Triadic Exploration and Exploration with Multiple Experts
TTriadic Explorationand Exploration with Multiple Experts
Maximilian Felde , and Gerd Stumme , Knowledge & Data Engineering Group, University of Kassel, Germany Interdisciplinary Research Center for Information System DesignUniversity of Kassel, Germany [email protected], [email protected]
Abstract
Formal Concept Analysis (FCA) provides a method called attribute exploration which helps a domain expert discover structural depen-dencies in knowledge domains that can be represented by a formal context(a cross table of objects and attributes). Triadic Concept Analysis is anextension of FCA that incorporates the notion of conditions. Many exten-sions and variants of attribute exploration have been studied but only fewattempts at incorporating multiple experts have been made. In this paperwe present triadic exploration based on Triadic Concept Analysis to explore conditional attribute implications in a triadic domain. We then adapt thisapproach to formulate attribute exploration with multiple experts thathave different views on a domain.
Keywords:
Formal Concept Analysis, Triadic Concept Analysis, Attribute Ex-ploration
Attribute exploration [4] is a well established knowledge acquisition method from thefield of Formal Concept Analysis (FCA) [8]. Attribute exploration works on domainsthat can be represented as binary tabular data of objects and attributes (also calledfeatures or properties). It helps a domain expert to uncover the dependency structureof attributes of the domain. For non-binary tabular data the method of conceptualscaling , cf. [7], can be used to transform non-binary attributes into binary ones.Attribute exploration is based on the idea that we extend domain informationthrough a domain expert. To this end, attribute exploration uses a question-answerscheme to extract dependency information about attributes. The questions arein the form of implications , for example, do attributes A and B imply attribute C? (also written as AB → C ? ). The expert’s task is to confirm or refute the validity ofsuch implications in the domain. If the expert refutes the validity of an implicationshe has to offer a counterexample, for example, in case of the question AB → C ? an object of the domain that has the attributes A and B but lacks attribute C .The attribute exploration algorithm asks these questions in an optimized mannersuch that the expert has to answer as few questions as possible until the validity of a r X i v : . [ c s . A I] F e b Maximilian Felde and Gerd Stumme every conceivable implication can be inferred from the answers given by the expert.This is the case when every implication either follows from the set of implicationsaccepted as valid or is contradicted by one of the examples given by the expert.The basic version of attribute exploration requires an all-knowing expert of thedomain, i.e. an expert who can answer any question about the domain correctly.It was introduced by Ganter in [4]. Since then, many variants and extensions ofattribute exploration have been studied. A good overview can be found in thebook
Conceptual Exploration by Ganter and Obiedkov [5]. These extensions andvariants notably include: Attribute exploration with background knowledge andexceptions [3, 18], where the idea is to support the exploration with prior knowledgeabout some of the relations between attributes, for example if one attribute is thenegation of another; attribute exploration with partial information [11–13], wherethe expert is not required to be all-knowing and is also allowed to answer
I do notknow in addition to confirming or refuting a question. Further, the expert is notrequired to fully specify a counterexample as long as the specified parts contradictthe implication in question; and a sketch of how to explore triadic formal contexts [5,6], where the idea of attribute exploration is transferred to triadic concept analysis(an extension of FCA with conditions [17]). We elaborate further on this in Section 3.However, most of the extensions and variants of attribute exploration that havebeen studied are based on the idea of a single expert answering the questions. As faras we know, there exist only a few papers that mention exploration with multipleexperts, notably: Paper [16] deals with how to perform exploration in parallel andpotentially offers a way to speed up the exploration with multiple experts; [10]addresses collaborative conceptual exploration based on the notions of local expertsfor subdomains of a given knowledge domain; and [2] studies attribute explorationin a collaborative exploration setting with multiple experts who share the sameview on the domain but only have partial knowledge thereof.When we explore a domain with multiple experts, one of the fundamentalproblems we face is that different views on a domain, for example different opinionswhether an object has an attribute or not, or whether an implication is valid or notin a domain , are impossible to resolve by combining different pieces of informationinto one. Either, because there is no clear right or wrong , e.g. in case of opinions, orsimply because we can not know which information to trust most. And, even if weused methods such as majority-voting on information, there is a reasonable chancethat the result is not always correct. Combined with the inherent non-robustnessof implication theories, i.e., small changes in the underlying data can lead to a verydifferent theory, this suggests that merging