A General Theory of Concept Lattice (II): Tractable Lattice Construction and Implication Extraction

TL;DR

This paper introduces a tractable method for constructing a general concept lattice that efficiently captures both its structure and logical implications, unifying formal-concept and rough-set lattices with a single scan.

Contribution

It presents a feasible construction approach for the general concept lattice and demonstrates how to derive all implication relations from a single formula, simplifying inference processes.

Findings

Single-scan construction of the general concept lattice is feasible.

All implications can be derived from a single contextual formula.

Formal-concept and rough-set implications are special cases within this framework.

Abstract

As the second part of the treatise 'A General Theory of Concept Lattice', this paper speaks of the tractability of the general concept lattice for both its lattice structure and logic content. The general concept lattice permits a feasible construction that can be completed in a single scan of the formal context, though the conventional formal-concept lattice and rough-set lattice can be regained from the general concept lattice. The logic implication deducible from the general concept lattice takes the form of {\mu}_1 {\rightarrow} {\mu}_2 where {\mu}_1,{\mu}_2 {\in} M^{\ast} are composite attributes out of the concerned formal attributes M. Remarkable is that with a single formula based on the contextual truth 1_{\eta} one can deduce all the implication relations extractable from the formal context. For concreteness, it can be shown that any implication A {\rightarrow} B (A, B being subsets of the formal attributes M) discussed in the formal-concept lattice corresponds to a special case of {\mu}_1 {\rightarrow} {\mu}_2 by means of {\mu}_1 = {\prod} A and {\mu}_2 = {\prod} B. Thus, one may elude the intractability due to searching for the Guigues-Duquenne basis appropriate for the implication relations deducible from the formal-concept lattice. Likewise, one may identify those {\mu}_1 {\rightarrow} {\mu}_2 where {\mu}_1 = {\sum} A and {\mu}_2 = {\sum} B with the implications that can be acquired from the rough-set lattice. (Here, the product {\prod} stands for the conjunction and the summation {\sum} the disjunction.)}

Tables

Table 1. Table 1: The formal context F ​ ( G , M ) 𝐹 𝐺 𝑀 F(G,M) with ( G , M ) = ( { 1 , 2 , 3 , 4 , 5 , 6 } , { a , b , c , d , e } ) 𝐺 𝑀 1 2 3 4 5 6 𝑎 𝑏 𝑐 𝑑 𝑒 (G,M)=(\{1,2,3,4,5,6\},\{a,b,c,d,e\}) and the formal context F ′ ​ ( G , M ) superscript 𝐹 ′ 𝐺 𝑀 F^{\prime}(G,M) with ( G , M ) = ( { a , b , c , d , e } , { 1 , 2 , 3 , 4 , 5 , 6 } ) 𝐺 𝑀 𝑎 𝑏 𝑐 𝑑 𝑒 1 2 3 4 5 6 (G,M)=(\{a,b,c,d,e\},\{1,2,3,4,5,6\}) Although transposing one of the contexts gives rise to the other one, there are two mathematically distinct concept lattices to be constructed. One may assume U = { 1 , 2 , 3 , 4 , 5 , 6 } 𝑈 1 2 3 4 5 6 U=\{1,2,3,4,5,6\} and V = { a , b , c , d , e } 𝑉 𝑎 𝑏 𝑐 𝑑 𝑒 V=\{a,b,c,d,e\} to be what are conventionally recognised as objects and attributes, respectively. According to the setting of Ref. YY04 , the choice F ​ ( G , M ) 𝐹 𝐺 𝑀 F(G,M) in which ( U , V ) ≡ ( G , M ) 𝑈 𝑉 𝐺 𝑀 (U,V)\equiv(G,M) is referred to as an object-oriented scheme and F ′ ​ ( G , M ) superscript 𝐹 ′ 𝐺 𝑀 F^{\prime}(G,M) ( ( V , U ) ≡ ( G , M ) 𝑉 𝑈 𝐺 𝑀 (V,U)\equiv(G,M) ) a property-oriented scheme.

$F(G,M)$
$M$	$a$	$b$	$c$	$d$	$e$
$G$
$1$	$\times$		$\times$	$\times$	$\times$
$2$	$\times$		$\times$
$3$		$\times$			$\times$
$4$		$\times$			$\times$
$5$	$\times$
$6$	$\times$	$\times$			$\times$

Table 2. Table 2: The factorisation of | M ∗ | superscript 𝑀 ∗ |M^{\ast}| attributes into 2 n F superscript 2 subscript 𝑛 𝐹 2^{n_{F}} equivalent classes, exemplified with F ​ ( G , M ) 𝐹 𝐺 𝑀 F(G,M) in Table 1 For convenience, it is assumed that all the members in M 𝑀 M are independent. Based on a given formal context it is possible, albeit tedious, to determine the object correspondence to all the attributes that can be constructed out of M 𝑀 M , i.e. M ∗ superscript 𝑀 ∗ M^{\ast} because this is nothing but the completion of a truth-value table. Accordingly, any μ 𝜇 \mu in M ∗ superscript 𝑀 ∗ M^{\ast} must end up with μ R ∈ E F superscript 𝜇 𝑅 subscript 𝐸 𝐹 \mu^{R}\in E_{F} ( μ R ⊆ G superscript 𝜇 𝑅 𝐺 \mu^{R}\subseteq G ). It turns out that M ∗ superscript 𝑀 ∗ M^{\ast} can be further categorised: since n F = 5 subscript 𝑛 𝐹 5 n_{F}=5 ( | E F | = 2 | n F | subscript 𝐸 𝐹 superscript 2 subscript 𝑛 𝐹 |E_{F}|=2^{|n_{F}|} ), the | M ∗ | = 2 2 5 = 4294967296 superscript 𝑀 ∗ superscript 2 superscript 2 5 4294967296 |M^{\ast}|=2^{2^{5}}=4294967296 generalised attributes can be grouped into 2 5 superscript 2 5 2^{5} equivalent classes ( [ X ] F subscript delimited-[] 𝑋 𝐹 [X]_{F} s). Indeed, | [ X ] F | subscript delimited-[] 𝑋 𝐹 |[X]_{F}| is even a constant over the GCL ( Corollary 4.4 ).

$⟵4294967296$ columns grouped into $32$ equivalent classes $⟶$

${\left[X\right]}_F$

${\left[\{1,2,5,6\}\right]}_F$

${\left[\{3,4,6\}\right]}_F$

${\left[\{1,2\}\right]}_F$

${\left[\left\{1\right\}\right]}_F$

${\left[\{1,3,4,6\}\right]}_F$

${\left[\varnothing \right]}_F$

${\left[G\right]}_F$

${\left[\left\{2\right\}\right]}_F$

${\left[\{3,4\}\right]}_F$

etc.

