e c d) = e4

e(d e c e) = e5

The equation for c is

f(c) = min(1 1minus e1 middot f(a) + 1minus e2 middot f(b))timesmin(1minus e3 middot f(x))timesmin(1 1minus e4 middot f(d) + 1minus e5 middot f(e))


Take ei = 1 and compare this with Fig 16

a b x d e

c

e1 e2e3e4 e5

Figure 22

Definition 2 A Bayesian network N has the form 〈S P 〉 where S is a set ofnodes sub S times S is an acyclic dependence relation and P gives probability distri-butions based on (S ) defined below The symbol Ψx will denote the set of parentsof the node x in N ie Ψx = y | (y x) isin

bull The elements of S are considered Boolean variables which can be in only oneof two states either true = gt = 1 or false = perp = 0

bull If x isin S is a source node ie Ψx = empty then P (x) isin [0 1] denotes theprobability that x = gt P (notx) = 1minusP (x) denotes the probability that X = perp

bull If Ψx 6= empty then P gives the conditional probability of x on Ψx = y1 ykdenoted by P (x|Ψx) This means the following

a First for each q isin S consider a new atom letter denoted by notq Readq0 def= notq and q1 def= q

b For each ε isin 2k (a vector of numbers ε(i) 1 le i le k from 01) we lookat the option

minusrarry (ε) = andiyε(i)i

where we read y0i as notyi or yi = perp and y1

i as yi or yi = gtc We can now state what P (x|Ψx) is P (x|Ψx) gives values P (xΨx ε) isin

[0 1] for each ε isin 2k

bull Let x be a node such that Ψx 6= empty Assuming that the probabilities P (yi) foryi isin Ψx are known the probability P (x) of x can be calculated as

P (x) =sumε

P (xΨx ε)times P (minusrarry (ε))

where P (minusrarry (ε)) isprodi P (yi)ε(i) and P (yi)0 = 1minus P (yi) and P (yi)1 = P (yi)

Once we know P (x) we also know P (notx) = 1minus P (x)


Remark 3 Note that certainly P (x) ge 0 but also that P (x) lesumε P (minusrarry (ε)) = 1

since all P (yi) are probabilities

Remark 4 Note that Definition 2 is restricted to Boolean values Variables can beeither in state 1 = true or in state 0 = false So any conditional probability for avariable x depending on a variable y needs to give real numbers for each state of ysee condition c of Definition 2 This is similar to Pearlrsquos definition in [19] Notehowever that we propagate values in the direction of the arrows ie towards descen-dant nodes Pearl allows for updating of probabilities in both directions (ancestorsand descendant nodes) at any point in the network (see Section 223 of [19]) Aswe are dealing with argumentation networks the restriction to boolean variables ismore natural and we need not be concerned about it However the propagation ofupdates in both directions is important and should be investigated not only in orderto be more faithful in translating Bayesian networks into argumentation networksbut even without any connection with Bayesian networks Just by looking at tra-ditional Dung networks we may wish to insist on a certain status for an argument(ie ldquoinrdquo ldquooutrdquo or ldquoundecidedrdquo) and propagate this result in both directions Ourwork on Bayesian networks may give us ideas on how to do that not only in the caseof Bayesian Argumentation Networks (ie reflecting their behaviour) but perhapsby also taking advantage of the argumentation environment in developing an updatetheory applicable in argumentation in general and not just restricted to the contextof translated Bayesian networks This requires extensive research and we leave it asfuture work

Example 2 In order to illustrate Definition 2 we consider the network in Fig 1once more with the probability values given in Section 2 We have P (X) = 001 soP (notX) = 099 P (Y ) = 002 so P (notY ) = 098

ΨA = XY We have that P (AΨA X and Y ) = 095 P (AΨA X and notY ) =094 P (AΨAnotX and Y ) = 029 and P (AΨAnotX and notY ) = 0001 According to


Definition 2

P (A) = P (AΨA X and Y )times P (X)times P (Y ) +P (AΨA X and notY )times P (X)times P (notY ) +P (AΨAnotX and Y )times P (notX)times P (Y ) +P (AΨAnotX and notY )times P (notX)times P (notY )


P (A) asymp 0016

We then get P (notA) = 1minus P (A) = 0984 The values of P (U) (resp P (notU)) andP (W ) (resp P (notW )) are calculated in a similar way

Definition 3 We now translate any Bayesian network N = 〈S P 〉 into aBayesian Argumentation Network 〈AR e〉

a Assume the elements of S to be positive atoms of the form qi Let S be anew set of atoms of the form S = q | q isin S and let A0 = S cup S

b For any x isin S such that Ψx has k elements define two sets of new atoms

C(xΨx) = c(xΨx ε) | ε isin 2k andD(xΨx) = d(xΨx ε) | ε isin 2k

Let A = A0 cup⋃xisinS (C(xΨx) cupD(xΨx))

c We now define R on A

(a) Have (q q) and (q q) be in R(b) For any x isin S such that Ψx has k elements let (c(xΨx ε) d(xΨx ε))

be in R and let (y1minusε(i)i c(xΨx ε)) be in R

(c) Let (D(xΨx) x) be in R

d We now define e We let

e(c(xΨx ε) d(xΨx ε) c(xΨx ε)) = P (xΨx ε)


Theorem 1 Let N = 〈S P 〉 be a Bayesian network and let 〈AR e〉 be its(argumentation) translation Let the source nodes of N be the set Ω sube S and let fbe a solution extension of (AR e) such that f(w) = P (w) for w isin Ω Then forevery s isin S f(s) = P (s) (remember that S sube A)

Proof The proof is done by induction on the distance (level) of nodes from ΩLevel 0 nodes w isin ΩLevel n+ 1 nodes s such that all nodes in Ψs are of level up to n with at least

one of them being of level nEvery node in S has a unique level because (S ) is acyclic So we prove by

induction on the level of a node s that f(s) = P (s)Consider a node s of level n+ 1 such as the one in Fig 23 Its translation into

(AR e) is Fig 24 The computation of P (s) in the Bayesian network is

P (s) =sumεisin2k

P (s ys ε)timesprodi

P (yi)ε(i)

Our inductive assumption is that P (yi) = f(yi) We want to show that P (s) = f(s)Let us calculate f(s) from Fig 24 First we have

f(yi) = P (yi)f(notyi) = f(y0

i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi

(1minus f(yi)1minusε(i))

=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So

f(d(sys ε) = 1minus e(s ys c(s ys ε))times f(c(s ys ε))= 1minus P (s ys ε)times

prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε

(1minus f(d(s ys ε))

= min(1sumε


P (yi)ε(i))

= P (s)


Remark 5 Note that the translation of a Bayesian network uses only a restrictedfragment of Definition 1 where each node s satisfies for all yis and for all yjs

(yis s) isin R and (yjs s) isin R implies either yis and yjs are singleton sets or yis = yjs

This means that translated Bayesian networks do not have the combination of jointattacks and single attacks such as the one in Fig 22 Nodes either have a uniquejoint attack or zero or more attacks by individual nodes

ys y1 yk

s

Figure 23 A node of level n+ 1

noty1 y1 y1minusε(i)i

notyk yk

c(s ys ε)

d(s ys ε)

s

P (s ys ε) = e(s ys c(s ys ε))


Remark 6 a Note that the class of Bayesian Argumentation Networks con-tains more networks than just translated images of original Bayesian net-works These are indeed a type of argumentation networks inspired by lookingat Bayesian networks Note also that images of Bayesian networks are faithfuland can translated back into Bayesian networks

b Another important point to observe is that we are not imposing an argumen-tation structure on top of a Bayesian network thus analysing the Bayesiannetwork from an argumentation point of view Such methods are common inthe argumentation community This will be discussed further in Section 6


4 A Comprehensive Translation Example

In this section we show in detail how a Bayesian network can be translated into aBayesian Argumentation Network For this we will use the network of Fig 1 as anexample The translation starts with the source nodes of the Bayesian network