different views on a domain is a badidea for attribute exploration. If we take a closer look at the publications mentionedbefore, we see that all three avoid this issue in their own way. In [16] the expertsall have the same complete knowledge about the domain; in [10] the local expertshave partial knowledge about the same consistent domain knowledge; and, in [2]the problem was also avoided by defining expert knowledge as partial knowledgeof some consistent domain knowledge.Attribute exploration where multiple experts can have truly different and evenopposing views on the domain has to the best of our knowledge not yet been studied. riadic Exploration and Exploration with Multiple Experts 3 To this end we develop triadic exploration based on ideas presented by Ganterand Obiedkov in [6]. We then adapt triadic exploration to the setting of multipleexperts with different views on a domain and thus provide a step in the directionof attribute exploration with multiple experts.The paper is structured as follows: We begin by giving a brief introduction tothe problem in Section 1. We recollect some fundamentals of Formal and TriadicConcept Analysis in Section 2, in particular formal and triadic contexts , attributeimplications , the relative canonical base and attribute exploration . In Section 3, wediscuss implications in the triadic setting, in particular, we focus on conditionalattribute implications . Subsequently, we formulate triadic exploration . In Section 4,we discuss how to adapt triadic exploration to model attribute exploration withmultiple experts with different views. Finally, Section 5 contains conclusion andoutlook. Note that for this paper we do not provide a separate section for relatedwork, instead we address related work throughout the paper whenever appropriate. In this section we recollect the fundamentals of (dyadic) Formal Concept Analysisand Triadic Formal Concept Analysis (TCA). We mostly rely on [8, 19] for FCA andon [17, 20] for TCA. We begin with the definition of formal contexts and associatednotions. We then introduce triadic contexts and give an example which will serve asour running example for the remainder of this paper. Afterwards, we briefly cover attribute implications , the relative canonical base and attribute exploration . Thisserves as a foundation for Section 3, where we look at implications in the triadicsetting and subsequently develop triadic exploration . Formal Concept Analysis was introduced by Wille in [19]. As the theory matured,Ganter and Wille compiled the mathematical foundations of the theory in [8]. A formal context K = ( G,M,I ) consists of a set G of objects, a set M of attributes andan incidence relation I ⊆ G × M with ( g,m ) ∈ I meaning object g has attribute m . Wedefine two derivation operators ( · ) (cid:48) : P ( M ) → P ( G ) and ( · ) (cid:48) : P ( G ) → P ( M ) in thefollowing way: For a set of objects A ⊆ G , the set of attributes common to the objectsin A is provided by A (cid:48) := { m ∈ M | ∀ g ∈ A : ( g,m ) ∈ I } . Analogously, for a set ofattributes B ⊆ M , the set of objects that have all the attributes from B is provided by B (cid:48) := { g ∈ G | ∀ m ∈ B : ( g,m ) ∈ I } . A formal concept of a formal context K = ( G,M,I ) is a pair ( A,B ) with A ⊆ G and B ⊆ M such that A (cid:48) = B and A = B (cid:48) . We call A the extent and B the intent of the formal concept ( A,B ) . The set of all formal concepts ofa context K is denoted by B ( K ) . Note that for any set A ⊆ G the set A (cid:48) is the intent ofa concept and for any set B ⊆ M the set B (cid:48) is the extent of a concept. The subconcept-superconcept relation on B ( K ) is formalized by: ( A ,B ) ≤ ( A ,B ) : ⇔ A ⊆ A ( ⇔ B ⊇ B ) . The set of concepts together with this order relation ( B ( K ) , ≤ ) forms acomplete lattice, the concept lattice . The vertical combination of two formal contexts Maximilian Felde and Gerd Stumme K i = ( G i ,M,I i ) ,i ∈ { , } without the same set of attributes M is called the subposi-tion of K and K . Formally, it is defined as ( ˙ G ∪ ˙ G ,M, ˙ I ∪ ˙ I ) , where ˙ G i := { i }× G and ˙ I i := { (( i,g ) , ( i,m )) | ( g,m ) ∈ I i } for i ∈ { , } . The subposition of a set of contextson the same set of attributes is defined analogously and we denote this by subpos ( · ) . Triadic Concept Analysis (TCA) was introduced by Lehmann and Wille in [17]as an extension to Formal Concept Analysis with conditions. In particular theyintroduced the notion of triadic concepts for which Wille proceeded to show thebasic theorem of triadic concept analysis in [20] – clarifying the connection betweentriadic concepts and complete tri-lattices, analogous to the dyadic case.The basic structure in TCA is a triadic context which is similar to the formalcontext in FCA. A triadic context is defined as a quadruple T = ( G,M,B,Y ) , where G,M and B are sets and Y ⊆ G × M × B is a ternary relation on these sets. Theelements of G,M and B are called objects, attributes and conditions respectively.For g ∈ G , m ∈ M and b ∈ B with ( g,m,b ) ∈ Y we say that object g has attribute m under condition b . The conditions are understood in a broad sense , cf. [20]:They comprise, amongst others, relations, interpretations, meanings, purposes andreasons concerning the connections of objects and attributes. Example . The following example will serve as our running example throughoutthe paper. It shows the situation of public transport at the train station Bf. Wil-helmshöhe with direction to the city center in Kassel. From