$M^{\ast }$

$a$

$\dots$

$b$

$\dots$

$c$

$\dots$

$d$

$\dots$

$e$

$\dots$

$\mathrm{𝟎}$

$\dots$

$\mathrm{𝟏}$

$\dots$

$\dots$

$\dots$

$\dots$

$\dots$

$\dots$

$G_{/R}$

$\left\{1\right\}$

$\times$

$\dots \times$

$\times$

$\dots \times$

$\times$

$\dots \times$

$\times$

$\dots \times$

$\times$

$\dots \times$

$\dots$

$\dots$

$\dots$

$\left\{2\right\}$

$\times$

$\dots \times$

$\times$

$\dots \times$

$\times$

$\dots \times$

$\times \dots \times$

$\dots$

$\dots$

$\dots$

$\{3,4\}$

$\times$

$\dots \times$

$\times$

$\dots \times$

$\times$

$\dots \times$

$\times \dots \times$

$\dots$

$\dots$

$\dots$

$\left\{5\right\}$

$\times$

$\dots \times$

$\times$

$\dots \times$

$\dots$

$\dots$

$\dots$

$\left\{6\right\}$

$\times$

$\dots \times$

$\times$

$\dots \times$

$\times$

$\dots \times$

$\times$

$\dots \times$

$\dots$

$\dots$

$\dots$

Equations

$$g\in G\mapsto g^R\in 2^M(m\in M\mapsto m^R\in 2^G),$$

$$l_i\ne l_j⟺\{\begin{array}{c}{X}_i\ne X_j\\ [X_i{]}_F\ne [X_j{]}_F\end{array},l_i<l_j⟺\{\begin{array}{c}{X}_i\subset X_j\\ [X_i{]}_F<[X_j{]}_F\end{array}$$

$$\rho (X)=\sum X^{{\square }^{\ast }}=\sum \{\mu \in M^{\ast }\mid {\mu }^R\subseteq X\}=\sum _{X_0\subseteq X}\left(\sum [X_0{]}_F\right)\equiv \sum [X{]}_F.$$

$$\eta (X)=\prod X^{I^{\ast }}=\prod \{\mu \in M^{\ast }\mid {\mu }^R\supseteq X\}=\prod _{X_0\supseteq X}\left(\prod [X_0{]}_F\right)\equiv \prod [X{]}_F.$$

$$X^{I^{\ast }}=\left\{\mu \in M^{\ast }\mid gR\mu ,g\in X\right\}=\bigcup _{X_0\supseteq X}\left\{\mu \in M^{\ast }\mid {\mu }^R=X_0\right\}=\bigcup _{X_0\supseteq X}[X_0{]}_F.$$

$$Y^{I^{\ast }}=(\bigcup _{X_0\supseteq X}[X_0{]}_F{)}^{I^{\ast }}=\bigcap _{X_0\supseteq X}([X_0{]}_F{)}^{I^{\ast }}=\bigcap _{X_0\supseteq X}{\left\{\mu \in M^{\ast }\mid {\mu }^R=X_0\right\}}^{I^{\ast }}=\bigcap _{X_0\supseteq X}{X}_0=X,$$

$$X^{{\square }^{\ast }}=\left\{\mu \in M^{\ast }\mid \forall g\in X,gR\mu ⟹g\in X\right\}=\bigcup _{X_0\subseteq X}\left\{\mu \in M^{\ast }\mid {\mu }^R=X_0\right\}=\bigcup _{X_0\subseteq X}[X_0{]}_F=Y.$$

$$(X,\rho (X),\eta (X))\equiv (X,[X{]}_F)=(X,[\eta (X),\rho (X)]),$$

$$\eta (X)=\prod \{\eta (G\backslash D_k)\mid G\backslash D_k\supseteq X\},$$

$$\eta (X)={\eta }_0(X)\cdot 1_{\eta },$$

$$\eta (G)\equiv 1_{\eta }=\prod [G^{+}{]}_F,$$

$${\eta }_0(G\backslash D_k)=\prod [(G\backslash D_k{)}^{+}{]}_F,$$

$$\eta (G)\equiv 1_{\eta }=\prod [G^{+}{]}_F$$

$$(a+b{)}^R=\dots =(b+d+\neg e{)}^R=(b+\neg d+e{)}^R=\dots =(\neg d+e{)}^R=G,$$

$$1_{\eta }=\sum _{k=1}^{n_F}{\Psi }^k=a\neg bcde+a\neg bc\neg d\neg e+\neg ab\neg c\neg de+a\neg b\neg c\neg d\neg e+ab\neg c\neg de.$$

$${\eta }_0(G\backslash D_1)\cdot 1_{\eta }$$

$$\eta (X)=\prod _{k\in K_0\backslash K}\eta (D_k^c)=\prod _{k\in K_0\backslash K}\sum _{k^{\prime }\ne k}\eta (D_{k^{\prime }})=\sum _{k^{\prime }\in {\bigcap }_{k\in K}{K}_0\backslash \{k\}}\eta (D_{k^{\prime }})=\sum _{k\in K}\eta (D_k),$$

$$\rho (X)=\neg \eta (X^c)=\neg \sum _{j\in K_0\backslash K}\eta (D_j)=\prod _{k\in K_0\backslash K}\neg \eta (D_k),$$

$$\bigcup _i{X}_i=\bigcup _i\bigcup _{k\in K^i}{D}_k$$

$$\bigcap _j{X}_j=\bigcap _j\bigcap _{k\in K^j}{D}_k^c$$

$$e_k\equiv e_k(M),$$

$$rank(M)$$

$$M^{\ast }=(b_{inf}(M^{\ast }){)}^{\ast }=(M^{\ast }{)}^{\ast },rank(M)=rank\left(b_{inf}(M^{\ast })\right)=rank(M^{\ast }),$$

$${{\rm Y}}_{\eta }^F=[\underset{\eta (D_1)}{\underbrace{a\neg bcde}},\underset{\eta (D_2)}{\underbrace{a\neg bc\neg d\neg e}},\underset{\begin{array}{c}\eta (D_3)\\ D_3=\{3,4\}\end{array}}{\underbrace{\neg ab\neg c\neg de}},\underset{\eta (D_4)}{\underbrace{a\neg b\neg c\neg d\neg e}},\underset{\eta (D_5)}{\underbrace{ab\neg c\neg de}}].$$

$$\eta (D_k)\equiv {\eta }_k=\prod {\Psi }^k:=e_k,k\in K_0=\{1,2,\dots ,n_F\}\subseteq K^{\ast },$$

$$X$$

$$\eta (X)$$

$$\rho (X)$$

$$E_F^{fcl}=\left\{\bigcap _{m\in M_0}{m}^R\mid M_0\subseteq M\right\},E_F^{rsl}=\left\{\bigcup _{m\in M_0}{m}^R\mid M_0\subseteq M\right\}$$

$$\forall\mu\in M^{\ast}\quad\begin{array}[]{cc}\mu\leq 1_{\eta}\iff\mu=\eta(X)&\mbox{for some}\ X\in E_{F}\\ \mu\geq 0_{\rho}\iff\mu=\rho(X)&\mbox{for some}\ X\in E_{F}\end{array},$$

$$\forall X\in E_F[X{]}_F=\{\mu \in M^{\ast }\mid \mu +0_{\rho }=\rho (X),\mu \cdot 1_{\eta }=\eta (X)\}.$$

$${\mu }_1\notin [X{]}_F⟹{\mu }_1^R\ne X\left(\because \{\begin{array}{c}({\mu }_1\cdot 1_{\eta }{)}^R={\mu }_1^R\cap G={\mu }_1^R\ne X\\ ({\mu }_1+0_{\rho }{)}^R={\mu }_1^R\cup \varnothing \ne X\end{array}\right)\text{mbox}whichimplies{\mu }_1\notin \{\mu \in M^{\ast }\mid \mu +0_{\rho }=\rho (X),\mu \cdot 1_{\eta }=\eta (X)\}.$$

$$\begin{array}[]{ccccc}\sum_{i=1}^{n}\mu_{i}&=&\sum_{i=1}^{n}\eta(X_{i})+\sum_{i=1}^{n}\tau_{i}=\eta\left(\bigcup_{i=1}^{n}X_{i}\right)+\tau&\in&\left[\bigcup_{i=1}^{n}X_{i}\right]_{F},\\ \prod_{i=1}^{n}\mu_{i}&=&\prod_{i=1}^{n}\eta(X_{i})+\prod_{i=1}^{n}\tau_{i}=\eta\left(\bigcap_{i=1}^{n}X_{i}\right)+\tau^{\prime}&\in&\left[\bigcap_{i=1}^{n}X_{i}\right]_{F}\end{array}$$

$$(a\cdot \neg e+c)\cdot 1_{\eta }$$

Full text

For concreteness, it can be shown that any implication $A\rightarrow B$ ($A,B$ being subsets of the formal attributes $M$) discussed in the formal-concept lattice corresponds to a special case of $\mu_{1}\rightarrow\mu_{2}$ by means of $\mu_{1}=\prod A$ and $\mu_{2}=\prod B$. Thus, one may elude the intractability due to searching for the Guigues-Duquenne basis appropriate for the implication relations deducible from the formal-concept lattice. Likewise, one may identify those $\mu_{1}\rightarrow\mu_{2}$ where $\mu_{1}=\sum A$ and $\mu_{2}=\sum B$ with the implications that can be acquired from the rough-set lattice. (Here, the product $\prod$ stands for the conjunction and the summation $\sum$ the disjunction.)