Consider a node α and its attackers Att(α) = β1 βk Assume that thenodes in Att(α) are all source nodes without any attackers When βi is a sourcenode it is given an initial probability P (βi) We model this in a BAN by adding anew node notβi for each βi such that βi and notβi attack each other Now in the newnetwork βi can assume any initial probability P (βi) we want with notβi obtainingP (notβi) = 1 minus P (βi) In order to represent all possible conjunctive expressions ofthe form

e2kminus1j=0 = andijβplusmnij

where β+ij

= βij and βminusij = notβij we need to create 2k intermediate points cj whoseattackers are all βi notβi such that ej |= βi and ej |= notβi respectively Note thatthe value of each cj will now correspond to a particular product

P (cj) =prodij

pij

where

pij =

1minus P (βi) if ej |= βiP (βi) if ej |= notβj

Because we also want to represent transmission values we now need to duplicateeach intermediate point cj with a corresponding xj and have cj attack xj withtransmission factor ej

In order to illustrate this let us recall the Bayesian network of Fig 1

X Y

A

U W

Consider the node A with attackers X and Y We first add points notX and notYsuch that each of X and notX and Y and notY attack each other If we give valueP (X) to X and value P (Y ) to Y then notX will get value 1 minus P (X) and notY willget value 1minus P (Y )


We now add points c00 c01 c10 and c11 corresponding to the conjunctive expres-sions

e0 c00 = X and Ye1 c01 = X and notYe2 c10 = notX and Ye3 c11 = notX and notY

Note that e0 |= X and e0 |= Y so we add attacks from X and Y into c00 and dothe same for e1ndashe3

X notX Y notY

c11 c10 c01 c00

We can now see that the probabilities of cij are given as follows

P (c00) = (1minus P (X)) middot (1minus P (Y ))P (c01) = (1minus P (X)) middot (1minus 1 + P (Y )) = (1minus P (X)) middot P (Y )P (c10) = (1minus 1 + P (X)) middot (1minus P (Y )) = P (X) middot (1minus P (Y ))P (c11) = (1minus 1 + P (X)) middot (1minus 1 + P (Y )) = P (X) middot P (Y )

Remember the initial parameters given for this network

P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us

P (c00) = 099times 098 = 09702P (c01) = 099times 002 = 00198P (c10) = 001times 098 = 00098P (c11) = 001times 002 = 00002

Now for each node cij we add an additional node xij so that we can incorporatethe transmission factors eij


X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

The resulting probabilities of the nodes xij are then given by the equations

P (x00) = 1minus ex00|ϕ00 middot c00




With our initial parameters we have that

ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence

P (x00) = 1minus 0001times 09702 = 09990P (x01) = 1minus 029times 00198 = 09942P (x10) = 1minus 094times 00098 = 09907P (x11) = 1minus 095times 00002 = 09998

The xij jointly attack A

X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1


This finally gives us the value of the probability of node A as

P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016

as before Therefore P (notA) asymp 0984Now to proceed to the next level all we need to do is to add a complementary

node to A notA such that A and notA attack each other This will give us P (notA) =1 minus P (A) As before we also need new intermediate nodes wA wnotA uA and unotAto incorporate transmission factors

The original attack of A on W is then realised by the joint attack of the newintermediate nodes wnotA and wA and the attack of notA on U is realised by the jointattack of the new intermediate nodes unotA and uA

X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1

ewnotA|notAewA|A eunotA|notAeuA|A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1

The values of wA wnotA uA and unotA are calculated as follows

P (wA) = 1minus ewA|A middot P (A)

P (wnotA) = 1minus ewnotA|notA middot P (notA)

P (uA) = 1minus euA|A middot P (A)

P (unotA) = 1minus eunotA|notA middot P (notA)


Since

ewA|A = 07ewnotA|notA = 001

euA|A = 09eunotA|notA = 005

We get

P (wA) = 1minus 07times 0016 asymp 0988

P (wnotA) = 1minus 001times 0984 asymp 0990

P (uA) = 1minus 09times 0016 asymp 0985

P (unotA) = 1minus 005times 0984 asymp 0950

We can finally calculate the values of W and U

P (W ) = min1 1minus P (wnotA) + 1minus P (wA) = min1 0009 + 00112 asymp 0021

P (U) = min1 1minus P (unotA) + 1minus P (uA) = min1 0049 + 00144 asymp 0063

These are exactly the same values we had before

5 Complexity Discussion

We need to consider two aspects involved in the complexity arising from the trans-lation of a Bayesian network into a Bayesian Argumentation Network The firstone is related to the translation itself whereas the second is related to the actualcomputation of the node values

It is easy to see that the translation of a Bayesian network into a BayesianArgumentation Network according to Definition 3 results in the creation of manynew nodes This addition of nodes is linear on the number of nodes of the origi-nal Bayesian network (item a of Definition 3) and exponential on the number ofancestors of each node of the original Bayesian network (item b of Definition 3)

The effect of the increase in the number of nodes in the actual computation ofnode values is deceptive

Although the translation does incur in the addition of many new nodes thecomplexity of the actual calculation of the node values remains basically the same


The nodes added in a Bayesian Argumentation Network simply encode explicitlythe intermediate calculations that would otherwise be implicitly done in the originalBayesian network

In conclusion a Bayesian Argumentation Network simply explicitly encodes asnodes the calculations that would have to be done anyway in the original Bayesiannetwork The new nodes name key values when using the conditional probabilitytables in the original calculations of the Bayesian network Therefore there is nosignificant additional cost

6 Related WorkLet us compare with several related papers We start with Vreeswijkrsquos ldquoArgumen-tation in bayesian belief networksrdquo [27] An important point to observe is that weare not using a meta-level device of extractingidentifying arguments and a notionof attack on the arguments from the Bayesian network and thus obtaining an argu-mentation network as is done in [27]

Vreeswijkrsquos approach does not give us a direct connectiontranslation betweenBayesian networks and argumentation networks It is more akin to imposing anargumentation structure on top of a Bayesian network thus analysing the Bayesiannetwork from an argumentation point of view Indeed Vreeswijk sees his approach[27] as

ldquoa proposal to look at Bayesian belief networks from the perspectiveof argumentation More specifically I propose an algorithm that enablesusers to start an argumentation process within the context of an existingBayesian belief network with some imagination the CPTs9 of theabove Bayesian network can be translated into the rule-base and evidence A next step towards argumentation is to chain rules into arguments What remains to be done to obtain a full-fledged argument systemis to define an attack relation between pairs of arguments To this end Ichoose to define the notion of attack on the basis of two notions that aremore elementary and (therefore) fall beyond the scope of a Dung-typeargument system viz the notion of counterargument and the notionof strength of an argument First I will discuss counter-arguments andthen I will discuss argument strength

Definition 4 (Attack) We say that argument a is attacked by argu-ment b written alarr b if it satisfies the following two conditions

9Conditional Probability Tables


1 Argument b is a counterargument of a sub-argument aprime of a2 Argument b is stronger than argument aprimerdquo

Such methods are common in the argumentation community For example tak-ing a logic program forming arguments using this logic program and thus generatingan argumentation network and then proving equivalence of the logic program withthe argumentation network (see for example [7 28])

The next related work we consider is the translation of various kinds of argu-mentation networks into Bayesian networks as is done for example in [16] The ideais beautifully described by the authors

ldquoThis paper presents a technique with which instances of argumentstructures in the Carneades model can be given a probabilistic semanticsby translating them into Bayesian networks The propagation of argu-ment applicability and statement acceptability can be expressed throughconditional probability tables This translation suggests a way to extendCarneades to improve its utility for decision support in the presence ofuncertaintyrdquo

Note that [27] identifies some argumentation structure in Bayesian networkswhilst [16] uses Bayesian networks to implement certain kinds of argumentationnetworks There is similarity but the approaches are different On the other handour approach is to faithfully represent Bayesian networks inside a newly motivatednumerical argumentation network