Bf. Wilhelmshöhe youcan travel by one of four bus lines (52, 55, 100 and 500), four tram lines (1, 3, 4 and7), one night tram (N3) and one regional tram (RT5) to the city center. These arethe objects G ex. of our context. The buses and trams leave the station at differenttimes throughout the day. The attributes M ex. of our context are the aggregatedleave-times, more specifically, we have split each day in five distinct time-slots: earlymorning (4:00 to 7:00), working hours (7:00 to 19:00), evening (19:00 to 21:00),late evening (21:00 to 24:00) and night (0:00 to 4:00). The conditions B ex. of ourcontext are the days of the week. A bus or tram line is related to a time-slot on aday if a bus or tram of this line leaves the station at least once during the time-sloton the day. This describes the ternary relation Y ⊆ G ex. × M ex. × B ex. . We haveaggregated Monday to Friday into a single condition, because the schedule is thesame for these days. Thus, we obtain the context T ex. = ( G ex. ,M ex. ,B ex. ,Y ) . Theresulting triadic context can be found in Figure 1.Naturally, we can view the triadic context as a family of formal contexts, whereeach context represents one condition, basically slicing the triadic context verticallyalong the conditions. In Figure 2 we provide the resulting context family of ourrunning example.Formally such a family of contexts representing a triadic context T = ( G,M,B,Y ) is a set of contexts K b , b ∈ B where K b := ( G,M,I b ) with ( g,m ) ∈ I b : ⇔ ( g,m,b ) ∈ Y .We will refer to the contexts K b as condition contexts of the triadic context T ; forour example these are K Mo-Fr , K Sat and K Sun . The example is similar to the one given in [5], which inspired it.riadic Exploration and Exploration with Multiple Experts 5 × × ×× × × ×× × × × ×× × × ×× × ×× × ×× × ×× × × ×× × × ××× × ×× × × ×× × ×× × × ×× × × ×× × × ×× × × ×× × ×× × × ×× ×× × × ×× × × × ×× × × ×
N317554350052RT5100 e a r l y - m o r n i n g w o r k i n g - h o u r s e v e n i n g l a t e - e v e n i n g n i g h t SunSatMo-Fr
Figure 1: Triadic context T ex. of Example 2.1 Mo-Fr e a r l y - m o r n i n g w o r k i n g - h o u r s e v e n i n g l a t e - e v e n i n g n i g h t × × × × × × × × × × × × × × × × × × × × × N3100 × × × × × × × × ×
RT5 × × × ×
Sat e a r l y - m o r n i n g w o r k i n g - h o u r s e v e n i n g l a t e - e v e n i n g n i g h t × × × × × × × × × × × × × × × × × × × × × × RT5 × × × ×
Sun e a r l y - m o r n i n g w o r k i n g - h o u r s e v e n i n g l a t e - e v e n i n g n i g h t × × × × × × × × × × × × × × × × × × × RT5 × × ×
Figure 2: The triadic context T ex. from Example 2.1represented as context family of the condition contexts K Mo-Fr , K Sat and K Sun
Attribute implications are used to describe dependencies between attributes ina formal context. In the following we give a brief introduction. Let M be a set ofattributes. (For a start, we do not require it to be related to a specific context.)An attribute implication over M is a pair of subsets A,B ⊆ M of M . We denote thisby A → B . We call A the premise and B the conclusion of the implication A → B .We denote the set of all implications over a set M by Imp M = { A → B | A,B ⊆ M } .A subset T ⊆ M respects an attribute implication A → B over M if A (cid:54)⊆ T or B ⊆ T . We then also call T a model of the implication. T respects a set L ofimplications if T respects all implications in L . An implication A → B holds in aset of subsets of M if each of these subsets respects the implication.For a formal context K = ( G,M,I ) we say that an implication A → B over M holds in the context if for every object g ∈ G the object intent g (cid:48) respects theimplication. We then also call A → B a valid implication of K . An implication A → B holds in K if and only if every object g ∈ G that has all attributes in A also has all attribute in B . Further, an implication A → B holds in K if and onlyif B ⊆ A (cid:48)(cid:48) , or equivalently B (cid:48) ⊆ A (cid:48) . An implication A → B follows from a set L ofimplications over M if each subset of M respecting L also respects A → B . A familyof implications is called closed if every implication following from L is alreadycontained in L . Closed sets of implications are also called implication theories . Relative Canonical Base.
The set of all implications that hold in a given context K have a canonical irredundant representation which is called the canonical base ,cf. [8, 9]. Stumme has generalized this representation to the case where some(background) implications are known [18], i.e. attribute implications that are knownto hold based on prior knowledge. Maximilian Felde and Gerd Stumme
Given an formal context K = ( G,M,I ) and a set of (background) implications L on M that hold in the context K . A pseudo-intent of K relative to L is a set P ⊆ M where P respects L , P (cid:54) = P (cid:48)(cid:48) and if Q ⊆ P, Q (cid:54) = P , is a relative psuedo-intent of K then Q (cid:48)(cid:48) ⊆ P . The set L K , L := { P → P (cid:48)(cid:48) | P relative pseudo-intent of K } is calledthe canonical base of K relative to L , or simply the relative canonical base . Allimplications in L K , L hold in K . Theorem 2.2 (see [6, 18]) . If all implications of L hold in K , then1. each implication that holds in K follows from L∪L , and2. L K , L is irredundant w.r.t. 1. The notion of a relative canonical base combined with Theorem 2.2 allows us toreduce the amount of questions that need to be posed during a triadic exploration.