General Concept Lattice; Contextual Truth; Pseudo Intent; Guigues-Duquenne Basis.

1 introduction

The general concept lattice (GCL) LLJD12-1 is a novel structure that brings together the formal-concept lattice (FCL) Wi82 ; GW99 ; Wi05 and the rough-set lattice (RSL) Ke96 ; GD02 ; DG03 ; YY04 on a common theoretical foundation. The GCL accomplishes the categorisation of whatever discernible object sets into general extents according to the general intents, which are whatever features the general extents can possess. The general intent can be represented by the pair of the generalised formal-concept property (Gfcp) and the generalised rough-set property (Grsp), which are respectively the generalisations for the intents of FCL and of RSL LLJD12-1 . There are well known challenges both for the construction of lattice Ku01 ; KO02 and for extracting the logic content in FCL GW99 ; GD86 ; Kso04 ; KO08 ; Sb09 ; Df10 ; DS11 ; BK13 . In this paper, one will show that the GCL is free from such intractability problems, the GCL is in practice much easier to handle than the FCL and RSL. To begin with, a remarkable principle that leads to the GCL is that the information content revealed by the formal context $F(G,M)$ should not be specific to the particular choice of formal attribute set $M$. In effect, the possible property characterisations for the objects may run over the generalised attribute set $M^{\ast}$. Hence, by consistency, the GCL turns out to depend on the extended formal context $F^{\ast}(G,M^{\ast})$, see Lemma 2.8 in Ref. LLJD12-1 , since $F^{\ast}(G,M^{\ast})$ is the direct consequence of $F(G,M)$. Intriguingly, despite the enormous increase of attribute freedom, owing to the extension from $M$ to $M^{\ast}$, one finds that the problem treatment is much simplified. Here, an interpretation for the GCL to be manageable is that $F^{\ast}(G,M^{\ast})$ unveils additional instructive relations which are not accessible otherwise. Clearly, such instructive relations can by no means be observed in the FCL and the RSL that only refer to $F(G,M)$.

In Section 2, one will clarify the technical origin why GCL is tractable both in constructing the lattice structure and in implementing the logic content. Basically, the GCL comprises $2^{n_{F}}$ nodes (general formal concepts) characterising all the distinct object classes (general extents) discernible by the formal context, see Proposition 3.4 and 3.5 of Ref. LLJD12-1 , thus, the efforts in deciding which object class is categorised are in fact inessential. Further, it will also become obvious that the identifications for the general intents in fact exhaust all the $|M^{\ast}|$ generalised attributes. In effect, corresponding to each general extent $X\in E_{F}$, see Proposition 3.5 and Proposition 3.6 of Ref. LLJD12-1 , the general intent amounts to the closed interval $[X]_{F}=\{\mu\in M^{\ast}\mid\ \mu^{R}=X\}=[\eta(X),\rho(X)]$, which has the upper bound $\rho(X)$ (Grsp) and the lower bound $\eta(X)$ (Gfcp). Note that the whole generalised attribute set $M^{\ast}$ is then divided into $2^{n_{F}}$ non-overlapping general intents on the GCL, in contrast to the cases for FCL and RSL where attributes in $M$ can be repeatedly used for the intents. Therefore, it is a simplicity to determine the general intents since each of them appears to be a unique closed interval up to its bounds, which are ordered according to Galois connection. Moreover, one notices that the complexity in determining all the bounds can still be halved by virtue of the conjugateness relation of Proposition 3.7 in Ref. LLJD12-1 . Remarkable is also that the the non-overlapping of general intents facilitates the implementations of logic implications. The idea, relating two attributes pertaining to two general intents whose general extents are ordered via set-inclusion relations by an implication, maps to the extraction of implications from the FCL GD86 and RSL. On the other hand, the logic content of GCL are then clarified by means of the simple equivalence among attributes grouped into the same general intent since the equivalent attributes are the property of the same object class.

Sec. 3 is devoted to the practical aspect in determining the GCL’s lattice structure. Unlike Ref. LLJD12-1 , one instead adopts the disjunctive normal form (DNF) for Gfcp and the conjunctive normal form (CNF) for Grsp. It will be demonstrated that the GCL subject to the formal context is characterised by a simple $\eta$-representation or $\rho$-representation which is obtained after one single glance over the formal context. From the $\eta$- or $\rho$-representation one may read out the general intent for any given general extent in $E_{F}$. Every $n_{F}$-bit binary string $B_{X}$ can then be employed to encode a general formal concept as $(X,\rho(X),\eta(X))=(X,[B_{X}]_{\rho},[B_{X}]_{\eta})$. The construction of GCL is thus as tractable as grouping the objects according to attributes in the formal context. One will also demonstrate that both the FCL and RSL can be rediscovered from the GCL as particular cases: The FCL categorises those object classes which are expressible in terms of an intersection of $m^{R}$s for some $m$s in $M$ where the whole object collection $G$ is also regarded as an FCL extent, cf. e.g. Ref. Wi82 ; the RSL categorises those object classes given in terms of a union of $m^{R}$s for some $m$s in $M$ where the empty object set $\emptyset$ is also regarded as an RSL extent, cf. e.g. Ref. YY04 .

In Sec. 4, an implication relation discussed in the frame of GCL is considered between two different attributes, say $\mu_{1}\rightarrow\mu_{2}$, which can be intuitively mapped onto the contextual Venn diagram (Definition 2.9 of Ref. LLJD12-1 ), giving rise to the set inclusion relation $\mu_{1}^{R}\subseteq\mu_{2}^{R}$. Interestingly, the relation $A{\rightarrow}B\ (A\subseteq M,B\subseteq M)$ developed from FCL GD86 , i.e. $A\stackrel{{\scriptstyle fcl}}{{\rightarrow}}B$, appears to be a special case of the logic implication from GCL. Likewise, one may also recognise the relation $A\stackrel{{\scriptstyle rsl}}{{\rightarrow}}B$, which is the one that could have been developed from RSL, as a special case of the GCL-based rules of implication. It is noteworthy that the logic implication arising from the GCL can be deduced from one single formula depending on the contextual truth $1_{\eta}$, see Definition 2.5 of Ref. LLJD12-1 . Remarkably, one may then forgo the prevalent process of finding pseudo-intents for the construction of the minimal implication bases, the so-called Guigues-Duquenne bases or stem bases GW99 ; GD86 ; Kso04 ; KO08 ; Sb09 ; Df10 ; DS11 ; BK13 . Not to mention that there are still implication relations deducible from the GCL which can be neither identified with $A\stackrel{{\scriptstyle fcl}}{{\rightarrow}}B$ nor with $A\stackrel{{\scriptstyle rsl}}{{\rightarrow}}B$.