Finally the approach used in [25 24] is similar in spirit to the one used in [27]The idea is to provide explanations of Bayesian networks in legal or medical domainsusing support graphs The nodes of the support graph can be labelled to providean argumentative interpretation of the Bayesian network

7 Conclusions and Future WorkThe perceptive reader from the Bayesian community may take no interest in thetranslation of Bayesian networks into argumentation They will point out justifiablythat we are just translating the Bayesian algorithms into argumentation and thenusing these same algorithms under the guise of argumentation This may be ofinterest from the abstract mathematical expressive power point of view but that isall

We would like to show that there are indeed other benefits to this translationboth to the argumentation community and to the Bayesian community


a By interpreting Bayesian networks in some extension of Dungrsquos networks andvice-versa the argumentation community who also deals with updates andchange stands to benefit from the updating algorithms in Bayesian networksWe can add nodes and change values of nodes and use Bayesian propagationto propagate the changesThis needs to be studied further

b Bayesian networks are acyclic they have difficulties with loops (see [26]) Theequational approach in argumentation can deal with loops without any prob-lems (see for instance the forthcoming special issue in the Journal of Logic andComputation on the Handling of Loops in Argumentation Networks and see[11]) We notice that Bayesian networks can adopt the equational approach[15] in case of loops and simply solve the equations which arise on the prob-abilities A solution always exists by Brouwerrsquos fixed-point theorem We arein fact surprised that this approach has never been taken (to the best of ourknowledge) by the Bayesian community One can then use loop handling tech-niques from argumentation to propagate updates and changes in the (cyclic)Bayesian network by simply solving equations By implementing Bayesian net-works in argumentation under the equational approach we can allow for loopsin Bayesian networks and still hopefully see a way to obtain results again thisrequires further research

c A third benefit to Bayesian networks is the possibility to develop proof theoryfor Bayesian networks The extension of argumentation networks which hoststhe translation of Bayesian networks is implemented in Łukasiewicz infinite-valued logic with product10 This Łukasiewicz logic has a proof theory (see[10]) We can therefore hope for the same for Bayesian networks Again thisneeds to be studied

In addition the following needs to be investigated in detail

a Take an example of a cyclic Bayesian network recognised as interesting inthe Bayesian community and translate it into argumentation See how argu-mentation handles the loops and try to find suitable algorithms for handlingcycles in Bayesian networks These algorithms should now be independent ofargumentation

b Develop algorithms of updating argumentation networks by looking at theexamples of updating used in the Bayesian domain and the way they implement

10This is actually shown in the current paper (see Remark 2)


the updates There are important papers investigating updates and revision ofargumentation networks by central figures in the argumentation communityincluding [4 22 21 5 8 3 9] Addressing these papers will be done in futurework

c Identify proof theoretic queries in Bayesian networks and import proof theoryfrom Łukasiewicz logic to model them

References[1] H Barringer D M Gabbay and J Woods Temporal dynamics of support and attack

networks In D Hutter and W Stephan editors Mechanizing Mathematical Reasoning2005 LNCS vol 2605

[2] Howard Barringer Dov M Gabbay and John Woods Temporal numerical and meta-level dynamics in argumentation networks Argument amp Computation 3(2-3)143ndash2022012

[3] R Baumann and G Brewka AGM meets abstract argumentation Expansion andrevision for dung frameworks In Proceedings of the Twenty-Fourth International JointConference on Artificial Intelligence IJCAI 2015 Buenos Aires Argentina July 25-312015 pages 2734ndash2740 2015

[4] G Brewka S Ellmauthaler H Strass J P Wallner and S Woltran Abstract di-alectical frameworks revisited In Proceedings of the Twenty-Third International JointConference on Artificial Intelligence IJCAI rsquo13 pages 803ndash809 AAAI Press 2013

[5] G Brewka and S Woltran GRAPPA A semantical framework for graph-based argu-ment processing In ECAI 2014 - 21st European Conference on Artificial Intelligence18-22 August 2014 Prague Czech Republic - Including Prestigious Applications of In-telligent Systems (PAIS 2014) pages 153ndash158 2014

[6] Gerhard Brewka and Stefan Woltran Abstract dialectical frameworks In Principlesof Knowledge Representation and Reasoning Proceedings of the Twelfth InternationalConference KR 2010 AAAI Press 2010

[7] Martin Caminada Samy Saacute Jo ao Alcacircntara and Wolfgang DvoAringŹaacutek On the equiv-alence between logic programming semantics and argumentation semantics Interna-tional Journal of Approximate Reasoning 5887 ndash 111 2015 Special Issue of the TwelfthEuropean Conference on Symbolic and Quantitative Approaches to Reasoning with Un-certainty (ECSQARU 2013)

[8] S Coste-Marquis S Konieczny J-G Mailly and P Marquis On the revision of ar-gumentation systems Minimal change of arguments statuses In 14th InternationalConference on Principles of Knowledge Representation and Reasoning (KRrsquo14) Vi-enna July 2014

[9] M Diller A Haret T Linsbichler S Ruumlmmele and S Woltran An extension-basedapproach to belief revision in abstract argumentation In Proceedings of the 24th Inter-


national Conference on Artificial Intelligence IJCAIrsquo15 pages 2926ndash2932 AAAI Press2015

[10] Francesc Esteva Lluiacutes Godo and Enrico Marchioni Fuzzy Logics with Enriched Lan-guage chapter VIII pages 627 ndash 711 Number 38 in Vol 2 College PublicationsLondon 2011

[11] D M Gabbay The handling of loops in argumentation networks Journal of Logic andComputation 2014

[12] D M Gabbay Theory of semi-instantiation in abstract argumentation Logica Uni-versalis To appear

[13] D M Gabbay and O Rodrigues Probabilistic argumentation An equational approachLogica Universalis pages 1ndash38 2015

[14] Dov M Gabbay Fibring argumentation frames Studia Logica 93(2-3)231ndash295 2009[15] Dov M Gabbay Equational approach to argumentation networks Argument amp Com-

putation 3(2-3)87ndash142 2012[16] M Grabmair T F Gordon and D Walton Probabilistic semantics for the carneades

argument model using bayesian networks In Proceedings of the 2010 Conference onComputational Models of Argument Proceedings of COMMA 2010 pages 255ndash266Amsterdam The Netherlands The Netherlands 2010 IOS Press

[17] Kevin B Korb and Ann E Nicholson Bayesian Artificial Intelligence Second EditionCRC Press Inc Boca Raton FL USA 2nd edition 2010

[18] Soslashren Holbech Nielsen and Simon Parsons A generalization of dungrsquos abstract frame-work for argumentation Arguing with sets of attacking arguments In Nicolas MaudetSimon Parsons and Iyad Rahwan editors ArgMAS volume 4766 of Lecture Notes inComputer Science pages 54ndash73 Springer 2006

[19] J Pearl Fusion propagation and structuring in belief networks Artificial Intelligence29(3)241ndash288 September 1986

[20] J Pearl Probabilistic Reasoning in Intelligent Systems Networks of Plausible Infer-ence Morgan Kaufmann Publishers Inc San Francisco CA USA 1988

[21] S Polberg and D Doder Probabilistic abstract dialectical frameworks In EduardoFermeacute and Joatildeo Leite editors Logics in Artificial Intelligence 14th European Confer-ence JELIA 2014 Funchal Madeira Portugal September 24-26 2014 Proceedingspages 591ndash599 Cham 2014 Springer International Publishing

[22] J Puumlhrer Realizability of three-valued semantics for abstract dialectical frameworksIn Proceedings of the 24th International Conference on Artificial Intelligence IJCAIrsquo15pages 3171ndash3177 AAAI Press 2015

[23] A Rose Formalisations of further alefsym0-valued Łukasiewicz propositional calculi Journalof Symbolic Logic 43207ndash210 6 1978