Attribute exploration ([4], cf. also [5, 8]) is a knowledge acquisition method basedon a question-answer scheme to obtain the implication theory of a domain.Let us consider a domain (a formal context) ( G,M,I ) that we do not knowcompletely and that we want to explore and a domain expert for this domain. Westart with a (possibly empty) set of known (background) implications L and a(possibly empty) set G E ⊆ G of known objects, represented as (possibly empty)formal context E = ( G E ,M,I E ) . In every step of the attribute exploration we have aset of already accepted implications L and a context of already provided counterex-amples E . The attribute exploration algorithm picks the next implication A → B that does not follow from L and that holds in E . It then asks the expert whetherthe implication truly holds in the domain. The expert can either confirm that theimplication holds or they can refute its validity by providing a counterexample,i.e., an object g ∈ G whose intent does not respect the implication. If the expertconfirms the implication’s validity in the domain, it is added to the set L , otherwisethe provided counterexample is added to the context of counterexamples E . Thisprocess is repeated until there is no implication left to be asked.After performing the attribute exploration we have the canonical base of impli-cations from which every valid implication in the domain follows. Furthermore, forevery implication that is not valid, the set of examples contains a counterexample. In this section we look at implications in the triadic setting, in particular, we formallyintroduce conditional attribute implications , and develop a triadic exploration forTriadic Concept Analysis as proposed by Ganter and Obiedkov in [5, 6].
In formal contexts (of type ( G,M,I ) ) the matter of implications is fairly straightfor-ward: There are attribute implications to describe dependencies between attributes riadic Exploration and Exploration with Multiple Experts 7 (and dually there are object implications ). In triadic contexts, the notion of impli-cation is not as simple. This manifests in a multitude of types of implications thathave been proposed: The earliest suggestion for a triadic implication came fromBiedermann [1], where he suggested the study of implications of the form ( R → S ) C which is interpreted as: If an object has all attributes from R under all conditionsfrom C, then it also has all attributes from S under all conditions from C .In [6], Ganter and Obiedkov studied some other types of implications for thetriadic setting. They introduced a stronger version of the triadic implication called conditional attribute implications to describe dependencies that hold for someconditions. The symmetry arising from the arbitrary choice of objects, attributesand conditions in a triadic context results in five more types of implications. Further,they introduced another generalization of Biedermann’s triadic implication called attribute × condition implication to express dependencies between combinationsof attributes and conditions. For the remainder of this paper we will focus on conditional attribute implications , because they best serve our goal of developingattribute exploration with multiple experts.Given a triadic context T = ( G,M,B,Y ) , a conditional attribute implication isan expression of the form R C −→ S where R,S ⊆ M , C ⊆ B , which reads as: R implies S under all conditions from C . A conditional attribute implication R C −→ S holds in a triadic context T iff for each condition c ∈ C it holds that if an object g ∈ G has all the attributes in R it also has all the attributes in S . This is the case if theimplication R → S holds in every conditional context K c for c ∈ C . Proposition 3.1.
Let T = ( G,M,B,Y ) and K c , c ∈ B , its respective conditioncontexts. For a conditional implication R C −→ S with R,S ⊆ M and C ⊆ B , thefollowing statements are equivalent:1. R C −→ S holds in T R → S holds in K c for every c ∈ C R → S holds in subpos ( { K c | c ∈ C } ) Proof. . ⇔ . follows directly from the definitions of holds in the triadic and dyadicsetting. . ⇔ . follows from the definition of subposition and that an implication R → S holds in a context if and only if for every object g the object intent respectsthe implication. Example . In the context family of Example 2.1 in Figure 2 we observe that theimplication early-morning → working-hours holds in all three condition contexts K Mo-Fr , K Sat and K Sun , hence, early-morning
Mo-Fr , Sat , Sun −−−−−−−−−→ working-hours holds in T ex. . Whereas the implication working-hours → evening only holds in the conditioncontext K Sun because tram line is a counterexample in K Sat and bus line is acounterexample in K Mo-Fr and thus working-hours
Sun −−→ evening holds in T ex. , but working-hours Mo-Fr , Sat , Sun −−−−−−−−−→ evening . does not.Clearly, if a conditional implication R C −→ S holds in a triadic context T then allconditional implications R D −→ S with D ⊆ C hold as well. Further, for every subset Maximilian Felde and Gerd Stumme early-morning → working-hoursworking-hours, night → late-evening, early-morning, eveninglate-evening → working-hours, eveningevening → working-hoursevening → working-hours,late-evening Sat working-hours → early-morningnight → working-hours, late-evening,early-morning, eveningMo-Frworking-hours → eveningSun Figure 3: The lattice of conditional implications of the running example T ex. withsimplified labels, which consist of the relative canonical base with respect to theimplications in all nodes below. We omit the top label of implications as the extentof this concept is always Imp M . C ⊆ B there is a set of conditional implications R C −→ S that hold in T . This set ofconditional implications for a fixed set of conditions C is the implication theoryof the subposition of condition contexts subpos ( { K c | c ∈ C } ) . Context of Conditional Implications.
