Manipulating Triadic Concept Analysis Contexts through Binary

Decision Diagrams

Kaio H. A. Ananias, Julio C. V. Neves, Pedro H. B. Ruas, Luis E. Z

arate and Mark A. J. Song

Pontiﬁcal Catholic University of Minas Gerais (PUC Minas),

Keywords:

Formal Concept Analisys,Triadic Concept Analisys, Binary Decision Diagram, TRIAS Algorithm.

Abstract:

Formal Concept Analysis (FCA) is an approach based on the mathematization and hierarchy of formal con-

cepts. Nowadays, with the increasing of social network for personal and professional usage, more and more

applications of data analysis on environments with high dimensionality (Big Data) have been discussed in

the literature. Through the Formal Concept Analysis and Triadic Concept Analysis, it is possible to extract

database knowledge in a hierarchical and systematized representation. It is common that the data set trans-

forms the extraction of this knowledge into a problem of high computational cost. Therefore, this paper has an

objective to evaluate the behavior of the algorithm for extraction triadic concepts using TRIAS in high dimen-

sional contexts. It was used a synthetic generator known as SCGaz (Synthetic Context Generator a-z). After

the analysis, it was proposed a representation of triadic contexts using a structure known as Binary Decision

Diagram (BDD).

1 INTRODUCTION

The discovery of valid, tacit, understandable and use-

ful information is the goal of several areas of knowl-

edge in Computer Science. The difﬁculty in reach-

ing this goal is aggravated as these bases become ever

larger. One of the challenges is the problem of ﬁnd-

ing relationships and rules that describe the behavior

of the elements. For example, the growing popular-

ization of social networks and the volume of data pro-

duced by their users. This is an application that cre-

ates a demand for new techniques to extract knowl-

edge in order to make explicit the interactions be-

tween users and to deﬁne patterns that represent the

behavior of the network.

A possible solution to the problem is the use of

Formal Concept Analysis (FCA), which is a technique

based on the mathematization of the notion of con-

cepts and the structuring of concepts into a concep-

tual hierarchy. With the use of FCA it is possible to

analyze the data through associations and dependen-

cies of objects and attributes formally described from

a real or synthetic dataset (Wille, 1982) (Bernhard and

Rudolf, 1999). The representation of the knowledge

contained in the base is done through a description of

the objects, attributes and the relations of incidence

between them, known as a formal context. In this tra-

ditional approach, called dyadic, information is rep-

resented by a triple (G, M, I), where G is the set of

objects, M is the set of attributes and the binary inci-

dence relation between G and M.

However, in several situations it is necessary to

describe the condition that establishes the relation be-

tween the different objects and their attributes. An ex-

tension of the classical (dyadic) FCA, called Triadic

Concept Analysis (TCA) was proposed with the goal

of dealing with this problem (Lehmann and Wille,

1995). Although it is from the FCA, the triadic ap-

proach is more complex because it deals with three-

dimensional data. TCA is based on the triadic re-

lationship between objects, attributes and conditions

deﬁned by the quadruple (K

, K

, Y) where K

, K

and K

are respectively the sets of objects, attributes,

and conditions, and Y the ternary relation between

them.

As with the FCA, the triadic approach has to deal

with problems in which databases are of high dimen-

sionality. Although several algorithms have been pro-

posed in the literature in order to extract information

from triadic concepts, neither one directly attacks the

high dimensionality problem (Jaschke et al., 2006)

(Cerf et al., 2009) (Trabelsi et al., 2012).

Regarding this scenario, the main goals of this pa-

per is:

182

Ananias, K., Neves, J., Ruas, P., Zárate, L. and Song, M.

Manipulating Triadic Concept Analysis Contexts through Binary Decision Diagrams.