[24] S T Timmer J-J Ch Meyer H Prakken S Renooij and B Verheij Explainingbayesian networks using argumentation In Symbolic and Quantitative Approaches toReasoning with Uncertainty - 13th European Conference ECSQARU 2015 CompiegravegneFrance July 15-17 2015 Proceedings pages 83ndash92 2015


[25] S T Timmer J-J Ch Meyer H Prakken S Renooij and B Verheij A structure-guided approach to capturing bayesian reasoning about legal evidence in argumentationIn Proceedings of the 15th International Conference on Artificial Intelligence and LawICAIL rsquo15 pages 109ndash118 New York NY USA 2015 ACM

[26] A L Tulupyev and SI Nikolenko Directed cycles in bayesian belief networks Proba-bilistic semantics and consistency checking complexity In A Gelbukh A de Albornozand H Terashima editors MICAI 2005 Lectune notes in Artificial Intelligence 3789pages 214ndash223 Springer-Verlag Berlin Heidelberg 2005

[27] G A W Vreeswijk Argumentation in bayesian belief networks In Proceedings of theFirst International Conference on Argumentation in Multi-Agent Systems ArgMASrsquo04pages 111ndash129 Berlin Heidelberg 2005 Springer-Verlag

[28] Yining Wu Martin Caminada and Dov M Gabbay Complete extensions in argu-mentation coincide with 3-valued stable models in logic programming Studia Logica93(2)383ndash403 2009

Received September 2015

Introduction

Background Discussion

A Common Ground for Bayesian Argumentation and Abstract Dialectical Frameworks

Combining It All

Formal Definitions

A Comprehensive Translation Example

Complexity Discussion

Related Work

Conclusions and Future Work


Bayesian Argumentation Networks We shall see later that the extra features areall well-known features existing in the literature in various contexts

In order to illustrate these ideas consider the network 〈SR〉 of Fig 1 Inthis figure the arrows just indicate parenthood If the arrows are considered asattacks then 〈SR〉 is a traditional abstract argumentation framework (hencefortha ldquoDung networkrdquo) and there is only one complete extension E = XY UW(with A = ldquooutrdquo)

X Y

A

U W

Figure 1 A sample argumentation network

The operating assumptions (which are violated in Bayesian networks) are

1 Since there is no connection (ie attack) going into X and into Y then Xand Y are ldquoinrdquo (intuitively meaning X = Y = gt)

2 Since U and V have the same parent A we treat U and V in the same way

3 We do not mind having cycles ie R need not be acyclic

There are other assumptions in the case of argumentation networks but let usconcentrate only on the ones above

Bayesian networks do not allow for cycles (R must be acyclic) and they do notdetermine the values of nodes without parents such as X and Y Moreover theyare not committed to treating nodes with the same parents (such as U and W ) thesame way Such a view is not new to argumentation In fact such a view is sharedby Abstract Dialectical Frameworks (ADFs) [6] In an ADF each node α dependson its parents say β1 βn via a Boolean formula Ψα specific to α Thus wehave that α depends on Ψα(β1 βn) in the ADF case and we want that

αharr Ψα(β1 βn)

In Dungrsquos networks the same constraint Ψα is imposed on all nodes α namely

αharrnandi=1notβi













e = andiαplusmni





summ π(m) = 1




α

πplusmn(α)












summ π(m) = 1





3See [13]







X Y v

gt gt eFA|11

gt perp eFA|10

perp gt eFA|01

perp perp eFA|00


A v

gt eFU|1

perp eFU|0


A v

gt eFW |1

perp eFW |0





Burglary Earthquake

Alarm




X Y v


A v


A v

gt eFW |1 = 070perp eFW |0 = 001






So


= 000019 + 0009212 + 0005742 + 00009702= 00161142 asymp 0016
















X Y

A





A eFA|00

eFA|10

eFA|11

eFA|01







a Attack formations





α

β

α

Xαβ

Yαβ

β

(L) (R)

Figure 4







α

F [α β] =



β

Input α

Output β

Figure 5





inputoutput

A

Figure 6

inputoutput F 1

inputoutput F 2

Input 1

Output 1

Input 2

Output 2



β1 β2 βk

α



f(α) =kprodi=1

(1minus f(βi)) (1)




f(α) =kprodi=1



c Joint attacks


β1 β2 βk

α

e1 e2 ek



f(α) = 1minusprodi

f(βi)


β1 βk

α






β1 β2

α= β1 middot β2

π product

Figure 11

β1 β2

x

α

and joint

Figure 12

β1 β2

α

and joint

Figure 13

β1 β2

y

α

π product

Figure 14










ϕ([X1]) = notX1

ϕ([X1 X2]) = X2 rarr ϕ([X1])



(1minusXi))

Then



= ϕ(X1 Xn+1)


ϕ(X1) = notX1




a b

c

Figure 15

a b x d e

c


Figure 16


2 c = min(1 12 + 1




and([X1 Xn]) = Xn

and([X1 Xnminus1])


β1 β2 β3

c


prodi

(1minus βi)

Figure 17

ϕ11 ϕ10 ϕ01 ϕ00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1


22 Combining It All



and

f(A) = min (1sum






notX X Y notY

c00

notX X Y notY

c01

notX X Y notY

c10

notX X Y notY

c11

1 1

1 1

1 1

1 1

F 00[notXnotY ] =


F 01[notXY ] =


F 10[XnotY ] =


F 11[XY ] =


Figure 19

















f(x) =mprodj=1

min(1kjsumi=1





h1 hk

x

e1 ek

H = h1 hk(Hx h1) = e1

(Hx h2) = e2

(Hx hk) = ek

Figure 20

H1 H i Hm

h11 h1

k1hm1 hmkm

x

e11 e1k1 em1 emkm

Figure 21



e(a b c a) = e1

e(a b c b) = e2

e(x c x) = e3

e(d e c d) = e4

e(d e c e) = e5





a b x d e

c

e1 e2e3e4 e5

Figure 22












P (x) =sumε











Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work














e = andiαplusmni





summ π(m) = 1




α

πplusmn(α)












summ π(m) = 1





3See [13]







X Y v

gt gt eFA|11

gt perp eFA|10

perp gt eFA|01

perp perp eFA|00


A v

gt eFU|1

perp eFU|0


A v

gt eFW |1

perp eFW |0





Burglary Earthquake

Alarm




X Y v


A v


A v

gt eFW |1 = 070perp eFW |0 = 001






So


= 000019 + 0009212 + 0005742 + 00009702= 00161142 asymp 0016
















X Y

A





A eFA|00

eFA|10

eFA|11

eFA|01







a Attack formations





α

β

α

Xαβ

Yαβ

β

(L) (R)

Figure 4







α

F [α β] =



β

Input α

Output β

Figure 5





inputoutput

A

Figure 6

inputoutput F 1

inputoutput F 2

Input 1

Output 1

Input 2

Output 2



β1 β2 βk

α



f(α) =kprodi=1

(1minus f(βi)) (1)




f(α) =kprodi=1



c Joint attacks


β1 β2 βk

α

e1 e2 ek



f(α) = 1minusprodi

f(βi)


β1 βk

α






β1 β2

α= β1 middot β2

π product

Figure 11

β1 β2

x

α

and joint

Figure 12

β1 β2

α

and joint

Figure 13

β1 β2

y

α

π product

Figure 14










ϕ([X1]) = notX1

ϕ([X1 X2]) = X2 rarr ϕ([X1])



(1minusXi))

Then



= ϕ(X1 Xn+1)


ϕ(X1) = notX1




a b

c

Figure 15

a b x d e

c


Figure 16


2 c = min(1 12 + 1




and([X1 Xn]) = Xn

and([X1 Xnminus1])


β1 β2 β3

c


prodi

(1minus βi)