A nice way to structure the conditionalimplications that hold in a triadic context T is to use the approach suggested by Gan-ter and Obiedkov, cf. [6], and to introduce a context of conditional implications : Givena triadic context T , we construct a formal context C imp ( T ) := (Imp M ,B,I ) , where theset of all possible implications on M is the object set, the set of conditions B of thetriadic context T is the set of attributes and the incidence relation I is determined by ( R → S ) Ic : ⇔ R c −→ S holds in T . The formal concepts of C imp ( T ) are pairs ( L , C ) , where L is a set of implicationsand C is a set of conditions, such that L is the set of all implications R → S for which R C −→ S holds, and C is the largest set of conditions for which this is the case. Theseconcepts structure the set of conditional implications in a lattice ordered by theconditions for which they hold. Their extents form a system of implication theories. Example . For our running example we present the concept lattice of C imp ( T ex. ) with simplified labels in Figure 3: The extent of the top node always contains the impli-cations that hold under the empty set of conditions, i.e., the whole set Imp M . We omitthis label. For the other nodes we give the relative canonical base with respect to setof implications from all nodes below. Looking at the implications from Example 3.2,we find the implication early-morning → working-hours at the bottom node , becauseit holds for all three conditions, whereas we find the implication working-hours → evening at the node for Sunday , because that is the only condition for which it holds. riadic Exploration and Exploration with Multiple Experts 9
Now, we develop
Triadic Exploration to explore the conditional implications ofa triadic domain.Previously, we have structured the conditional implications of a triadic domain T as a system of implication theories by utilizing the context of conditional impli-cations C imp ( T ) . This was possible because we had complete information about thedomains implications in the context T . However, it is easy to imagine a situationwhere we can access the information about a domain only indirectly through adomain expert and where an attribute exploration might be useful. For our runningexample, imagine someone with a bus and train schedule where the informationcan be looked up but is not fully available at once. Now the question is: How toexplore the complete system of conditional implications?A naive approach is to explore the implication theory for each fixed subsetof the conditions, essentially exploring each node of the system of implicationtheories independently. But, this is clearly not a good idea; it means answeringmany questions multiple times for each condition.A better approach might be to only explore the implication theory for every condi-tion, each providing one column in the context of conditional implications C imp . Thenwe can compute the concept lattice without any further interactions with the expert.However, there are some points to consider that suggest a different approach,cf. [5]: First, to stay in the triadic setting, a complete counterexample to a questionshould describe the new object by the attributes it has for each of the conditions ,and not only for the one, that is currently under consideration. And second, someimplications may hold for several conditions and the domain expert might wantto confirm each of them for multiple conditions at once.Thus, we come back to the context of conditional implications. Ganter andObiedkov suggested to explore the triadic domain by exploring the nodes in thelattice of conditional implications from the bottom up; using the already knownvalid implications as background knowledge. Hence, as we explore the system ofconditional implications, we successively fill the context of conditional implications.In the following we describe the nested process of exploring the nodes of the con-cept lattice of conditional implications with the help of two algorithms: Algorithm 1for the exploration of the conditional implications for a fixed set of conditions andAlgorithm 2 that uses this algorithm as a subroutine to explore all conditionalimplications of the triadic domain. Explore Conditional Implications for a fixed set of Conditions.
For afixed set of conditions D ⊆ B in a triadic domain T = ( G,M,B,Y ) , the explorationalgorithm is an adapted version of the algorithm for attribute exploration withbackground implications and exceptions, see [5, 18]. In Algorithm 1 we present animplementation for the exploration in pseudo-code.The algorithm starts with some background knowledge, in particular: A triadiccontext E = ( G E ,M,B,Y E ) , that contains some examples from the domain T , and aset of implications L that are known to hold for all conditions in D ; both of thesecan be empty. The rest of the domain can only be accessed by the algorithm through interaction with the domain expert. In each step, the algorithm determines the nextimplication A → A (cid:48)(cid:48) to ask the expert. To determine the next question A → A (cid:48)(cid:48) thealgorithm uses both the information from the examples in E and the known validimplications in L . It automatically skips questions that follow from the implicationsin L or for which E already contains a counterexample. More precisely, A is thenext relative pseudo-intent in subpos ( { K d | d ∈ D } ) , i.e., the lectically smallest set A closed under the set of known valid implications and background implications L thatis not already closed in the subposition context of examples for the conditions in D .Essentially, this algorithm is an attribute exploration with background impli-cations on the subposition of the condition contexts. Additionally, it tracks whichimplications hold for which conditions in D . This enables us to reduce the amountof interaction required from the expert in subsequent explorations by preventingto ask the same question multiple times for different subposition contexts. Theproof of correctness for Algorithm 1 is a straightforward adaption of the proof of[18, Theorem 6] and we therefore omit the details.Note that we chose to collect all implications that are asked about and thesubset of conditions of D for which they hold in Line 13 instead of only addingthe implications that hold for all conditions in the context C . Hence, if there is acounterexample, i.e., the implication does not hold for D , we track for which subsetof D (if any) the implication does hold. This further reduces the number of questionsposed in later explorations. The trade-off is that the background knowledge we have isnot just of nodes below the currently explored one in the lattice but may also containimplications that first hold for the conditions of the current node. This has no effecton the implication theory of the node but somewhat complicates the labeling of thenode – we cannot simply use the relative canonical base with respect to the knowledgewe have. In contrast, if we only added the implications that hold for all conditionsin the current exploration then the labels are exactly the implications of the relativecanonical base, but, we might have to ask some questions multiple times for some ofthe conditions. For our running example this approach further reduces the numberof questions posed to the expert from fifteen to twelve, cf. Example 3.5 in Section 3.3. The Order of Explorations.