DOI: 10.5220/0007716101820189

In Proceedings of the 21st International Conference on Enterprise Information Systems (ICEIS 2019), pages 182-189

ISBN: 978-989-758-372-8

• Evaluate the behavior of the algorithm in a high

dimensional triadic bases (speciﬁcally TRIAS

(Jaschke et al., 2006));

• Propose modiﬁcations in the synthetic dyadic

contexts generator called SCGaz (Rimsa et al.,

2013) to be used in order to generate triadic con-

texts;

• Generate synthetic contexts that allow the analy-

sis of TRIAS algorithm behavior to extract triadic

concepts (main point is to understand the behav-

ior of this algorithm when processing high dimen-

sional databases);

• Represent triadic contexts using BDDs (Binary

Decision Diagram) in order to store and manipu-

late high-dimensional contexts efﬁciently (Akers,

1978). In this case, a set of boolean operations

was implemented to be able to retrieve objects, at-

tributes and conditions.

This paper is divided as following: the Section 2

presents the theoretical basis; Section 3 has the re-

lated works; Section 4 the proposed approach, tests

and analyzes are showed; and ﬁnally, in Section 5,

conclusions and future works.

2 BACKGROUND

2.1 Formal Concepts Analysis (FCA)

Developed by Rudolf Wille in the 1980s, Formal

Analysis of Concepts (FCA) is a branch of applied

mathematics based on concept mathematics and the

conceptual hierarchy (Wille, 1982) (Bernhard and

Rudolf, 1999). The formalization of the concepts

should be transparent and simple, but also compre-

hensive, so that the main aspects of a concept can

have their explicit references in the formal model

(Lehmann and Wille, 1995).

The dyadic approach is based on the primitive no-

tion of a formal context which is a triple (G, M, I),

where G is the set of objects, M the set of attributes

and i is the binary incidence relation between G and

M, indicating that a G object has a certain m attribute

of G of M. The Table 1 represents a dyadic context.

Formal concepts and rules of implication can be de-

ﬁned from dyadic contexts.

Table 1: Dyadic context represented by a cross-table.

G/M m

× ×

A formal concept of a formal context (G, M, I) is

deﬁned by a pair (A, B) which A ⊆ G, B ⊆ M. The pair

(A, B) that deﬁnes the concept follows the conditions

A = B’ and B = A’, deﬁned by the derivation operator

(

): A’ = {g ∈ G | gIm ∀ m ∈ B} and B’ = {m ∈ M |

gIm ∀ g ∈ G} - the extent A contains each object of G

which has all the attributes of B, and the intent B con-

tains all attributes of M which belongs to all objects

of A.

Implications are dependencies between elements

of a set obtained from a formal context. For example,

given the context (G, M, I) the implication rules are

of the form B → C, if and only if, B, C ⊆ M and B’

⊆ C’. An implication rule B → C is considered valid

if and only if, every object that has the attributes of B

also have the attributes of C.

2.2 Triadic Concept Analysis (TCA)

The TCA was introduced by Lehmann and Wille

(Lehmann and Wille, 1995), extends the classic FCA,

but a new dimension was added. The primitive notion

of a triadic formal context is deﬁned by a quadruple

, K

, Y) where K

, K

and K

are sets and Y

the ternary relation between K

, K

and K

. The el-

ements of K

, K

and K

are known as objects, at-

tributes and conditions respectively and (o

, a

, c

) ∈

Y is interpreted as the object o

that has the attribute

under the condition c

(Lehmann and Wille, 1995)

(Wille, 1995). The Table 2 presents a triadic context

where the incidences are represented through the rela-

tion between the objects o

, attributes a

and concepts

of the context, assigned or not, marked by a ×.

Table 2: Triadic context represented by a cross-table.

-K

× × × ×

× × ×

Although it comes from the FCA, the triadic ap-

proach has concept deﬁnitions, implication rules and

derivation much more complex than the dyadic ap-

proach. It’s due to the third dimension that was added,

which includes, for example, quality metrics (trust

and support) to the formally deﬁned implication rules.