Figure 17

ϕ11 ϕ10 ϕ01 ϕ00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1


22 Combining It All



and

f(A) = min (1sum






notX X Y notY

c00

notX X Y notY

c01

notX X Y notY

c10

notX X Y notY

c11

1 1

1 1

1 1

1 1

F 00[notXnotY ] =


F 01[notXY ] =


F 10[XnotY ] =


F 11[XY ] =


Figure 19

















f(x) =mprodj=1

min(1kjsumi=1





h1 hk

x

e1 ek

H = h1 hk(Hx h1) = e1

(Hx h2) = e2

(Hx hk) = ek

Figure 20

H1 H i Hm

h11 h1

k1hm1 hmkm

x

e11 e1k1 em1 emkm

Figure 21



e(a b c a) = e1

e(a b c b) = e2

e(x c x) = e3

e(d e c d) = e4

e(d e c e) = e5





a b x d e

c

e1 e2e3e4 e5

Figure 22












P (x) =sumε











Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work











summ π(m) = 1





3See [13]







X Y v

gt gt eFA|11

gt perp eFA|10

perp gt eFA|01

perp perp eFA|00


A v

gt eFU|1

perp eFU|0


A v

gt eFW |1

perp eFW |0





Burglary Earthquake

Alarm




X Y v


A v


A v

gt eFW |1 = 070perp eFW |0 = 001






So


= 000019 + 0009212 + 0005742 + 00009702= 00161142 asymp 0016
















X Y

A





A eFA|00

eFA|10

eFA|11

eFA|01







a Attack formations





α

β

α

Xαβ

Yαβ

β

(L) (R)

Figure 4







α

F [α β] =



β

Input α

Output β

Figure 5





inputoutput

A

Figure 6

inputoutput F 1

inputoutput F 2

Input 1

Output 1

Input 2

Output 2



β1 β2 βk

α



f(α) =kprodi=1

(1minus f(βi)) (1)




f(α) =kprodi=1



c Joint attacks


β1 β2 βk

α

e1 e2 ek



f(α) = 1minusprodi

f(βi)


β1 βk

α






β1 β2

α= β1 middot β2

π product

Figure 11

β1 β2

x

α

and joint

Figure 12

β1 β2

α

and joint

Figure 13

β1 β2

y

α

π product

Figure 14










ϕ([X1]) = notX1

ϕ([X1 X2]) = X2 rarr ϕ([X1])



(1minusXi))

Then



= ϕ(X1 Xn+1)


ϕ(X1) = notX1




a b

c

Figure 15

a b x d e

c


Figure 16


2 c = min(1 12 + 1




and([X1 Xn]) = Xn

and([X1 Xnminus1])


β1 β2 β3

c


prodi

(1minus βi)

Figure 17

ϕ11 ϕ10 ϕ01 ϕ00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1


22 Combining It All



and

f(A) = min (1sum






notX X Y notY

c00

notX X Y notY

c01

notX X Y notY

c10

notX X Y notY

c11

1 1

1 1

1 1

1 1

F 00[notXnotY ] =


F 01[notXY ] =


F 10[XnotY ] =


F 11[XY ] =


Figure 19

















f(x) =mprodj=1

min(1kjsumi=1





h1 hk

x

e1 ek

H = h1 hk(Hx h1) = e1

(Hx h2) = e2

(Hx hk) = ek

Figure 20

H1 H i Hm

h11 h1

k1hm1 hmkm

x

e11 e1k1 em1 emkm

Figure 21



e(a b c a) = e1

e(a b c b) = e2

e(x c x) = e3

e(d e c d) = e4

e(d e c e) = e5





a b x d e

c

e1 e2e3e4 e5

Figure 22












P (x) =sumε











Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work








X Y v

gt gt eFA|11

gt perp eFA|10

perp gt eFA|01

perp perp eFA|00


A v

gt eFU|1

perp eFU|0


A v

gt eFW |1

perp eFW |0





Burglary Earthquake

Alarm




X Y v


A v


A v

gt eFW |1 = 070perp eFW |0 = 001






So


= 000019 + 0009212 + 0005742 + 00009702= 00161142 asymp 0016
















X Y

A





A eFA|00

eFA|10

eFA|11

eFA|01







a Attack formations





α

β

α

Xαβ

Yαβ

β

(L) (R)

Figure 4







α

F [α β] =



β

Input α

Output β

Figure 5





inputoutput

A

Figure 6

inputoutput F 1

inputoutput F 2

Input 1

Output 1

Input 2

Output 2



β1 β2 βk

α



f(α) =kprodi=1

(1minus f(βi)) (1)




f(α) =kprodi=1



c Joint attacks


β1 β2 βk

α

e1 e2 ek



f(α) = 1minusprodi

f(βi)


β1 βk

α






β1 β2

α= β1 middot β2

π product

Figure 11

β1 β2

x

α

and joint

Figure 12

β1 β2

α

and joint

Figure 13

β1 β2

y

α

π product

Figure 14










ϕ([X1]) = notX1

ϕ([X1 X2]) = X2 rarr ϕ([X1])



(1minusXi))

Then



= ϕ(X1 Xn+1)


ϕ(X1) = notX1




a b

c

Figure 15

a b x d e

c


Figure 16


2 c = min(1 12 + 1




and([X1 Xn]) = Xn

and([X1 Xnminus1])


β1 β2 β3

c


prodi

(1minus βi)

Figure 17

ϕ11 ϕ10 ϕ01 ϕ00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1


22 Combining It All



and

f(A) = min (1sum






notX X Y notY

c00

notX X Y notY

c01

notX X Y notY

c10

notX X Y notY

c11

1 1

1 1

1 1

1 1

F 00[notXnotY ] =


F 01[notXY ] =


F 10[XnotY ] =


F 11[XY ] =


Figure 19

















f(x) =mprodj=1

min(1kjsumi=1





h1 hk

x

e1 ek

H = h1 hk(Hx h1) = e1

(Hx h2) = e2

(Hx hk) = ek

Figure 20

H1 H i Hm

h11 h1

k1hm1 hmkm

x

e11 e1k1 em1 emkm

Figure 21



e(a b c a) = e1

e(a b c b) = e2

e(x c x) = e3

e(d e c d) = e4

e(d e c e) = e5





a b x d e

c

e1 e2e3e4 e5

Figure 22












P (x) =sumε











Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work





Burglary Earthquake

Alarm




X Y v


A v


A v

gt eFW |1 = 070perp eFW |0 = 001






So


= 000019 + 0009212 + 0005742 + 00009702= 00161142 asymp 0016
















X Y

A





A eFA|00

eFA|10

eFA|11

eFA|01







a Attack formations





α

β

α

Xαβ

Yαβ

β

(L) (R)

Figure 4







α

F [α β] =



β

Input α

Output β

Figure 5





inputoutput

A

Figure 6

inputoutput F 1

inputoutput F 2

Input 1

Output 1

Input 2

Output 2



β1 β2 βk

α



f(α) =kprodi=1

(1minus f(βi)) (1)




f(α) =kprodi=1



c Joint attacks


β1 β2 βk

α

e1 e2 ek



f(α) = 1minusprodi

f(βi)


β1 βk

α






β1 β2

α= β1 middot β2

π product

Figure 11

β1 β2

x

α

and joint

Figure 12

β1 β2

α

and joint

Figure 13

β1 β2

y

α

π product

Figure 14










ϕ([X1]) = notX1

ϕ([X1 X2]) = X2 rarr ϕ([X1])



(1minusXi))

Then



= ϕ(X1 Xn+1)


ϕ(X1) = notX1




a b

c

Figure 15

a b x d e

c


Figure 16


2 c = min(1 12 + 1




and([X1 Xn]) = Xn

and([X1 Xnminus1])


β1 β2 β3

c


prodi

(1minus βi)