To determine the sequence in which the nodes ofthe lattice of conditional implications are explored, Ganter and Obiedkov furthersuggested to follow a linear extension of the lattice of conditional implications,see [6], and later specified this to follow the
NextExtent-Algorithm , i.e.,
NextClosure on the extents, on the context of conditional attribute implications, see [5].However, in our setting the
NextExtent-Algorithm does not fit. The problemis that we may not have the necessary information to correctly determine the nextnode to explore. This is because the questions that are asked during the explorationof a node are not guaranteed to discriminate between the conditions that are beingexplored. Questions that would discriminate between conditions are not asked ifthere already exists a counterexample for any one of the conditions. This mightresult in not exploring all nodes of the lattice. For the same reason, the nested application of NextClosure for computing all concepts ofa triadic context, as described in [14, 15], cannot serve as a base for the triadic exploration.riadic Exploration and Exploration with Multiple Experts 11
Algorithm 1: explore-conditions
Input: a set of conditions D ⊆ B , a triadic context E = ( G E ,M,B,Y E ) of examples(possibly empty) and a set L of background implications known to holdfor all conditions in D (also possibly empty) Interactive Input : ( (cid:63) ) The expert confirms or rejects an implication to holdfor the set of conditions D . Upon rejection the expertprovides a counterexample g from the domain togetherwith its relation to all conditions and all attributes, i.e., thecontext K g := ( M,B,I ) where ( m,b ) ∈ I ⇔ g has m under thecondition b in the domain. Output: the relative canonical base
L \ L of implications that hold for allconditions in D with respect to L , a possibly enlarged triadic contextof counterexamples E and the formal context C of asked implicationsand the conditions for which they hold. L := L A := ∅ C := ( ∅ ,B, ∅ ) while A (cid:54) = M do while A (cid:54) = A (cid:48)(cid:48) in S where S := ( G S ,M,J ) = subposition of K d for d ∈ D with K d := ( G E ,M,I d ) where ( g,m ) ∈ I d ⇔ ( g,m,d ) ∈ Y E do Ask the expert if A → A (cid:48)(cid:48) holds for all conditions d ∈ D ( (cid:63) ) if A → A (cid:48)(cid:48) holds then L := L∪{ A → A (cid:48)(cid:48) } else extend E with the counterexample provided by the expert ( (cid:63) ) extend C with the object A → A (cid:48)(cid:48) and its relation to all conditions d ∈ D ( (cid:63) ) end A := NextClosure( A,M, L ) /* computes the next closure of A in M with respect to the implications in L ; see for example [5, 8] */ end return L\L , E and C Example . Let us illustrate the problem with a small example: Take a look atthe domain given by the triadic context T in Figure 4. If we explore this contextand begin with the bottom node, i.e., the implications that hold for all conditionswithout any background knowledge, then the first question posed to the expertis ∅ → ab ? , which the expert refutes with a counterexample – object with allits attributes under all conditions. It substantiates that implication holds forneither of the conditions. The second question that is posed to the expert is b → a ? which the expert confirms. This concludes the exploration of the bottom nodein this example. If we now compute the next extent in the resulting context ofconditional implications C in order to determine which node to explore next, weobtain NextExtent( ∅ ) = { b → a } with intent { d ,d } which we just explored andthen NextExtent( { b → a } ) = G C with intent ∅ which concludes the exploration.However, clearly the implication ∅ → a holds in d but not in d and is missing in C . d a b1 × d a b1 T b → a ∅ → ad a → bd B ( C imp ( T )) C d d ∅ → abb → a × × C b → ad ,d B ( C ) Figure 4: A triadic context T , the lattice of conditional implications of T , thecontext C after exploring the conditional implications of T using the NextExtent-Algorithm to determine the next conditions to explore, and the lattice of C And, the question ∅ → a ? was not posed to the expert because there already existeda counterexample for condition d after the first question. Similarly, the implication a → b holds in d but not in d and is also missing. In Figure 4, we present boththe lattice of C and the lattice of C imp ( T ) . Hence, an exploration that uses the NextExtent-Algorithm to determine which nodes of the conditional implicationslattice to explore next does not necessarily explore all nodes of the lattice.To circumvent this problem, we use the suggested strategy of exploring thelattice node by node from the bottom up with the already known valid conditionalimplications in C imp as background knowledge. But, instead of using the NextExtent-Algorithm to incrementally determine the next combination of conditions to explore,we simply follow a linear extension of ( P ( B ) \∅ , ⊇ ) . Which means, we walk throughall subsets of B sorted by their cardinality from biggest to smallest and stop whenwe have explored all subsets of cardinality one. At first glance this might lookas if we explore more nodes than necessary, because the implication theory of acondition might be included in another ones and thus is explored at least twice– once in combination and once alone. But, because we only ask questions aboutimplications that are unknown with respect to the knowledge we already have whenthe condition is explored alone, these questions won’t be asked again.In Algorithm 2 we present the algorithm for triadic exploration in pseudo-code:We walk through ( P ( B ) \∅ , ⊇ ) , i.e. the subsets of conditions, in Line 1. For eachset of conditions D ⊆ B we determine the implications L that are known to holdfor all conditions in D in Line 2. We compute the canonical base relative to L inthe subposition of condition context of D Line 3. And, update the known examples E and the known implications in Lines 4 and 5. Example . We now give a brief example for a triadic exploration of the domainof our running example (Example 2.1): Let us assume we only have a triadic expertfor this domain and not the whole domain information – imagine someone withaccess to a search interface for the bus and train schedule of Figure 2. In Figure 5we have listed all interactions with the expert. Each row shows one interaction andthe order of interactions is from top to bottom. The resulting lattice of conditional riadic Exploration and Exploration with Multiple Experts 13
Algorithm 2: triadic-exploration
