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generated from big data problems and they are com-
putationally difficult to handle - most algorithms do
not perform accordingly in these contexts and they are
usually unable to process large amounts of data.
So, in this paper, we present a new algorithm,
ProperImplicBDD, which is able to handle high-
dimensional formal contexts. It is based on PropIm
algorithm proposed in (Silva et al., 2017), but our al-
gorithm uses a known data structure named binary
decision diagram (BDD) to store and manipulate ef-
ficiently formal contexts (Neto et al., 2018). There-
fore, it can process a greater volume of data if compa-
red to the algorithms proposed in (Silva et al., 2017)
and (Taouil and Bastide, 2001). The main objectives
of this work are:
• to use BDD’s as a data structure in order to ex-
tract proper implications from high dimensional
contexts;
• to process a greater volume of data compared to
the algorithms proposed in (Silva et al., 2017)
and (Taouil and Bastide, 2001);
• to handle contexts in order to partially meet those
proposed by the challenge (Priss, 2006), such as
context containing 120,000 objects and 70,000 at-
tributes.
In order to analyze the performance of the Pro-
perImplicBDD algorithm, we performed tests using
only synthetic contexts. We have decided to use synt-
hetic contexts instead of real ones in order to control
the experiments and produce concrete results. We can
control and vary the number of attributes and context
density to evaluate our approach.
The variation in attributes and densities were used
to analyze the size and time of searches using the
BDD. The number of objects was fixed in 120,000 to
meet partially the challenge proposed in (Priss, 2006).
The experiments shown that ProperImplicBDD
has a better perfomance – up to 8 times faster – than
PropIm, regardless of the number of attributes, obje-
tcts and densities.
This paper is organized as follows: Section II pre-
sents the basic concepts of the FCA and BDD approa-
ches. Section III describes the related works on BDD
and implications. Section IV describes the methodo-
logy. In Section V, the experiments are presented and
discussed. Finally, Section VI presents the conclusi-
ons and future works.
2 BACKGROUND
2.1 Formal Concept Analysis
The formal concept analysis (FCA) is a field of mat-
hematics that allows the identification of associations
(concepts) and dependencies (implications) of a data
set represented as a formal context (Ganter and Wille,
1997).
Details about formal context, formal concept and
implication rules, basic elements of FCA, are descri-
bed in the following subsections.
2.1.1 Formal Context
A formal context is formed by a triple (G, M,I),
where G is a set of objects (rows), M is a set of at-
tributes (columns) and I is defined as the binary rela-
tionship (incidence relation) between objects and at-
tributes where I ⊆ G × M (Ganter et al., 2005).
An example of a context is shown in Table 1. The
objects (rows) are: Whale, Owl, Human, Shark and
Penguin. The attributes (columns) are : Aquatic, Ter-
restrial, Irrational, Feathered. The incidences (“X”)
are the relationship between objects and attributes:
Whale contains Aquatic and Irrational attributes, Owl
contains Terrestrial, Irrational and Feathered attribu-
tes, etc. For example, the owl has the following cha-
racteristics: terrestrial, irrational and feathered.
Table 1: A simple formal context.
Aquatic Terrestrial Irrational Feathered
Whale X . X .
Owl . X X X
Human . X . .
Shark X . X .
Penguin . X X X
2.1.2 Formal Concept
A formal concept is defined by a pair (A,B) where
A ⊆ G is called extension and B ⊆ M is called
intention. This pair must follow the conditions where
A = B
0
and B = A
0
(Ganter et al., 2005). The relation
is defined by the derivation operator (
0
):
A
0
= {m ∈ M | gIm ∀ g ∈ A}
B
0
= {g ∈ G | gIm ∀ m ∈ B}
A formal concept seeks to identify the set of
attributes (intention) that delimit and characterize an
object (extension). Using the Table 1 as an example,
the generated concepts would be:
({Whale, Owl, Human, Shark, Penguin}, {})
An Approach to Extract Proper Implications Set from High-dimension Formal Contexts using Binary Decision Diagram
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