Figure 17

ϕ11 ϕ10 ϕ01 ϕ00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1


22 Combining It All



and

f(A) = min (1sum






notX X Y notY

c00

notX X Y notY

c01

notX X Y notY

c10

notX X Y notY

c11

1 1

1 1

1 1

1 1

F 00[notXnotY ] =


F 01[notXY ] =


F 10[XnotY ] =


F 11[XY ] =


Figure 19

















f(x) =mprodj=1

min(1kjsumi=1





h1 hk

x

e1 ek

H = h1 hk(Hx h1) = e1

(Hx h2) = e2

(Hx hk) = ek

Figure 20

H1 H i Hm

h11 h1

k1hm1 hmkm

x

e11 e1k1 em1 emkm

Figure 21



e(a b c a) = e1

e(a b c b) = e2

e(x c x) = e3

e(d e c d) = e4

e(d e c e) = e5





a b x d e

c

e1 e2e3e4 e5

Figure 22












P (x) =sumε











Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work







So


= 000019 + 0009212 + 0005742 + 00009702= 00161142 asymp 0016
















X Y

A





A eFA|00

eFA|10

eFA|11

eFA|01







a Attack formations





α

β

α

Xαβ

Yαβ

β

(L) (R)

Figure 4







α

F [α β] =



β

Input α

Output β

Figure 5





inputoutput

A

Figure 6

inputoutput F 1

inputoutput F 2

Input 1

Output 1

Input 2

Output 2



β1 β2 βk

α



f(α) =kprodi=1

(1minus f(βi)) (1)




f(α) =kprodi=1



c Joint attacks


β1 β2 βk

α

e1 e2 ek



f(α) = 1minusprodi

f(βi)


β1 βk

α






β1 β2

α= β1 middot β2

π product

Figure 11

β1 β2

x

α

and joint

Figure 12

β1 β2

α

and joint

Figure 13

β1 β2

y

α

π product

Figure 14










ϕ([X1]) = notX1

ϕ([X1 X2]) = X2 rarr ϕ([X1])



(1minusXi))

Then



= ϕ(X1 Xn+1)


ϕ(X1) = notX1




a b

c

Figure 15

a b x d e

c


Figure 16


2 c = min(1 12 + 1




and([X1 Xn]) = Xn

and([X1 Xnminus1])


β1 β2 β3

c


prodi

(1minus βi)

Figure 17

ϕ11 ϕ10 ϕ01 ϕ00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1


22 Combining It All



and

f(A) = min (1sum






notX X Y notY

c00

notX X Y notY

c01

notX X Y notY

c10

notX X Y notY

c11

1 1

1 1

1 1

1 1

F 00[notXnotY ] =


F 01[notXY ] =


F 10[XnotY ] =


F 11[XY ] =


Figure 19

















f(x) =mprodj=1

min(1kjsumi=1





h1 hk

x

e1 ek

H = h1 hk(Hx h1) = e1

(Hx h2) = e2

(Hx hk) = ek

Figure 20

H1 H i Hm

h11 h1

k1hm1 hmkm

x

e11 e1k1 em1 emkm

Figure 21



e(a b c a) = e1

e(a b c b) = e2

e(x c x) = e3

e(d e c d) = e4

e(d e c e) = e5





a b x d e

c

e1 e2e3e4 e5

Figure 22












P (x) =sumε











Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work











X Y

A





A eFA|00

eFA|10

eFA|11

eFA|01







a Attack formations





α

β

α

Xαβ

Yαβ

β

(L) (R)

Figure 4







α

F [α β] =



β

Input α

Output β

Figure 5





inputoutput

A

Figure 6

inputoutput F 1

inputoutput F 2

Input 1

Output 1

Input 2

Output 2



β1 β2 βk

α



f(α) =kprodi=1

(1minus f(βi)) (1)




f(α) =kprodi=1



c Joint attacks


β1 β2 βk

α

e1 e2 ek



f(α) = 1minusprodi

f(βi)


β1 βk

α






β1 β2

α= β1 middot β2

π product

Figure 11

β1 β2

x

α

and joint

Figure 12

β1 β2

α

and joint

Figure 13

β1 β2

y

α

π product

Figure 14










ϕ([X1]) = notX1

ϕ([X1 X2]) = X2 rarr ϕ([X1])



(1minusXi))

Then



= ϕ(X1 Xn+1)


ϕ(X1) = notX1




a b

c

Figure 15

a b x d e

c


Figure 16


2 c = min(1 12 + 1




and([X1 Xn]) = Xn

and([X1 Xnminus1])


β1 β2 β3

c


prodi

(1minus βi)

Figure 17

ϕ11 ϕ10 ϕ01 ϕ00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1


22 Combining It All



and

f(A) = min (1sum






notX X Y notY

c00

notX X Y notY

c01

notX X Y notY

c10

notX X Y notY

c11

1 1

1 1

1 1

1 1

F 00[notXnotY ] =


F 01[notXY ] =


F 10[XnotY ] =


F 11[XY ] =


Figure 19

















f(x) =mprodj=1

min(1kjsumi=1





h1 hk

x

e1 ek

H = h1 hk(Hx h1) = e1

(Hx h2) = e2

(Hx hk) = ek

Figure 20

H1 H i Hm

h11 h1

k1hm1 hmkm

x

e11 e1k1 em1 emkm

Figure 21



e(a b c a) = e1

e(a b c b) = e2

e(x c x) = e3

e(d e c d) = e4

e(d e c e) = e5





a b x d e

c

e1 e2e3e4 e5

Figure 22












P (x) =sumε











Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work




a Attack formations





α

β

α

Xαβ

Yαβ

β

(L) (R)

Figure 4







α

F [α β] =



β

Input α

Output β

Figure 5





inputoutput

A

Figure 6

inputoutput F 1

inputoutput F 2

Input 1

Output 1

Input 2

Output 2



β1 β2 βk

α



f(α) =kprodi=1

(1minus f(βi)) (1)




f(α) =kprodi=1



c Joint attacks


β1 β2 βk

α

e1 e2 ek



f(α) = 1minusprodi

f(βi)


β1 βk

α






β1 β2

α= β1 middot β2

π product

Figure 11

β1 β2

x

α

and joint

Figure 12

β1 β2

α

and joint

Figure 13

β1 β2

y

α

π product

Figure 14










ϕ([X1]) = notX1

ϕ([X1 X2]) = X2 rarr ϕ([X1])



(1minusXi))

Then



= ϕ(X1 Xn+1)


ϕ(X1) = notX1




a b

c

Figure 15

a b x d e

c


Figure 16


2 c = min(1 12 + 1




and([X1 Xn]) = Xn

and([X1 Xnminus1])


β1 β2 β3

c


prodi

(1minus βi)

Figure 17

ϕ11 ϕ10 ϕ01 ϕ00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1


22 Combining It All



and

f(A) = min (1sum






notX X Y notY

c00

notX X Y notY

c01

notX X Y notY

c10

notX X Y notY

c11

1 1

1 1

1 1

1 1

F 00[notXnotY ] =


F 01[notXY ] =


F 10[XnotY ] =


F 11[XY ] =


Figure 19

















f(x) =mprodj=1

min(1kjsumi=1





h1 hk

x

e1 ek

H = h1 hk(Hx h1) = e1

(Hx h2) = e2

(Hx hk) = ek

Figure 20

H1 H i Hm

h11 h1

k1hm1 hmkm

x

e11 e1k1 em1 emkm

Figure 21



e(a b c a) = e1

e(a b c b) = e2

e(x c x) = e3

e(d e c d) = e4

e(d e c e) = e5





a b x d e

c

e1 e2e3e4 e5

Figure 22












P (x) =sumε











Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work




α

F [α β] =



β

Input α

Output β

Figure 5





inputoutput

A

Figure 6

inputoutput F 1

inputoutput F 2

Input 1

Output 1

Input 2

Output 2



β1 β2 βk

α



f(α) =kprodi=1

(1minus f(βi)) (1)




f(α) =kprodi=1



c Joint attacks


β1 β2 βk

α

e1 e2 ek



f(α) = 1minusprodi

f(βi)


β1 βk

α






β1 β2

α= β1 middot β2

π product

Figure 11

β1 β2

x

α

and joint

Figure 12

β1 β2

α

and joint

Figure 13

β1 β2

y

α

π product

Figure 14










ϕ([X1]) = notX1

ϕ([X1 X2]) = X2 rarr ϕ([X1])



(1minusXi))

Then



= ϕ(X1 Xn+1)


ϕ(X1) = notX1




a b

c

Figure 15

a b x d e

c


Figure 16


2 c = min(1 12 + 1




and([X1 Xn]) = Xn

and([X1 Xnminus1])


β1 β2 β3

c


prodi

(1minus βi)

Figure 17

ϕ11 ϕ10 ϕ01 ϕ00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1


22 Combining It All



and

f(A) = min (1sum






notX X Y notY

c00

notX X Y notY

c01

notX X Y notY

c10

notX X Y notY

c11

1 1

1 1

1 1

1 1

F 00[notXnotY ] =


F 01[notXY ] =


F 10[XnotY ] =


F 11[XY ] =


Figure 19

















f(x) =mprodj=1

min(1kjsumi=1





h1 hk

x

e1 ek

H = h1 hk(Hx h1) = e1

(Hx h2) = e2

(Hx hk) = ek

Figure 20

H1 H i Hm

h11 h1

k1hm1 hmkm

x

e11 e1k1 em1 emkm

Figure 21



e(a b c a) = e1

e(a b c b) = e2

e(x c x) = e3

e(d e c d) = e4

e(d e c e) = e5





a b x d e

c

e1 e2e3e4 e5

Figure 22












P (x) =sumε











Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work



inputoutput

A

Figure 6

inputoutput F 1

inputoutput F 2

Input 1

Output 1

Input 2

Output 2



β1 β2 βk

α



f(α) =kprodi=1

(1minus f(βi)) (1)




f(α) =kprodi=1



c Joint attacks


β1 β2 βk

α

e1 e2 ek



f(α) = 1minusprodi

f(βi)


β1 βk

α






β1 β2

α= β1 middot β2

π product

Figure 11

β1 β2

x

α

and joint

Figure 12

β1 β2

α

and joint

Figure 13

β1 β2

y

α

π product

Figure 14










ϕ([X1]) = notX1

ϕ([X1 X2]) = X2 rarr ϕ([X1])



(1minusXi))

Then



= ϕ(X1 Xn+1)


ϕ(X1) = notX1




a b

c

Figure 15

a b x d e

c


Figure 16


2 c = min(1 12 + 1




and([X1 Xn]) = Xn

and([X1 Xnminus1])


β1 β2 β3

c


prodi

(1minus βi)

Figure 17

ϕ11 ϕ10 ϕ01 ϕ00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1


22 Combining It All



and

f(A) = min (1sum






notX X Y notY

c00

notX X Y notY

c01

notX X Y notY

c10

notX X Y notY

c11

1 1

1 1

1 1

1 1

F 00[notXnotY ] =


F 01[notXY ] =


F 10[XnotY ] =


F 11[XY ] =


Figure 19

















f(x) =mprodj=1

min(1kjsumi=1





h1 hk

x

e1 ek

H = h1 hk(Hx h1) = e1

(Hx h2) = e2

(Hx hk) = ek

Figure 20

H1 H i Hm

h11 h1

k1hm1 hmkm

x

e11 e1k1 em1 emkm

Figure 21



e(a b c a) = e1

e(a b c b) = e2

e(x c x) = e3

e(d e c d) = e4

e(d e c e) = e5





a b x d e

c

e1 e2e3e4 e5

Figure 22












P (x) =sumε











Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work



β1 β2 βk

α



f(α) =kprodi=1

(1minus f(βi)) (1)




f(α) =kprodi=1



c Joint attacks


β1 β2 βk

α

e1 e2 ek



f(α) = 1minusprodi

f(βi)


β1 βk

α






β1 β2

α= β1 middot β2

π product

Figure 11

β1 β2

x

α

and joint

Figure 12

β1 β2

α

and joint

Figure 13

β1 β2

y

α

π product

Figure 14










ϕ([X1]) = notX1

ϕ([X1 X2]) = X2 rarr ϕ([X1])



(1minusXi))

Then



= ϕ(X1 Xn+1)


ϕ(X1) = notX1




a b

c

Figure 15

a b x d e

c


Figure 16


2 c = min(1 12 + 1




and([X1 Xn]) = Xn

and([X1 Xnminus1])


β1 β2 β3

c


prodi

(1minus βi)

Figure 17

ϕ11 ϕ10 ϕ01 ϕ00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1


22 Combining It All



and

f(A) = min (1sum






notX X Y notY

c00

notX X Y notY

c01

notX X Y notY

c10

notX X Y notY

c11

1 1

1 1

1 1

1 1

F 00[notXnotY ] =


F 01[notXY ] =


F 10[XnotY ] =


F 11[XY ] =


Figure 19

















f(x) =mprodj=1

min(1kjsumi=1





h1 hk

x

e1 ek

H = h1 hk(Hx h1) = e1

(Hx h2) = e2

(Hx hk) = ek

Figure 20

H1 H i Hm

h11 h1

k1hm1 hmkm

x

e11 e1k1 em1 emkm

Figure 21



e(a b c a) = e1

e(a b c b) = e2

e(x c x) = e3

e(d e c d) = e4

e(d e c e) = e5





a b x d e

c

e1 e2e3e4 e5

Figure 22












P (x) =sumε











Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work



β1 β2 βk

α

e1 e2 ek



f(α) = 1minusprodi

f(βi)


β1 βk

α






β1 β2

α= β1 middot β2

π product

Figure 11

β1 β2

x

α

and joint

Figure 12

β1 β2

α

and joint

Figure 13

β1 β2

y

α

π product

Figure 14










ϕ([X1]) = notX1

ϕ([X1 X2]) = X2 rarr ϕ([X1])



(1minusXi))

Then



= ϕ(X1 Xn+1)


ϕ(X1) = notX1




a b

c

Figure 15

a b x d e

c


Figure 16


2 c = min(1 12 + 1




and([X1 Xn]) = Xn

and([X1 Xnminus1])


β1 β2 β3

c


prodi

(1minus βi)

Figure 17

ϕ11 ϕ10 ϕ01 ϕ00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1


22 Combining It All



and

f(A) = min (1sum






notX X Y notY

c00

notX X Y notY

c01

notX X Y notY

c10

notX X Y notY

c11

1 1

1 1

1 1

1 1

F 00[notXnotY ] =


F 01[notXY ] =


F 10[XnotY ] =


F 11[XY ] =


Figure 19

















f(x) =mprodj=1

min(1kjsumi=1





h1 hk

x

e1 ek

H = h1 hk(Hx h1) = e1

(Hx h2) = e2

(Hx hk) = ek

Figure 20

H1 H i Hm

h11 h1

k1hm1 hmkm

x

e11 e1k1 em1 emkm

Figure 21



e(a b c a) = e1

e(a b c b) = e2

e(x c x) = e3

e(d e c d) = e4

e(d e c e) = e5





a b x d e

c

e1 e2e3e4 e5

Figure 22












P (x) =sumε











Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work



β1 β2

α= β1 middot β2

π product

Figure 11

β1 β2

x

α

and joint

Figure 12

β1 β2

α

and joint

Figure 13

β1 β2

y

α

π product

Figure 14










ϕ([X1]) = notX1

ϕ([X1 X2]) = X2 rarr ϕ([X1])



(1minusXi))

Then



= ϕ(X1 Xn+1)


ϕ(X1) = notX1




a b

c

Figure 15

a b x d e

c


Figure 16


2 c = min(1 12 + 1




and([X1 Xn]) = Xn

and([X1 Xnminus1])


β1 β2 β3

c


prodi

(1minus βi)