Input: a triadic context E = ( G E ,M,B,Y E ) of examples (possibly empty) and acontext C = ( G C ,B,I C ) of implications known to hold for some conditions Output: a triadic context of counterexamples E and the context of conditional im-plications C , from which all valid conditional implications can be inferred for D in linear extension of ( P ( B ) \∅ , ⊇ ) do L := D (cid:48) (in C ) L D , E D , C D := explore-conditions( D, E , L ) E := E D C := C ∪ C D = ( G C ∪ G C D ,B,I C ∪ I C D ) end return E and C Conditions Question – does the implication hold? Question holds for AnswerMo-Fr, Sat, Sun ∅ → working-hours, late-evening, early-morning, night, evening ∅ RT5Mo-Fr, Sat, Sun ∅ → working-hours, late-evening, evening ∅ → working-hours, late-evening Sat, Sun 7Mo-Fr, Sat, Sun evening → working-hours Mo-Fr, Sat, Sun trueMo-Fr, Sat, Sun night → working-hours, late-evening, early-morning, evening Mo-Fr N3Mo-Fr, Sat, Sun early-morning → working-hours Mo-Fr, Sat, Sun trueMo-Fr, Sat, Sun late-evening → working-hours, evening Mo-Fr, Sat, Sun trueMo-Fr, Sat, Sun working-hours, night → late-evening, early-morning, evening Mo-Fr, Sat, Sun trueMo-Fr, Sun working-hours → evening Sun 55Mo-Fr, Sat working-hours, evening → early-morning Mo-Fr 500Sun working-hours, late-evening, early-morning, evening → night ∅ → early-morning Mo-Fr true Figure 5: Triadic Exploration of the running example. Each row represents oneinteraction with the expert. It comprises the set of conditions that is explored, thequestion posed in form of an implication, the conditions for which the implicationholds, and, the answer given by the expert.implications is exactly the lattice shown in Figure 3. The extent of each conceptof this lattice is a generating set for the implication theory of implications that holdfor all conditions of the intent which follows from Theorem 2.2, and, because weiteratively computed relative canonical bases. Thus, we know that, for each concept, the implications in its extent are complete, but – as a union of “stacked” relativecanonical bases – not necessarily irredundant.
In this section we discuss how to adapt triadic exploration to a setting where we havemultiple experts with different views on a domain (i.e., a set of attributes). In Sec-tion 1, we have briefly discussed the problem of exploration with multiple experts withdifferent views and concluded that combining answers from different experts is not agood strategy for attribute exploration in general. We have further established thatall previous methods for multi-expert exploration avoided this problem by assumingthat the experts’ knowledge is derived from some consistent domain knowledge.
Here, we suggest a different approach that allows for a group of experts withdifferent, opposing views on a domain. The basic idea is to accept all answers equallyand look for the subset of knowledge that all experts agree on. To explore the domainwe then explore the agreed-upon knowledge of different subsets of the expert group.If all experts know about the same objects of the domain we can regard the groupof experts as a triadic domain where each experts view is expressed as one condition.In our running example, imagine that there are three experts for the bus and trainschedule: One for Monday-Friday, one for Saturday and one for Sunday. The threeexperts will have different opinions about the implication theory of the time slots.To explore the dependencies of attributes in this triadic multi-expert domain,we utilize triadic exploration. To ask about a conditional implication then means toask all experts if the implication holds in their view. However, a simple translationback to the triadic case means that each time an expert gives a counterexample to aquestion, all experts must be consulted about their view on the counterexample tostay within the triadic setting (because we need the full slice of the triadic context).This is not ideal, but we can adapt the triadic exploration to avoid this issue: Sincewe do not rely on any specific properties of the triadic context other than beingable to form the subposition of the condition contexts K d , we can simply leave thetriadic setting behind and transfer the idea of conditional attribute implicationsto a setting where we replace the triadic context with a context family on the sameset of attributes (but not necessarily the same set of objects), i.e., a context family { K e = ( G e ,M,I e ) | e ∈ E } for a group of experts E .Note that we could also explore the implication theory for each context in sucha context family independently and combine the results afterwards, as initiallysuggested in Section 3. It is not obvious how this approach compares to the triadicone. However, the triadic approach also allows to only explore a subset of the systemof implication theories.A real world example for such a context family can be found in the BSI-IT-Grundschutzkatalog , a publication by the German Federal Office for InformationSecurity , which contains security recommendations on a wide variety of IT topics.There, a general set of elementary threats is defined and for topics where thesethreats are present (for example organizational, infrastructure and personnel) aset of measures is defined where each measure combats one or multiple of thethreats. Hence, if we regard the elementary threats as attributes, the topics asconditions/experts and the measures as objects, we have a context family on thesame set of attributes but with different object sets.To explore the domain of such a context family, where the set G varies for thedifferent conditions, we have to slightly alter Algorithms 1 and 2. In particular, weneed to replace the triadic contexts with context families and the conditions with theexperts. The triadic context of counterexamples becomes a context family of coun-terexamples where each expert has their own context of counterexamples and theobjects between them can differ. Hence, in Algorithm 1, A (cid:48)(cid:48) is computed on the subpo-sition of the respective contexts of counterexamples and asking about the implication riadic Exploration and Exploration with Multiple Experts 15 A → A (cid:48)(cid:48) means asking each of the experts. Now, counterexamples from one expert canbe accepted without having to ask all other experts about their view on the example.If we explore a triadic domain in this more abstract setting of context families onthe same set of attributes, the trade-off is that we obtain less complete informationabout the counterexamples. However, we still obtain the same knowledge in termsof conditional attribute implications that hold in the domain.In addition, we gain the ability to explore context families that do not fit thetriadic setting or only do so after some modifications, as for example, the contextfamily of the BSI-IT-Grundschutzkatalog . Another example can be derived fromthe running example: If we look at K Mo-Fr in Figure 2, imagine that instead of onecontext (and thus one expert) of bus and tram lines we had one for bus lines andone for tram lines. Clearly, this family of two contexts could be transformed intoa triadic context, however, to do so we would have to add bus lines to the tramlines context and vice versa – mixing domains that might be perceived as different.