A formal triadic concept is deﬁned by a triple

(A, B, C), where A ⊆ K

, B ⊆ K

and C ⊆ K

and

A × B × C ⊆ Y. The set A, B, C are called objects, at-

tributes and mode respectively (Lehmann and Wille,

1995). The set of all the concepts of a partially or-

dered triadic context form a complete lattice called

conceptual lattice (Missaoui and Kwuida, 2011).

Manipulating Triadic Concept Analysis Contexts through Binary Decision Diagrams

183

2.3 Synthetic Generator SCGaz

Using a synthetic database for generating formal con-

texts becomes interesting due to the complexity of

the databases obtained from real scenarios. Real

databases usually require preprocessing, a task that

can, if not done correctly, directly interfere in the re-

sults. Considering that, using tools for database sim-

ulation becomes interesting and extremely useful in

comparative analyzes between algorithms, as realized

in (de Moraes et al., 2016) (Santos et al., 2018).

The SCGaz tool proposed in (Rimsa et al., 2013)

is a random synthetic generator of dyadic formal con-

texts with density control. Through SCGaz it is pos-

sible to specify the amount of objects and attributes

desired in a formal context, as well as density, to gen-

erate irreducible contexts. Density values for a given

context vary according to their dimensions and/or can

be speciﬁed in advance. The generated context is ir-

reducible, that is, there are no attributes that are not

shared by at least one object or attributes that are

shared by all objects. The same occurs with objects

that do not have any attributes or objects that share

all the attributes of the context. Objects that share the

same attributes, in FCA, are considered redundant and

therefore are not inserted into the context.

2.4 TRIAS Algorithm

The authors in (Jaschke et al., 2006) deﬁne the prob-

lem of mining all the most frequent triadic concepts

of a formal context and proposes a solution called

TRIAS, based on dyadic projections to resolve the

problem. The authors adapt the dyadic notion of min-

ing all item sets of a formal dyadic context, deﬁned in

(Pei et al., 2000) for a triadic approach.

Given K = (U, T, R, Y) a triadic context, the prob-

lem of extracting all the common triadic concepts

from a context is to determine all triples (A, B, C)

of the context K such as

≥ u-minsup,

≥

t-minsup and

≥ r-minsup. TRIAS algorithm ﬁrst

constructs a dyadic context L = (U, T × R, Y

) where

its columns correspond to pairs of elements that be-

longs to T and R and via projection, it extracts all the

formal concepts. The second step consists of, for each

formal concept, check if they are closed in relation to

U. The main feature of the algorithm is to explore the

subsets of newly computed triadic concepts to see if

these are new concepts.

2.5 Binary Decision Diagram

Introduced by (Akers, 1978) and further developed by

(Bryant, 1986), binary decision diagrams (BDD) pro-

vide a canonical representation for much more com-

pact boolean formulas than normal conjunctive and

disjunctive forms. Additional to that, it is more efﬁ-

cient to handling data.

It is possible to get a BDD from a binary deci-

sion tree (Figure 1) which the dotted strokes represent

zero transitions. In other words, the value 1 and the

solid strokes represent positive transitions with value

1. The main idea of decision binary diagrams is to

merge sub-trees of the binary decision tree and elimi-

nate identical (redundant) nodes resulting in a canon-

ical representation. The result of the optimizations

gives us an acyclic directed graph, as represented in

Figure 2, where the dotted transitions represent a null

transition. Note that the node affected by the edge has

a null value and the nodes bound by edges in bold

have a positive value.

A BDD is a directed acyclic graph with two types

of nodes: terminals and non-terminals. Non-terminal

nodes represent the variables of the boolean formula

and the only two terminal nodes represent the values

0 or 1 when the function assumes true or false value.

Even as in the representation of the decision tree, the

dotted and continuous transitions represent false and

positive transitions, respectively.

Figure 1: F(X, Y, Z) = XY + Y Z + XY Z represented by a

decision tree.

Figure 2: F(X, Y, Z) = XY + Y Z + XY Z represented by a

binary decision diagram.

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3 RELATED WORK

Several papers used BDDs with different goals. In

(Salleb et al., 2002) the BDDs were used to store

transaction logs as a truth table and to ﬁnd common

patterns in large transactional data sets. The use of

BDDs allowed authors to load all transactions into the

main memory, avoiding database processing on disk.

In (Neto et al., 2018) the authors used the bi-

nary decision diagram to deal with high dimensional

dyadic contexts in order to extract formal dyadic con-

cepts. The authors proposed modiﬁcations in the

NextClosure algorithm and In-Close2 through BDD

to manipulate objects in a dyadic context.

For the tests, they used the NextClosure algorithm

and dyadic contexts with 50,000 objects and 25 at-

tributes, generated through the SCGaz tool. They ob-

tained signiﬁcant gains using BDD, taken results up to

4 times better than the original implementation. In ad-

dition, the use of diagrams allowed authors to explore

contexts with greater dimensionality, such as 50,000

objects, 20 attributes and 70% density.

The authors also explored the In-Close2 algorithm

with BDD. Once again, the approach proved to be ef-

ﬁcient in many cases. They obtained speedup of up to

2 in contexts with 500,000 objects and maximum den-

sity. In several situations the approach with BDD was

able to generate concepts while the original algorithm

stopped the execution due to memory overﬂow.

In (Santos et al., 2018) the authors proposed mod-

iﬁcations in the algorithm of extraction of dyadic im-

plications PropIm, using BDDs as the data structure

in order to manipulate and extract proper rules from

dyadic contexts. The ImplicPBDD algorithm pre-

sented signiﬁcantly better run time. The tests varied

the number of objects, attributes and their density.

The results showed that the version using BDD

has a better performance − up to 80% faster − than

its original algorithm, regardless of the number of at-

tributes, objects and densities. The authors were also

able to explore contexts with higher dimensionality

than the original algorithm was able to process, ex-

panding the horizon of applications.

4 THE APPROACH, TESTS AND

ANALYZES

This paper aimed to analyze the behavior of the

TRIAS algorithm in high dimensional triadic contexts

that was generated from a synthetic tool (SCGaz). It

was also proposed a representation of triadic contexts

using BDDs. This approach can be used in future

work as the main structure of the triadic algorithms

such as TRIAS, as explained in previous sections.

Modiﬁcations in the dyadic synthetic contexts

generator SCGaz were performed in order to gener-

ate triadic contexts by adding a third dimension, not

computed by the tool. The rules of reducibility de-

ﬁned in (Rimsa et al., 2013) were maintained for the

triadic contexts. A third dimension chosen by the user

is added and the objects of the dyadic context are then

replicated to the triadic context subject to the previ-

ously deﬁned conditions.

From the previous modiﬁcations, evaluations on

the behavior of the TRIAS algorithm were per-

formed using the contexts previously generated by the

SCGaz. Average execution time, number of concepts

found, dimensions and density were evaluated in the

initial tests of this paper in order to ﬁnd the limits of

the algorithm. Through projections of the triadic con-

text in dyadic, boolean functions are generated from

the context and then the BDD was built.

4.1 Triadic Contexts using SCGaz

The synthetic generator SCGaz proposed in (Rimsa

et al., 2013) provides a dyadic approach to generating

random contexts. However, the triadic TCA approach

has a third dimension commonly called conditions.

This dimension provides a greater characterization of

the objects, since they are now related to a given at-

tribute under a condition.

In this paper, the tool SCGaz was extended by

adding dimension of conditions in the generated con-

texts. The amount of conditions is set by the user.

Given an irreducible formal context (G, M, I), gen-

erated through of the SCGaz, a dyadic incidence is

deﬁned by gIm ⊆ I, where g ∈ G and m ∈ M. A tri-

adic context (K

, K

, Y) is generated in K

= G

and each attribute a

∈ K

is deﬁned by the Equation

(1). Therefore, the attributes of the new context are

given by the modular relation between the attributes

of the dyadic context and the size of the third dimen-

sion deﬁned by the user.

= m

mod |K

| (1)

Given an incidence gIm ∈ I, where g ∈ G and m ∈ M,

from a irreducible formal context (G, M, I), obtained

from the SCGaz, a triadic context (K

, K

, Y) is

generated, and the rule that adds the incidence gIm

linked to the condition c

is deﬁned by the ratio of

the original context attribute and the size of the third

dimension that was deﬁned by the user. The Equation

(2) represents the obtaining of the conditions of the

triadic context.

(2)

Manipulating Triadic Concept Analysis Contexts through Binary Decision Diagrams

185

4.2 TRIAS Algorithm Evaluation

From the random synthetic generator SCGaz, several

contexts were generated in order to evaluate the be-

havior of the TRIAS triadic concept extraction algo-

rithm. Synthetic triadic contexts with arbitrary num-

ber of dimensions and density were generated in order

to understand the behavior of the algorithm.

Initially, the number of attributes and conditions

were ﬁxed, maximizing the number of objects in order

to obtain a greater number of incidents. Contexts with

500, 1.500, 3.000, 5.000 and 10.000 objects were gen-

erated with 15 attributes and 5 conditions. The den-

sity was set at 30 % for all contexts because the main

objective was to understand the boundary dimensions

for the TRIAS algorithm (the amount of attributes,

objects and conditions supported).

The tests were run on an Intel Core i7-4790

3.60GHz with 4 cores, 8 threads, 32Gb RAM and an

Ubuntu 14.04 LTS operating system. Table 3 presents

the results initially considering contexts with reduced

dimensions according to (Old and Priss, 2006). It is

possible to notice that even with a reduced number of

objects, attributes and conditions, the algorithm took

approximately 40 minutes to compute all the concepts

of the ﬁrst synthetic context. Note that the test with

10,000 objects, 15 attributes and 5 conditions required

a time greater than 7 days and was not properly com-

puted.

Table 3: TRIAS Algorithm Results for Minor Contexts.

Context

(Objects X Attributes X Conditions

Incidences

TRIAS

(Minutes)

500 x 15 x 5 13,500 42.68

1,500 x 15 x 5 33,750 212.4

3,000 x 15 x 5 67,500 376.2

5,000 x 15 x 5 112,500 768.8

10,000 x 15 x 5 225,000 -

It is interesting note that the high dimensionality

characteristics in triadic applications may differ from

the same dyadic ones investigated. In 2006, the In-

ternational Conference on Formal Concept Analysis

(ICFCA) at Desdren (Old and Priss, 2006) discussed

the main challenges of formal analysis, including the

need to deal with dense and high dimensional formal

contexts, 120,000 objects and 70,000 attributes, these

are considerably larger than the tests observed here.

New tests were performed, with contexts char-

acterized by high dimensionality. Table 4 presents

the results obtained with contexts of 120,000 objects,

varying attributes and conditions respectively. It is

possible to note that in none of the cases of high di-

mensionality was possible to reach the end of the ex-

ecution of the algorithm, keeping it running for more

than 7 days and no conclusion was taken.

The results obtained in the implementation of

TRIAS showed the high computational cost of the

extraction of knowledge from triadic contexts. The

impossibility of using TCA with high-dimensional

databases is notorious. This fact certainly demands

more investigation and new proposals to make feasi-

ble the use of TCA algorithms in this context.

Table 4: TRIAS Algorithm Results for High-dimensional

Contexts.

Context

(Objects X Attributes X Conditions

Incidences

TRIAS

(Days)

120,000 x 15 x 5 2,999,984 >7

120,000 x 10 x 5 1,776,769 >7

120,000 x 5 x 10 1,776,769 >7

4.3 TRIAS Algorithm Results for Real

Datasets Contexts

The use of synthetic databases is a great help in test-

ing algorithms and knowledge extraction tools. How-

ever, understand the behavior of the algorithm in real

scenarios is extremely important to understand its ef-

ﬁciency.

Considering that, we applied the algorithm to an

extensive database of movie ratings called Movie-

Lens

. The database consists of ratings of more

than 6,000 anonymous users in approximately 4,000

movies. In the more than 1 million records, users

rate movies with 1 to 5 ratings. This base is con-

sidered sparse, meaning despite the huge number of

ratings, there is no guarantee that users rated the same

movies, consequently generating sparse triadic con-

texts regarding the number of incidents.

The contexts generated from the database have as

a set of objects the users who have made classiﬁca-

tions, the attributes of the context are the classiﬁed

movies and the conditions are the notes received. So,

our triadic context can be deﬁned as K = (U, T, R, Y)

where U are the set of users in the dataset, T are set

of movies, R the set of evaluations given by users and

Y are the relation between users and movies and their

respective evaluations.

Figure 3 represents one example of a triadic con-

cept extract from the dataset movie. Note that in ﬁrst

concept users 646, 1015 rated the same movie with

the same note.

Table 5 shows the results of the algorithm applied

to contexts generated from the real dataset Movie-

Lens. It was decided to ﬁx the number of objects

https://grouplens.org

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Figure 3: Example of triconcepts from MovieLens Dataset.

(users), using the maximum number of objects avail-

able in the dataset, and vary the amount of attributes

(movies) of the context.

Table 5: TRIAS Algorithm results for MovieLens dataset.

Context

(Objects X Attributes X Conditions

Incidences

TRIAS

(Days)

6,000 x 100 x 5 105,986 >7

6,000 x 150 x 5 158,029 >7

6,000 x 200 x 5 203,898 >7

As opposed to synthetic contexts, where we have

control of the dimensions and density of the context,

as shown in Section 4.1, in real databases we can not

guarantee that. In the context generated, it can not

be determined that a movie has been classiﬁed by all

users of the context, making it extremely sparse. In

order to increase the number of incidents, we vary the

number of attributes so that even in sparse contexts,

these can express the behavior of the algorithm.

As said before, the dataset contains 4,000 movies

that was classiﬁed by the 6,000 users. In number of

incidences, this dataset has high dimensional and a

fully context generated from that will have 1 million

of incidences.

In tests performed with synthetic data presented in

Table 4, TRIAS algorithms showed to be inefﬁcient

for contexts with this number of incidences. Even for

a reduced set of the original data, like as shown in

Table 5, the algorithm showed to be inefﬁcient and

time prohibitive for use in large real scenarios.

Notice that in each test for the Table 5 the al-

gorithm did not ﬁnish the computation of all formal

concepts, running for more than seven days with-

out conclusion. The biggest formal context generate

from MovieLens (6.000x200x5) has approximately

only 20% of the full database, reinforcing that the tri-

adic approach using the algorithm TRIAS is not efﬁ-

cient, even in small subsets of real data.

4.4 Triadic Contexts using BDD

Some FCA applications use BDD as main struc-

tures for efﬁcient storage and manipulation of objects

(Salleb et al., 2002) (Neto et al., 2018) (Santos et al.,

2018). However, there are few triadic approaches that

use the efﬁciency provided by binary decision dia-

grams. Therefore, this paper proposed a triadic rep-

resentation of contexts using BDDs, through dyadic

projections.

Given a formal triadic context (K

, K

, Y)

where K

, K

and K

are called objects, attributes and

condition respectively and Y the ternary relation be-

tween K

, K

and K

, a projection can be performed

in the triadic context (Table 6) resulting in a dyadic

context (K

, K

× K

, Y) (Table 7).

Table 6: Triadic Context (K

, K

, Y).

-K

× × ×

× × × ×

The projection results from the combination of at-

tributes and conditions where each attribute is re-

named according to the condition to which it belongs.

The retrieval and manipulation of attributes and con-

ditions can be done from the label assigned to each

attribute. In the context represented by Table 7 the

dyadic incidence given by the tuple (o1, a1c1) is

equivalent to the triadic incidence given by the triple

(o1, a1, c1) of the context presented in Table 6.

Table 7: Dyadic Context Projection (K

, K

× K

, Y).

×K

× × ×

× × × ×

Once projected, the triadic context, now described by

a dyadic context, can be represented by a binary deci-

sion diagram converting the context to a boolean for-

mula used to generate the corresponding BDD. Table

7 describes the triadic context projected in a dyadic

context and Equation 3 represents it through conjunc-

tive and disjunctive operations between objects and

attributes. The symbols accented by a slash represent

the negation of the attribute.

f (a

, a

) = (a

· ¯a

· a

·)

( ¯a

+ a

· ¯a

· a

· ¯a

) + (a

· ¯a

· a

· ¯a

)

(3)

Manipulating Triadic Concept Analysis Contexts through Binary Decision Diagrams

187

Figure 4 represents the dyadic projection of the tri-

adic context deﬁned above. This approach allows ma-

nipulating triadic contexts using a BDD, providing ef-

ﬁcient manipulation and storage.

Figure 4: Context (K

, K

× K

, Y) represented by a BDD.

Given a triadic context projected and represented

by a BDD, it is interesting to provide techniques for

recovering objects, attributes and conditions, since

any application that uses this representation will need

features that allow the recovery and efﬁcient alter-

ation of these elements.

Consider the context presented in the Table 6, re-

trieval of objects can be done, for example, from log-

ical operations AND or OR under Equation 3 of the

context. If it is necessary to obtain all the objects of

the context represented in Table 7 that have the at-

tribute a

, it can create a BDD object that represents

such an attribute and apply a logical operation AND

between the BDDs. Figure 5 represents such an oper-

ation, returning in a new BDD with the objects o

and

since both share the attribute a

In some situations, if it is necessary to retrieve all

the objects they have, for example, the attributes a

and a

of the context presented in Table 7, a BDD

with both attributes must be created and the logical

AND operation between this new BDD with the con-

text BDD must be performed. This returns only the

objects that is required. Figure 6 illustrates such an

operation.

5 CONCLUSION AND FUTURE

WORKS

The task of extracting triadic concepts from a triadic

context is more complex than in the classic approach

of FCA. The representation of the data in three di-

Figure 5: Logical operation between the attribute a

and

the BDD Context.

Figure 6: Logical operation between the attribute a

and

and the BDD Context.

mensions leads to the greater dimensionality of the

databases. Considering that and with the growth of

the contexts, techniques like the TRIAS algorithm be-

come inefﬁcient to extract information, as presented

in this paper.

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In order to improve the performance of these algo-

rithms, the representation of contexts through struc-

tures such as BDD has showed to be an interesting

and efﬁcient alternative in information retrieval.

The paper presented a number of challenges, such

as generating triadic contexts through a synthetic gen-

erator of dyadic contexts. In addition, contexts must

respect the formal triadic deﬁnition that all conditions

have the same amount of attributes.

Through this study, it is noticed that the problem

of high dimensionality in triadic contexts already hap-

pens with a reduced number of objects, attributes (and

conditions) when compared to the dyadic approach.

The tests performed showed that the TRIAS algo-

rithm, for example, can not handle dimensions char-

acterized as dyadic high dimensionality and showed

to be inefﬁcient when used with larger context for the

triadic approach.

The retrieval of objects, attributes and conditions

showed to be efﬁcient since operations are performed

under contexts and variables represented as BDDs.

Unlike conventional structures such as lists, queues,

and stacks, where computational complexity is re-

quired, BDDs provide attribute extraction through

unique logical operations that reduce the retrieval

time of elements in a context.

As a future work, we intend to implement a BDD

version of the TRIAS algorithm. The objective is to

reduce the time of the queries performed in order to

classify the subset of newly discovered concepts and

consequently increase the extractive power of more

frequent triadic concepts in a triadic context. It is

also expected to reduce execution time since the re-

sults presented in Table 4 proved to be infeasible.

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