Figure 17

ϕ11 ϕ10 ϕ01 ϕ00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1


22 Combining It All



and

f(A) = min (1sum






notX X Y notY

c00

notX X Y notY

c01

notX X Y notY

c10

notX X Y notY

c11

1 1

1 1

1 1

1 1

F 00[notXnotY ] =


F 01[notXY ] =


F 10[XnotY ] =


F 11[XY ] =


Figure 19

















f(x) =mprodj=1

min(1kjsumi=1





h1 hk

x

e1 ek

H = h1 hk(Hx h1) = e1

(Hx h2) = e2

(Hx hk) = ek

Figure 20

H1 H i Hm

h11 h1

k1hm1 hmkm

x

e11 e1k1 em1 emkm

Figure 21



e(a b c a) = e1

e(a b c b) = e2

e(x c x) = e3

e(d e c d) = e4

e(d e c e) = e5





a b x d e

c

e1 e2e3e4 e5

Figure 22












P (x) =sumε











Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work










ϕ([X1]) = notX1

ϕ([X1 X2]) = X2 rarr ϕ([X1])



(1minusXi))

Then



= ϕ(X1 Xn+1)


ϕ(X1) = notX1




a b

c

Figure 15

a b x d e

c


Figure 16


2 c = min(1 12 + 1




and([X1 Xn]) = Xn

and([X1 Xnminus1])


β1 β2 β3

c


prodi

(1minus βi)

Figure 17

ϕ11 ϕ10 ϕ01 ϕ00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1


22 Combining It All



and

f(A) = min (1sum






notX X Y notY

c00

notX X Y notY

c01

notX X Y notY

c10

notX X Y notY

c11

1 1

1 1

1 1

1 1

F 00[notXnotY ] =


F 01[notXY ] =


F 10[XnotY ] =


F 11[XY ] =


Figure 19

















f(x) =mprodj=1

min(1kjsumi=1





h1 hk

x

e1 ek

H = h1 hk(Hx h1) = e1

(Hx h2) = e2

(Hx hk) = ek

Figure 20

H1 H i Hm

h11 h1

k1hm1 hmkm

x

e11 e1k1 em1 emkm

Figure 21



e(a b c a) = e1

e(a b c b) = e2

e(x c x) = e3

e(d e c d) = e4

e(d e c e) = e5





a b x d e

c

e1 e2e3e4 e5

Figure 22












P (x) =sumε











Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work



a b

c

Figure 15

a b x d e

c


Figure 16


2 c = min(1 12 + 1




and([X1 Xn]) = Xn

and([X1 Xnminus1])


β1 β2 β3

c


prodi

(1minus βi)

Figure 17

ϕ11 ϕ10 ϕ01 ϕ00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1


22 Combining It All



and

f(A) = min (1sum






notX X Y notY

c00

notX X Y notY

c01

notX X Y notY

c10

notX X Y notY

c11

1 1

1 1

1 1

1 1

F 00[notXnotY ] =


F 01[notXY ] =


F 10[XnotY ] =


F 11[XY ] =


Figure 19

















f(x) =mprodj=1

min(1kjsumi=1





h1 hk

x

e1 ek

H = h1 hk(Hx h1) = e1

(Hx h2) = e2

(Hx hk) = ek

Figure 20

H1 H i Hm

h11 h1

k1hm1 hmkm

x

e11 e1k1 em1 emkm

Figure 21



e(a b c a) = e1

e(a b c b) = e2

e(x c x) = e3

e(d e c d) = e4

e(d e c e) = e5





a b x d e

c

e1 e2e3e4 e5

Figure 22












P (x) =sumε











Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work



β1 β2 β3

c


prodi

(1minus βi)

Figure 17

ϕ11 ϕ10 ϕ01 ϕ00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1


22 Combining It All



and

f(A) = min (1sum






notX X Y notY

c00

notX X Y notY

c01

notX X Y notY

c10

notX X Y notY

c11

1 1

1 1

1 1

1 1

F 00[notXnotY ] =


F 01[notXY ] =


F 10[XnotY ] =


F 11[XY ] =


Figure 19

















f(x) =mprodj=1

min(1kjsumi=1





h1 hk

x

e1 ek

H = h1 hk(Hx h1) = e1

(Hx h2) = e2

(Hx hk) = ek

Figure 20

H1 H i Hm

h11 h1

k1hm1 hmkm

x

e11 e1k1 em1 emkm

Figure 21



e(a b c a) = e1

e(a b c b) = e2

e(x c x) = e3

e(d e c d) = e4

e(d e c e) = e5





a b x d e

c

e1 e2e3e4 e5

Figure 22












P (x) =sumε











Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work



notX X Y notY

c00

notX X Y notY

c01

notX X Y notY

c10

notX X Y notY

c11

1 1

1 1

1 1

1 1

F 00[notXnotY ] =


F 01[notXY ] =


F 10[XnotY ] =


F 11[XY ] =


Figure 19

















f(x) =mprodj=1

min(1kjsumi=1





h1 hk

x

e1 ek

H = h1 hk(Hx h1) = e1

(Hx h2) = e2

(Hx hk) = ek

Figure 20

H1 H i Hm

h11 h1

k1hm1 hmkm

x

e11 e1k1 em1 emkm

Figure 21



e(a b c a) = e1

e(a b c b) = e2

e(x c x) = e3

e(d e c d) = e4

e(d e c e) = e5





a b x d e

c

e1 e2e3e4 e5

Figure 22












P (x) =sumε











Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work














f(x) =mprodj=1

min(1kjsumi=1





h1 hk

x

e1 ek

H = h1 hk(Hx h1) = e1

(Hx h2) = e2

(Hx hk) = ek

Figure 20

H1 H i Hm

h11 h1

k1hm1 hmkm

x

e11 e1k1 em1 emkm

Figure 21



e(a b c a) = e1

e(a b c b) = e2

e(x c x) = e3

e(d e c d) = e4

e(d e c e) = e5





a b x d e

c

e1 e2e3e4 e5

Figure 22












P (x) =sumε











Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work



h1 hk

x

e1 ek

H = h1 hk(Hx h1) = e1

(Hx h2) = e2

(Hx hk) = ek

Figure 20

H1 H i Hm

h11 h1

k1hm1 hmkm

x

e11 e1k1 em1 emkm

Figure 21



e(a b c a) = e1

e(a b c b) = e2

e(x c x) = e3

e(d e c d) = e4

e(d e c e) = e5





a b x d e

c

e1 e2e3e4 e5

Figure 22












P (x) =sumε











Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work




a b x d e

c

e1 e2e3e4 e5

Figure 22












P (x) =sumε











Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work









Definition 2



P (A) asymp 0016



















P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work








P (s) =sumεisin2k


P (yi)ε(i)



i ) = 1minus f(yi)

Thus

f(c(sys ε) =prodi


=prodi

f(yε(i)i ) =prodi

P (yi)ε(i)

So


prodi

P (yi)ε(i)

Therefore

f(s) = min(1sumε


= min(1sumε


P (yi)ε(i))

= P (s)





ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work






ys y1 yk

s



notyk yk

c(s ys ε)

d(s ys ε)

s










where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work







where β+ij


P (cj) =prodij

pij

where

pij =




X Y

A

U W






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work






X notX Y notY

c11 c10 c01 c00




P (X) = 001P (notX) = 099

P (Y ) = 002P (notY ) = 098

This gives us




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11







ex00|ϕ00 = 0001ex01|ϕ01 = 029ex10|ϕ10 = 094ex11|ϕ11 = 095

and hence



X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

A

ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1



P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work




P (A) = min1sumij

(1minus P (xij))

= min1 0001 + 00058 + 00093 + 00002asymp 0016




X notX Y notY

c11 c10 c01 c00

x11 x10 x01 x00

wA wnotA uA unotA

A notA

W U

1 1


ex00|ϕ00ex01|ϕ01ex10|ϕ10ex11|ϕ11

1







Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work



Since



We get











































































Introduction



Combining It All

Formal Definitions



Related Work































































Introduction



Combining It All

Formal Definitions



Related Work


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