In this paper, we addressed the problem of multi-expert attribute exploration in For-mal Concept Analysis. To this end, we developed triadic exploration – an analogue toattribute exploration – for Triadic Concept Analyis, which extends Formal ConceptAnalysis with the notion of conditions. Triadic exploration helps a triadic domain ex-pert to explore the structure of the conditional attribute implications of the domain.We adapted triadic exploration to a multi-expert setting by considering theexperts’ views of a domain as conditions in a triadic setting. We discussed the ram-ifications of this approach and subsequently suggested to adapt triadic explorationto the more general setting of context families on the same set of attributes.This paper is a step towards multi-expert exploration where experts can havedifferent views on a domain. In contrast to the few prior works on this subject,here the experts can have opposing views. A next step is the combination of thisapproach with the notion of partial expert knowledge and a more in depth studyof context families as a foundation for multi-expert explorations.
References [1] K. Biedermann. “A foundation of the theory of trilattices.” Dissertation TUDarmstadt, Shaker. Aachen, 1998.[2] M. Felde and G. Stumme. “Interactive Collaborative Exploration usingIncomplete Contexts.” In:
CoRR abs/1908.08740 (2019). arXiv: .[3] B. Ganter. “Attribute exploration with background knowledge.” In:
Theoreti-cal Computer Science
Conceptual Exploration . Springer, 2016. [6] B. Ganter and S. A. Obiedkov. “Implications in Triadic Formal Contexts.” In:
Proc. Intl. Conf. on Conceptual Structures . Vol. 3127. LNCS. Springer, 2004,pp. 186–195.[7] B. Ganter and R. Wille. “Conceptual scaling.” In:
Applications of combina-torics and graph theory to the biological and social sciences . Ed. by F. Roberts.Springer-Verlag, 1989, pp. 139–167.[8] B. Ganter and R. Wille.
Formal Concept Analysis: Mathematical Foundations .Berlin/Heidelberg: Springer-Verlag, 1999.[9] J.-L. Guigues and V. Duquenne. “Familles minimales d’implications infor-matives résultant d’un tableau de données binaires.” In:
Mathématiques etSciences Humaines
95 (1986), pp. 5–18.[10] T. Hanika and J. Zumbrägel. “Towards Collaborative Conceptual Explo-ration.” In:
Proc. Intl. Conf. on Conceptual Structures . Ed. by P. Chapman,D. Endres, and N. Pernelle. Vol. 10872. LNCS. Springer, 2018, pp. 120–134.[11] R. Holzer. “Knowledge acquisition under incomplete knowledge using methodsfrom formal concept analysis: Part I.” In:
Fundamenta Informaticae
Fundamenta Informaticae
Web Semant.
Proc. 6th ICDM conference . Hong Kong, Dec. 2006.[16] F. Kriegel. “Parallel Attribute Exploration.” In:
Proc. Intl. Conf. on Concep-tual Structures . Ed. by O. Haemmerlé, G. Stapleton, and C. Faron-Zucker.Vol. 9717. LNCS. Springer, 2016, pp. 91–106.[17] F. Lehmann and R. Wille.
A triadic approach to formal concept analysis .Vol. 954. LNCS. Springer, 1995.[18] G. Stumme. “Attribute Exploration with Background Implications and Excep-tions.” In:
Data Analysis and Information Systems. Statistical and Conceptualapproaches. Proc. GfKl’95. Studies in Classification, Data Analysis, andKnowledge Organization 7 . Ed. by H.-H. Bock and W. Polasek. Heidelberg:Springer, 1996, pp. 457–469.[19] R. Wille. “Restructuring lattice theory: An approach based on hierarchies ofconcepts.” In:
Ordered Sets . Ed. by I. Rival. Reidel, Dordrecht-Boston, 1982,pp. 445–470.[20] Rudolf Wille. “The Basic Theorem of triadic concept analysis.” In: