SCIENTIA SINICA Informationis, Volume 50 , Issue 12 : 1817(2020) https://doi.org/10.1360/N112018-00272

## Concept reduction and concept characteristics in formal concept analysis

Ling WEI 1,4,*, Li CAO 1,4, Jianjun QI 2,4,
• AcceptedAug 9, 2019
• PublishedNov 18, 2020
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### Abstract

Formal concept analysis, which is based on formal contexts and concept lattices, is a data analysis method. Formal concepts reflect the relationship between objects and attributes and realize a formal description of concepts in philosophy. To simplify the information description, this paper proposes concept reduction in the formal concept analysis framework and studies the related theories on concept reduction preserving binary relations. In addition, characteristics analysis of three types of concepts that play different roles in reduction processing is performed from the perspective of operators and a Boolean matrix. Finally, a method to calculate concept reduction is provided.

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• Figure 1

The concept lattice corresponding to example 1

• Figure 2

A concept reduct $\mathcal{F}_{1}$ of example 1

• Table 1   The context of living beings and water $(G,~M,~I)$
 $G$ $a$ $b$ $c$ $d$ $e$ $f$ $g$ $h$ $i$ $1$ $1$ $1$ $0$ $0$ $0$ $0$ $1$ $0$ $0$ $2$ $1$ $1$ $0$ $0$ $0$ $0$ $1$ $1$ $0$ $3$ $1$ $1$ $1$ $0$ $0$ $0$ $1$ $1$ $0$ $4$ $1$ $0$ $1$ $0$ $0$ $0$ $1$ $1$ $1$ $5$ $1$ $1$ $0$ $1$ $0$ $1$ $0$ $0$ $0$ $6$ $1$ $1$ $1$ $1$ $0$ $1$ $0$ $0$ $0$ $7$ $1$ $0$ $1$ $1$ $1$ $0$ $0$ $0$ $0$ $8$ $1$ $0$ $1$ $1$ $0$ $1$ $0$ $0$ $0$
• Table 2   The concept lattice corresponding to the formal context in example 1
 Label Concept Label Concept Label Concept Label Concept Label Concept $c_{1}$ $(\emptyset,M)$ $c_{5}$ $(6,~abcdf)$ $c_{9}$ $(36,~abc)$ $c_{13}$ $(234,~agh)$ $c_{17}$ $(12356,~ab)$ $c_{2}$ $(4,~acghi)$ $c_{6}$ $(34,~acgh)$ $c_{10}$ $(678,~acd)$ $c_{14}$ $(568,~adf)$ $c_{18}$ $(5678,~ad)$ $c_{3}$ $(3,~abcgh)$ $c_{7}$ $(23,~abgh)$ $c_{11}$ $(68,~acdf)$ $c_{15}$ $(1234,~ag)$ $c_{19}$ $(G,~a)$ $c_{4}$ $(7,~acde)$ $c_{8}$ $(123,~abg)$ $c_{12}$ $(56,~abdf)$ $c_{16}$ $(34678,~ac)$
• Table 3   The classification of example 1
 Type Concept Core concept $c_{2}$, $c_{4}$ Relatively necessary concept $c_{3}$, $c_{6}$, $c_{7}$, $c_{8}$, $c_{9}$, $c_{10}$, $c_{11}$, $c_{12}$, $c_{13}$, $c_{14}$, $c_{15}$, $c_{16}$, $c_{17}$ Unnecessary concept $c_{1}$, $c_{5}$, $c_{18}$, $c_{19}$
• Table 4   A part of pairs and concepts
 Serial number $k$ Pair Concept set ${\rm~CS}_{k}$ Serial number $k$ Pair Concept set ${\rm~CS}_{k}$ 1 $(1,~b)$ $\{c_{8},~c_{17}\}$ 8 $(3,~g)$ $\{c_{3},~c_{6},~c_{7},~c_{8},~c_{13},~c_{15}\}$ 2 $(1,~g)$ $\{c_{8},~c_{15}\}$ 9 $(3,~h)$ $\{c_{3},~c_{6},~c_{7},~c_{13}\}$ 3 $(2,~b)$ $\{c_{7},~c_{8},~c_{17}\}$ 10 $(5,~b)$ $\{c_{12},~c_{17}\}$ 4 $(2,~g)$ $\{c_{7},~c_{8},~c_{13},~c_{15}\}$ 11 $(5,~f)$ $\{c_{12},~c_{14}\}$ 5 $(2,~h)$ $\{c_{7},~c_{13}\}$ 12 $(8,~c)$ $\{c_{10},~c_{11},~c_{16}\}$ 6 $(3,~b)$ $\{c_{3},~c_{7},~c_{8},~c_{9},~c_{17}\}$ 13 $(8,~f)$ $\{c_{11},~c_{14}\}$ 7 $(3,~c)$ $\{c_{3},~c_{6},~c_{9},~c_{16}\}$
• Table 5   Pairs and concepts corresponding to the minimum concept intervals
 Serial number $k$ Minimum concept interval Pair Concept set ${\rm~CS}_{k}$ 1 $\mathcal{I}_{1b}$ $(1,~b)$ $\{c_{8},~c_{17}\}$ 2 $\mathcal{I}_{1g}$ $(1,~g)$ $\{c_{8},~c_{15}\}$ 5 $\mathcal{I}_{2h}$ $(2,~h)$ $\{c_{7},~c_{13}\}$ 7 $\mathcal{I}_{3c}$ $(3,~c)$ $\{c_{3},~c_{6},~c_{9},~c_{16}\}$ 10 $\mathcal{I}_{5b}$ $(5,~b)$ $\{c_{12},~c_{17}\}$ 11 $\mathcal{I}_{5f}$ $(5,~f)$ $\{c_{12},~c_{14}\}$ 12 $\mathcal{I}_{8c}$ $(8,~c)$ $\{c_{10},~c_{11},~c_{16}\}$ 13 $\mathcal{I}_{8f}$ $(8,~f)$ $\{c_{11},~c_{14}\}$

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