2 generality versus tractablity

Following the convention of the previous work, the formal context is denoted as $F(G,M)$, where $G$ represents the formal objects and $M$ represents the formal attributes. The fundamental operation $R$ defines the map from an object element to an attribute subset and from an attribute element to an object subset:

[TABLE]

based on which the derivation operators $I,\Box$ and $\Diamond$ are given by Eq. (4)-(6) and (8) of Ref. LLJD12-1 . Note that the treatment of objects and attributes are formally different. The objects are distinct entities while the attributes can overlap per conjunction, a point of view giving rise to the formal context extension from $F(G,M)$ to $F^{\ast}(G,M^{\ast})$ (Lemma 2.8 of Ref. LLJD12-1 ). However, there are then two distinct formal contexts to be considered subject to the same data structure YY04 , as are exemplified and explained in Table 1. Now that if a set of given parameters is regarded as objects it can no more be regarded as attributes and vice versa, it remains an open question whether or how the both formal contexts can co-exist in the same treatment, in what follows one will concentrate on a definite choice of objects and attributes.

Since the GCL makes reference on $F^{\ast}(G,M^{\ast})$, it is intriguing whether the problem complexity of GCL can be reduced despite the domain extension from $M$ to $M^{\ast}$. To this end, it is rather instructive to take into account two ordering systems one may establish among attributes. Without loss of generality, one could assume that no constraint has been pre-imposed on the given attribute set $M$. As the first ordering system, one then ends up the conventional Venn diagram ${\cal V}_{M}^{0}$, which is divided into $2^{|M|}$ disjoint regions in the sense that no pair of attributes in $M$ has an empty intersection. Moreover, as the second ordering system, the contextual Venn diagram ${\cal V}_{M}^{F}$ (Definition 2.9 in Ref. LLJD12-1 ) is employed to govern the attribute relations inferred from the formal context. In Fig. 1, the $n_{F}$ disjoint regions on ${\cal V}_{M}^{F}$ correspond to the $n_{F}$ object classes which are discernible subject to the formal context. The dimension reduction from $|M^{\ast}|$ to $2^{n_{F}}$ can thus be relaised by mapping the $|M^{\ast}|$ generalised attributes in ${\cal V}_{M}^{0}$ onto $2^{n_{F}}$ attribute classes in ${\cal V}_{M}^{F}$, as is illustrated in Table 2. Notice that each of the $2^{n_{F}}$ classes is given with respect to a definite general extent $X\in E_{F}$ as $[X]_{F}=\{\mu\in M^{\ast}\mid\mu^{R}=X\}$ (Definition 2.11 in Ref. LLJD12-1 ), where $E_{F}$ includes all the possible unions of the members in $G_{/R}$ ($|G_{/R}|=n_{F}$ by Definition 2.4 of Ref. LLJD12-1 ). Adopting $[X]_{F}$ as the intent of general concept turns out to be more intuitive than adopting the pair $(\rho(X),\eta(X))$ since $[X]_{F}$ is itself an equivalent class.

Proposition 2.1**.**

Subject to the formal context $F(G,M)$, the 2-tuple $(X,[X]_{F})$ with $X\in E_{F}$ is more appropriate than $(X,\rho(X),\eta(X))$ (Proposition 3.4 in Ref. LLJD12-1 ) for the general concept framework, where $[X]_{F}$ is referred to as the general intent. The following statements concerning the general concept framework are in order.

• •

$[X]_{F}$* can be deduced from $\rho(X)$ and $\eta(X)$ without any additional assumption.*

$[X]_{F}=[\eta(X),\rho(X)]$, where $[\eta(X),\rho(X)]=\{\mu\in M^{\ast}\mid\ \eta(X)\leq\mu\leq\rho(X)\}$.

• •

The GCL exhausts whatever attributes from $M^{\ast}$, $M^{\ast}=\bigcup_{X\in E_{F}}[X]_{F}$.

• •

Different general intents do not overlap: $\forall X_{i}\forall X_{j}\in E_{F}\ [X_{i}]_{F}\cap[X_{j}]_{F}=\emptyset$ iff $X_{i}\neq X_{j}$.

• •

The general concept $(X,[X]_{F})$ can be employed as the node on the lattice since the ordering can be unambiguously defined. Denoting the nodes $l_{i}$ and $l_{j}$ with $\left\{\begin{smallmatrix}l_{i}=(X_{i},[X_{i}]_{F}).\\ l_{j}=(X_{j},[X_{j}]_{F})\end{smallmatrix}\right.$,

[TABLE]

Proof.

The closed interval $[\eta(X),\rho(X)]$ is well-defined since $\forall X\in E_{F}$ $\eta(X)=\prod[X]_{F}\leq\sum[X]_{F}=\rho(X)$. Here, the general intent also includes all the attributes lying between $\eta(X)$ and $\rho(X)$.

• •

$\forall\mu\in[X]_{F}=\{\mu\in M^{\ast}\mid\mu^{R}=X\}\ \rho(X)\geq\mu\geq\eta(X)\ \therefore\mu\in[X]_{F}\implies\mu\in[\eta(X),\rho(X)]$. On the other hand, $\mu\in[\eta(X),\rho(X)]\implies\left\{\begin{smallmatrix}\mu^{R}\supseteq\eta(X)^{R}=X\\ \mu^{R}\subseteq\rho(X)^{R}=X\end{smallmatrix}\right.$ (Proposition 3.4 in Ref. LLJD12-1 ), which implies that $\mu^{R}=X$, i.e., $\mu\in[X]_{F}$. Therefore, $\mu\in[X]_{F}\iff\eta(X)\leq\mu\leq\rho(X)$.

• •

Since $\forall\mu\in M^{\ast}\ \mu^{R}\subseteq G$ (Definition 2.7, Lemma 2.8 in Ref. LLJD12-1 ) and $E_{F}=\left\{\mu^{R}\mid\mu\in M^{\ast}\right\}$, $\bigcup_{X\in E_{F}}[X]_{F}=\bigcup_{\mu\in M^{\ast}}[\mu^{R}]_{F}=M^{\ast}$.

• •

$[X_{i}]_{F}\cap[X_{j}]_{F}=\emptyset$ contradicts $X_{i}=X_{j}$, which implies that $[X_{i}]_{F}=[X_{j}]_{F}$, $\therefore\ [X_{i}]_{F}\cap[X_{j}]_{F}=\emptyset\implies X_{i}\neq X_{j}$. On the other hand, if $X_{i}\neq X_{j}$ then $\mu_{i}^{R}\neq\mu_{j}^{R}$ and hence $\mu_{i}\neq\mu_{j}$ (cf. Lemma 2.10 in Ref. LLJD12-1 ), $\forall\mu_{i}\in[X_{i}]_{F}\ \forall\mu_{j}\in[X_{j}]_{F}$. Consequently, $X_{i}\neq X_{j}\implies[X_{i}]_{F}\cap[X_{j}]_{F}=\emptyset$. Therefore, $[X_{i}]_{F}\cap[X_{j}]_{F}=\emptyset\iff X_{i}\neq X_{j}$.

• •

$(X_{i},[X_{i}]_{F})\neq(X_{j},[X_{j}]_{F})\iff X_{i}\neq X_{j}$ and $[X_{i}]_{F}\neq[X_{j}]_{F}$ since $[X_{i}]_{F}\cap[X_{j}]_{F}=\emptyset\iff X_{i}\neq X_{j}$. Consider then $[X_{i}]_{F}<[X_{j}]_{F}$ as $\left\{\begin{smallmatrix}\rho(X_{i})<\rho(X_{j})\\ \eta(X_{i})<\eta(X_{j})\end{smallmatrix}\right.$ since $X_{i}\subset X_{j}\iff\left\{\begin{smallmatrix}\rho(X_{i})<\rho(X_{j})\\ \eta(X_{i})<\eta(X_{j})\end{smallmatrix}\right.$(Proposition 3.14 in Ref. LLJD12-1 ). Therefore, $l_{i}<l_{j}\iff\left\{\begin{smallmatrix}X_{i}\subset X_{j}\\ [X_{i}]_{F}<[X_{j}]_{F}\end{smallmatrix}\right.$.

∎

While the general extents are all the object classes discernible from the perspective of the formal context, every general intent $[X]_{F}$ collects the attributes all the members of the general extent $X$ possess in common, where $\rho(X)$ and $\eta(X)$ happen to be the upper and lower bound of $[X]_{F}$, respectively. The general concept $(X,\rho(X),\eta(X))$ thus far relates to the RSL-concept and FCL-concept as follows.

• •

If $X$ appears to be an RSL extent, its corresponding intent $X^{\Box}=\{m\in M\mid\ m^{R}\subseteq X\}$, see Eq. (8) of Ref. LLJD12-1 , collects all the unique attributes in $M$ which are not observed on the members of $X^{c}$. Since carrying any part of these attributes suffices to ensure that an object belongs to $X$, the rough-set property as the logical OR of the members in $X^{\Box}$ faithfully characterises the RSL intent. Subsequently, the general rough-set property (Grsp) can be deduced via the extension from $F(G,M)$ to $F^{\ast}(G,M^{\ast})$ as

[TABLE]

• •

If $X$ is an FCL extent $X$, its intent collects all the attributes in $M$ possessed in common by $X$, say $X^{I}=\{m\in M\mid\ m^{R}\supseteq X\}$. Since members in $X$ essentially possess all these attributes, the formal-concept property as the logical AND of the members in $X^{I}$ faithfully characterises the FCL intent. Likewise, the general formal-concept property (Gfcp) is obtained via the extension from $F(G,M)$ to $F^{\ast}(G^{\ast},M)$ as

[TABLE]

• •

Moreover, the conjugate relation $\eta(X^{c})=\neg\rho(X)$ (Proposition 3.7 in Ref. LLJD12-1 ) can have a natural interpretation in terms of the conventional modal logics. With $\rho(X)$, it is not possible that any object $x$ in $X^{c}$ possess the property $\rho$ since otherwise $x\in X$. Hence, by “not possible $=$ definitely not” any object $x\in X^{c}$ definitely has the property NOT $\rho(X)$, also, any object in $X^{c}$ definitely possesses $\eta(X^{c})$. Therefore, $\eta(X^{c})=\neg\rho(X)$. Note that in Eq. (2) and (3) the operators $I^{\ast}$ and $\Box^{\ast}$ are respectively obtained from $I$ and $\Box$ by means of extending the attribute range to $M^{\ast}$, and the same relationship also holds between $\Diamond^{\ast}$ and $\Diamond$ (Definition 3.3 in Ref. LLJD12-1 ).

It should however be clear that the emergence of GCL need not be based on the RSL and FCL, although the GCL can be acquired as a common generalisation from RSL and FCL, see Lemma 2.8 and 3.1 of Ref. LLJD12-1 . Indeed, one can use the content of GCL to construct several extensions of the RSL and FCL.

Proposition 2.2**.**

*Subject to $F(G,M)$, a family of concept-lattice generalisations can be given as follows. *

• •

The generalised FCL (gFCL) can be accomplished by the gFCL concept $(X,Y)_{\mbox{\tiny gFCL}}$ which satisfies $X^{I^{\ast}}=Y$ and $Y^{I^{\ast}}=X$, where $Y=\bigcup_{X_{0}\supseteq X}[X_{0}]_{F}\ \forall X\in E_{F}$.

• •

The generalised RSL (gRSL) can be accomplished by the gRSL concept $(X,Y)_{\mbox{\tiny gRSL}}$ which satisfies $X^{\Box^{\ast}}=Y$ and $Y^{\Diamond^{\ast}}=X$, where $Y=\bigcup_{X_{0}\subseteq X}[X_{0}]_{F}\ \forall X\in E_{F}$.

• •

The complementary generalised RSL (cgRSL) can be furnished by the cgRSL concept $(X,Y)_{\mbox{\tiny cgRSL}}$ which satisfies $X^{\Diamond^{\ast}}=Y$ and $Y^{\Box^{\ast}}=X$, where $Y=M^{\ast}\backslash\bigcup_{X_{0}\subseteq G\backslash X}[X_{0}]_{F}\ \forall X\in E_{F}$.

Proof.

Note that the relations among the derivation operators $I,\Box$ and $\Diamond$ (Eq. (4)-(7) in Ref. LLJD12-1 ) are preserved under the substitutions $\left\{\begin{smallmatrix}I\rightarrow I^{\ast}\\ \Box\rightarrow\Box^{\ast}\\ \Diamond\rightarrow\Diamond^{\ast}\end{smallmatrix}\right.$ since $F^{\ast}(G,M^{\ast})$ can be regarded as a new formal context.

• •

Upon employing Eq. (9) of Ref. LLJD12-1 with the generalisation from $M$ to $M^{\ast}$,

[TABLE]

Accordingly, consider $Y=X^{I^{\ast}}=\bigcup_{X_{0}\supseteq X}[X_{0}]_{F}$, where

[TABLE]

namely, $Y^{I^{\ast}}=X$.

• •

Likewise, by Eq. (9) of Ref. LLJD12-1 ,

[TABLE]

Moreover, $Y^{\Diamond^{\ast}}=(\bigcup_{X_{0}\supseteq X}[X_{0}]_{F})^{\Diamond^{\ast}}=\bigcup_{X_{0}\subseteq X}([X_{0}]_{F})^{\Diamond^{\ast}}=\bigcup_{X_{0}\subseteq X}X_{0}=X$.

• •

As the generalisation of result in Ref. YY04 , if $(X,Y)$ satisfies $\left\{\begin{smallmatrix}X^{\Box^{\ast}}=Y\\ Y^{\Diamond^{\ast}}=X\end{smallmatrix}\right.$ then $\left\{\begin{smallmatrix}(X^{c})^{\Diamond^{\ast}}=Y^{c}\\ (Y^{c})^{\Box^{\ast}}=X^{c}\end{smallmatrix}\right.$, where apparently $\left\{\begin{smallmatrix}X^{c}=G\backslash X\\ Y^{c}=M^{\ast}\backslash Y\end{smallmatrix}\right.$ since $X\subseteq G$ and $Y\subseteq M^{\ast}$. Then, the expression $\forall X\ (X^{c},(\bigcup_{X_{0}\subseteq X}[X_{0}]_{F})^{c})_{\mbox{\tiny cgRSL}}$ can be used to construct a concept lattice since expressions in this form are equipped with well-defined partial order. Subsequently, because both $X$ and $X^{c}$ belong to $E_{F}$ and one prefers using the expression with respect to $X$, the cgRSL thus ranges over all the $X\in E_{F}$ with $(X,(\bigcup_{X_{0}\subseteq X^{c}}[X_{0}]_{F})^{c})_{\mbox{\tiny cgRSL}}\equiv(X,M^{\ast}\backslash(\bigcup_{X_{0}\subseteq G\backslash X}[X_{0}]_{F}))_{\mbox{\tiny cgRSL}}$.

∎

Certainly, on returning to the conventional scope based on $F(G,M)$, the gFCL will be restricted to the FCL, the gRSL will be restricted to the RSL, and the cgRSL will be restricted to the property-oriented RSL YY04 . Nevertheless, while both the gFCL and gRSL can be thought of as object-oriented, the cgRSL here cannot be regarded as a property-oriented lattice that generalises the property-oriented RSL.

Notably, apart from Proposition 2.2, still more concept lattices could have been generated from the GCL by means of the general intents. For instance, one could also consider $(X,Y)_{\mbox{new}}=(X,M^{\ast}\backslash(\bigcup_{X_{0}\supseteq X}[X_{0}]_{F}))\ \forall X\in E_{F}$, where $Y$ is the complementary set of the gFCL intent. However, for all those constructions it remains fundamental to have the intelligibility about why to relate the object class and its associating property in particular ways. It is based on such intelligibility that the analytics accompanied with lattice can be performed. In this regard, the gFCL deals with the necessary feature an object in any given class should exhibit and the gRSL the sufficient feature by which an object can be categorised into a definite class, while the artificial constructions like cgRSL or $(X,Y)_{\mbox{new}}$ could become less significant. Another point is that the lattices introduced in Proposition 2.2 are nested in the sense that the attributes in $M^{\ast}$ are used as intents in a repeated manner. Clearly, the fact that the intents overlap can render the analysis difficult. Nevertheless, both the gFCL and the gRSL can be regarded as a half of the GCL, whose intents are then disjoint (Proposition 2.1). For concreteness, if $(X,Y_{1})$ is a gFCL concept and $(X,Y_{2})$ is a gRSL concept then $(X,Y_{1}\cap Y_{2})$ is a general concept. The tractability problem for the GCL content will then be resolved as follows.

• •

For the lattice construction:

The general extents are known in advance subject to $F(G,M)$, which are the $2^{n_{F}}$ members in $E_{F}=\{\mu^{R}\mid\mu\in M^{\ast}\}=\sigma(\{m^{R}\mid m\in M\})$. The generalised attribute-set $M^{\ast}$ is distributed to the $2^{n_{F}}$ nodes as general intents, each of which is expressible in terms of a closed interval. Thus, one only needs the $2\times 2^{n_{F}}$ bounds (Proposition 2.1) for fixing down the general intents, which further reduces to $2^{n_{F}}$ attributes since $\forall X\in E_{F}\ \eta(X^{c})=\neg\rho(X)$ and $X\in E_{F}\iff X^{c}\in E_{F}$ (Proposition 3.7 in Ref. LLJD12-1 ). Basically, these $2^{n_{F}}$ attributes can be determined using a fundamental $n_{F}$ attribute construction (Proposition 3.9, 3.11 and 3.12; Corollary 3.13 in Ref. LLJD12-1 ). However, it will be further shown in the coming section that they can be directly read out from the formal context; the practical construction for GCL is as tractable as listing out the formal context.

• •

For the implication relations:

The GCL supports the logic deduction by characterising any object class of interest in terms of non-overlapping general intent, whose upper and lower bounds corresponds to the sufficient and necessary properties of the given object class, respectively. To be concrete, attributes grouped into the same general intent are logically equivalent since they correspond to the property of the same object class. One will show in Sec. 4 that a unique formula based on the contextual truth $1_{\eta}$ or falsity $0_{\rho}$ (Proposition 3.12 in Ref. LLJD12-1 ) suffices to generate all such logic implications established between any attribute pairs in $M^{\ast}$.

3 lattice construction per read out

To proceed with the construction of the general concept (Proposition 2.1)

[TABLE]

let $X=\bigcup_{D_{k}\subseteq X}D_{k}$ and $X=\bigcap_{G\backslash D_{k}\supseteq X}G\backslash D_{k}$ for any general extent, where $D_{k}$’s are the smallest object sets discernible by the formal context (Definition 2.4, Proposition 3.5 in Ref. LLJD12-1 ). After Proposition 3.7 and 3.9 of Ref. LLJD12-1 , one may end up with the relations

[TABLE]

where $X^{+}$ ($X^{\times}$) is referred to as an $X$-irreducible disjunction (conjunction) (Definition 2.14, Corollary 3.13 in Ref. LLJD12-1 ). In addition, $\rho(\emptyset)=0_{\rho}$ and $\eta(G)=1_{\eta}$ since $\rho(\emptyset)$ plays the role of falsity for all the upper bounds and $\eta(G)$ plays the role of truth for all the lower bounds (Proposition 3.12 in Ref. LLJD12-1 ). As an example, Fig. 2 illustrates how ${\rho_{0}(D_{k})}$ and ${\eta_{0}(G\backslash D_{k})}$ are determined from the irreducible expressions given in Eq. (5). Specifically, in the same manner,

[TABLE]

which conjuncts all the irreducible disjunctions of the attributes in $\breve{M}=M\cup\{\neg m\mid m\in M\}$. Notably,

[TABLE]

where, e.g., $(b+c+\neg d)\in[G^{+}]_{F}$ is irreducible because $(b+c+\neg d)^{R}=G$ but none of $\left\{(b+c)^{R},(c+\neg d)^{R},(b+\neg d)^{R}\right\}$ can be identified with $G$.

Note that the approach based on Eq. (5) in general renders $\rho(X)$ in DNF and $\eta(X)$ in CNF, which is instructive for the generality of GCL in relation to the other lattices since one has in effect obtained $\rho(X)$ in the style of RSL and $\eta(X)$ in the style of FCL. However, such an approach is rather tedious thus should not be recommeded in actual practice. Now one proceeds to show that adopting $\eta(X)$ in DNF and $\rho(X)$ in CNF will provide a simpler construction which potentially leads to the full determination of general concepts per read out.

Proposition 3.1**.**

Subject to a formal context $F(G,M)$, $\eta(G)\equiv 1_{\eta}=\sum_{k=1}^{n_{F}}\eta(D_{k})$. In effect, $1_{\eta}$ is obtained by summing up all the lower bounds of intents corresponding to disjoint regions on the contextual Venn diagram, cf. Fig. 1.

Proof.

$\forall k\in[1,n_{F}]\ \eta(G)>\eta(D_{k})$ by Proposition 3.10, 3.14 in Ref. LLJD12-1 $\therefore\eta(G)\geq\sum_{k=1}^{n_{F}}\eta(D_{k})$. On the other hand, $(\sum_{k=1}^{n_{F}}\eta(D_{k}))^{R}=\bigcup_{k=1}^{n_{F}}\eta(D_{k})^{R}=G\ \therefore\sum_{k=1}^{n_{F}}\eta(D_{k})\in[G]_{F}$. Hence, $\eta(G)\leq\sum_{k=1}^{n_{F}}\eta(D_{k})$ since $\eta(G)=\prod[G]_{F}$ is the lower bound for $[G]_{F}$ (Proposition 2.1).∎

Since $\eta(D_{k})=\prod\Psi^{k}$, where $\Psi^{k}=\{m\in M\mid m\in D_{k}^{I}\}\cup\{\neg m\mid m\not\in D_{k}^{I},m\in M\}$ by Proposition 3.10 in Ref. LLJD12-1 , Eq. (6) can also be computed as

[TABLE]

Lemma 3.2**.**

Given a formal context $F(G,M)$, it can be shown that

• •

$\eta(G\backslash D_{k})\equiv\eta_{0}(G\backslash D_{k})\cdot 1_{\eta}=\sum_{j\neq k}\eta(D_{j})$,

• •

$\rho(D_{k})\equiv\rho_{0}(D_{k})+0_{\rho}=\prod_{j\neq k}\rho(G\backslash D_{j})$.

Proof.

The fact that $\eta(G\backslash D_{k})=\eta_{0}(G\backslash D_{k})\cdot 1_{\eta}$ and $\rho(D_{k})=\rho_{0}(D_{k})+0_{\rho}$ is the direct consequence of Eq. (5).

• •

$\eta(G\backslash D_{k})\geq\sum_{j\neq k}\eta(D_{j})$ since $\eta(G\backslash D_{k})\geq\eta(D_{j})$ by $G\backslash D_{k}\supseteq\eta(D_{j})$ $\forall j\neq k$ due to Proposition 3.14 in Ref. LLJD12-1 . On the other hand, $\eta(G\backslash D_{k})\leq\mu\ \forall\mu\in[G\backslash D_{k}]_{F}$, implying that $\eta(G\backslash D_{k})\leq\sum_{j\neq k}\eta(D_{j})$ by $(\sum_{j\neq k}\eta(D_{j}))^{R}=G\backslash D_{k}$. Therefore, $\eta(G\backslash D_{k})=\sum_{j\neq k}\eta(D_{j})$.

• •

By the conjugateness of Proposition 3.7 in Ref. LLJD12-1 , $\rho(D_{k})=\neg\eta(G\backslash D_{k})=\neg\sum_{j\neq k}\eta(D_{j})$. It follows then $\neg\sum_{j\neq k}\eta(D_{j})=\prod_{j\neq k}\neg\eta(D_{j})=\prod_{j\neq k}\rho(G\backslash D_{j})$.

∎

Let $D_{1}=\{1\}$, $D_{2}=\{2\}$, $D_{3}=\{3,4\}$, $D_{4}=\{5\}$ and $D_{5}=\{6\}$. In Fig. 2, it can be checked that

[TABLE]

Proposition 3.3**.**

Given a formal context $F(G,M)$, one may express the general extent in terms of $X=\bigcup_{k\in K}D_{k}\equiv\bigcap_{k\in K_{0}\backslash K}G\backslash D_{k}$ for some $K\subseteq K_{0}$, where the index set $K_{0}:=\{1,\cdots,\ n_{F}\}$. The upper and lower bounds for the corresponding general intents are as follows.

• •

$\eta(X)=\sum_{k\in K}\eta(D_{k})$* and $\rho(X)=\prod_{k\in K_{0}\backslash K}\rho(D_{k}^{c})$, which is in contrast to Proposition 3.9 of Ref. LLJD12-1 that can be restated as $\rho(X)=\sum_{k\in K}\rho(D_{k})$ and $\eta(X)=\prod_{k\in K_{0}\backslash K}\eta(D_{k}^{c})$.*

• •

\left\{\begin{array}[]{ccc}\eta(\bigcup_{i}X_{i})=\sum_{i}\eta(X_{i}),&&\rho(\bigcap_{j}X_{j})=\prod_{j}\rho(X_{j})\\ \rho(\bigcup_{i}X_{i})=\sum_{i}\rho(X_{i}),&&\eta(\bigcap_{j}X_{j})=\prod_{j}\eta(X_{j})\end{array}\right.* in which $X_{i},X_{j}\in E_{F}$.*

Proof.

• •

For $X\in E_{F}$, $X=\bigcup_{D_{k}\subseteq X}D_{k}=\bigcap_{D_{k}^{c}\supseteq X}G\backslash D_{k}$, as is given in Proposition 3.5 of Ref. LLJD12-1 , can be rewritten as $X=\bigcup_{k\in K}D_{k}=\bigcap_{k\in K_{0}\backslash K}G\backslash D_{k}$ for some $K\subseteq K_{0}$, where $G=\bigcup_{k\in K_{0}}D_{k}$. Hence,

[TABLE]

where the first equality is based on Proposition 3.9 in Ref. LLJD12-1 , the second one makes use of ** Lemma 3.2** and the third one is due to the fact that $\eta(D_{k})\cdot\eta(D_{k^{\prime}})=0$ for $k\neq k^{\prime}$. Moreover, since $\eta(X)=\sum_{k\in K}\eta(D_{k})$, one has $\eta(X^{c})=\eta(\bigcap_{k\in K}D_{k}^{c})=\sum_{j\in K_{0}\backslash K}\eta(D_{j})$. Therefore, by conjugateness (Proposition 3.7 in Ref. LLJD12-1 ),

[TABLE]

where $\neg\eta(D_{k})=\rho(D_{k}^{c})$.

• •

Let $\forall X_{l}\in E_{F}$ $X_{l}=\bigcup_{k\in K^{l}}D_{k}=\bigcap_{k\in K_{0}\backslash K^{l}}G\backslash D_{k}$, where $K^{l}\subseteq K_{0}$. Thus,

[TABLE]

where the above results as well as Proposition 3.9 of Ref. LLJD12-1 are recursively utilised.

∎

Another significant point is about the decompositon of freedom incorporated in $M^{\ast}$. If the attributes in $M$ are independent (not exclusive) then any attribute pairs in $M$ intersect non-trivially, giving rise to $2^{|M|}$ disjoint regions on the Venn diagram ${\cal V}_{M}^{0}$. $|M^{\ast}|$ reaches the maximum value $2^{2^{|M|}}$ because any of the generalised attributes corresponds to a disjunction of some of the $2^{|M|}$ disjoint regions on ${\cal V}_{M}^{0}$. In general, the members in $M$ may be intrinsically related such that $r(\geq 0)$ of the disjoint regions vanish on the Venn diagram ${\cal V}_{M}$, thus, $|M^{\ast}|=2^{2^{|M|}}/2^{r}\leq 2^{2^{|M|}}$. Notably, disjoint regions on the Venn diagram correspond to the $M^{\ast}$-atoms collected in $b_{inf}(M^{\ast})=\{\mu\in M^{\ast}\mid\mu\succ_{M}{\bf 0}\}$ (Lemma 2.12, Corollary 2.13 in Ref. LLJD12-1 ). One may henceforth adopt $e_{k}\in b_{inf}(M^{\ast})$, where $e_{k}\cdot e_{k^{\prime}}\stackrel{{\scriptstyle\scriptscriptstyle k\neq k^{\prime}}}{{=}}{\bf 0}$, as a convention for basis:

[TABLE]

Note that $rank(M)$ is a particular integer describing the freedom in $M^{\ast}$ subject to the intrinsic constraint held among members in $M$ and $e_{k}\equiv e_{k}(M)$ is to mark that $e_{k}$ is also constructed out of $M$. However, it turns out formally

[TABLE]

highlighting that the set $M$ which gives rise to the generalised attribute set $M^{\ast}$ with $rank(M^{\ast})=2^{|M|}-r$ need not be unique. Indeed, there are abundant choices of $M^{\ast}$ fulfilling such a requirement, which will furnish the framework for analysing the reparametisation of the formal context in the next paper. Here, an immediate example appears to be the choice $M_{0}\equiv b_{inf}(M^{\ast})$ such that $M_{0}^{\ast}=M^{\ast}$ whenever $M\neq b_{inf}(M^{\ast})$. Certainly, $M_{0}$ then manifests a intrinsic constraint in the sense that it only consists mutually exclusive members.

Turning back to the lattice construction, a primary concern is about how the GCL presents itself.

Definition 3.4**.**

The GCL subject to a formal context will be henceforth referred to as $\Gamma_{F}(G,M)$, cf. Poposition 3.16 in Ref. LLJD12-1 , which is uniquely prescribed by either of

• •

the $\eta$-representation ${\Upsilon}_{\eta}^{F}(G,M):=[\eta(D_{1}),\ldots,\eta(D_{n_{F}})]\equiv[\eta_{1},\ldots,\eta_{n_{F}}]$,

• •

the $\rho$-representation ${\Upsilon}_{\rho}^{F}(G,M):=[\rho(G\backslash D_{1}),\ldots,\rho(G\backslash D_{n_{F}})]\equiv[\neg\eta_{1},\ldots,\neg\eta_{n_{F}}]$,

where $n_{F}$ denotes the number of subclasses discernible from the point of view of $F(G,M)$.

According to Proposition 3.1, obtaining the $\eta$-representation or the $\rho$-representation is a simple one-scan task, on the formal context. For the $F(G,M)$ in Table 1, it is straightforward to write down

[TABLE]

Note that Proposition 3.3 in fact provides simple identifications for the whole GCL structure. For convenience, assume that it is the first $n_{F}$ $M^{\ast}$-atoms in the convention of Eq. (8) which enter $\eta(X)$ as constituents:

[TABLE]

where $\Psi^{k}=\{m\in M\mid m\in D_{k}^{I}\}\cup\{\neg m\mid m\not\in D_{k}^{I},m\in M\}$ (cf. Eq. (7)). Then, $\eta(X)=\eta(\bigcup_{k\in K}D_{k})=\sum_{k\in K}\eta_{k}$, which picks up a subset $K\subseteq K_{0}$ from the expression $1_{\eta}=\sum_{k\in K_{0}}\eta_{k}$ (Proposition 3.1). Likewise, $\rho(\bigcap_{k\in K_{0}\backslash K}G\backslash D_{k})=\prod_{k\in K_{0}\backslash K}\neg\eta_{k}$ picks up all the components which are not in $K$ from the expression $0_{\rho}=\prod_{k\in K_{0}}\neg\eta_{k}$ since $\rho(G\backslash D_{k})=\neg\eta_{k}\ \forall k\in K$. It is then straightforward to extract the general extents and intents via $n_{F}$-bit binary masks from a known GCL structure as follows. Let $B_{X}$ be the binary string whose $k$th bit $B_{X}^{k}$ is given as $\left\{\begin{smallmatrix}B_{X}^{k}=1\ \mbox{if}\ D_{k}\subseteq X\\ B_{X}^{k}=0\ \mbox{if}\ D_{k}\not\subseteq X\end{smallmatrix}\right.$. Then, for any $X\in E_{F}$ there is a binary string $B_{X}$ marking the $D_{k}$’s that $X$ contains; any of $n_{F}$-bit binary strings corresponds to a definite general extent. One may thus write down that

[TABLE]

Thus, one has $B_{G}=$“11111” which implies that $\eta(G)=[11111]_{\eta}=1_{\eta}$ in the above example, where $1_{\eta}=\sum{\Upsilon}_{\eta}^{F}$ sums up all the components of the $\eta$-representation of GCL (Definition 3.4). Likewise, $B_{\emptyset}$=“00000” tells that $\rho(\emptyset)=[00000]_{\rho}=0_{\rho}$ where $0_{\rho}=\neg 1_{\eta}=\prod{\Upsilon}_{\rho}^{F}$ is the product of all the components of the $\rho$-representation. Let one also consider that $\rho(X)=\prod_{k}\neg e_{k}\equiv\prod_{k}\neg e_{k}\cdot{\bf 1}$ and $\eta(X)=\sum_{k}e_{k}\equiv\sum_{k}e_{k}+{\bf 0}$. In such a manner, $\rho(G)=[11111]_{\rho}=\bf{1}$ because one picks up no term in $\{\neg e_{1},\ldots,\neg e_{5}\}$ due to the absence of 0 in $B_{G}$. Similarly, $\eta(\emptyset)=[00000]_{\eta}=\bf{0}$ due to the absence of 1 in $B_{\emptyset}$.

It is also particularly interesting to rediscover the FCL and RSL within a known GCL structure (Fig. 3). After Proposition 3.15 in Ref. LLJD12-1 , two steps are in order. Firstly, re-express the given $\eta(X)\ \mbox{in {\bf CNF}}$ and $\rho(X)\ \mbox{in {\bf DNF}}$. Then, one may collect from these expressions those attributes belonging to $M$ to form a candidate FCL intent $Y_{c-fcl}(X)$ and a candidate RSL intent $Y_{c-rsl}(X)$, respectively. Secondly, if $(Y_{c-fcl}(X))^{I}=X$ then $(X,Y_{c-fcl}(X))$ is accepted as an FCL concept, whereas if $(Y_{c-rsl}(X))^{\Diamond}=X$ then $(X,Y_{c-rsl}(X))$ is accepted as an RSL concept. Note that such constructions could be less intuitive for particular nodes. For instance, at $X=\emptyset$, one should imagine $\eta(\emptyset)={\bf 0}=a\cdot b\cdot c\cdot d\cdot e\cdot\ldots$ (Step 1) so one ends up with the FCL concept $(\emptyset,\{a,b,c,d,e\})$, based on $a^{R}\cap b^{R}\cap c^{R}\cap d^{R}\cap e^{R}=\emptyset$ (Step 2). Similarly, $\rho(G)={\bf 1}=a+b+c+d+e+\ldots$ and $a^{R}\cup b^{R}\cup c^{R}\cup d^{R}\cup e^{R}=\{1,2,3,4,5,6\}=G$ imply that $(\{1,2,3,4,5,6\},\{a,b,c,d,e\})$ is a RSL concept. It seems by Proposition 3.15 of Ref. LLJD12-1 that

[TABLE]

provide all the FCL and RSL extents. However, conventionally, one also regards the object class $G$ as an FCL extent and $\emptyset$ as an RSL extent, while, as object classes, $G\not\in E_{F}^{fcl}$ and $\emptyset\not\in E_{F}^{rsl}$. The point is that the FCL concept corresponding to $G$ do not always satisfy the condition $\left\{\begin{smallmatrix}X^{I}=Y\\ Y^{I}=X\end{smallmatrix}\right.$ and the RSL concept corresponding to $\emptyset$ do not always satisfy $\left\{\begin{smallmatrix}X^{\Box}=Y\\ Y^{\Diamond}=X\end{smallmatrix}\right.$. In effect, e.g. in Fig. 3, $(G,\emptyset)$ is determined via $(\bigcup_{X\in E_{F}^{fcl}}X,\bigcap_{X\in E_{F}^{fcl}}Y(X))$ and $(\emptyset,\emptyset)$ is via $(\bigcap_{X\in E_{F}^{rsl}}X,\bigcap_{X\in E_{F}^{rsl}}Y(X))$, which are artificially appended as the lattice supremum and infimum respectively, for the sake of the completeness of lattices Wi82 ; YY04 . It should however be clear that there is no need of such artificial completions for the gFCL and gRSL (Proposition 2.2). The conditions $\left\{\begin{smallmatrix}X^{I^{\ast}}=Y\\ Y^{I^{\ast}}=X\end{smallmatrix}\right.$ and $\left\{\begin{smallmatrix}X^{\Box^{\ast}}=Y\\ Y^{\Diamond^{\ast}}=X\end{smallmatrix}\right.$ are well defined because these conditions are all based on the general intents which are never empty sets. Indeed, the cardinality of general intent even remains constant over $E_{F}$, as will be demonstrated in Corollary 4.4.

4 implication relations

It is the object-attribute relationship resulted from the categorisation that leads to the logic significance implemented by the GCL structure. Roughly speaking, one’s inspections of attributes are essentially restricted in a definite object domain thus the attributes receive additional ordering prescriptions, which are the origins of the logic implication in GCL. Here, as a primary observation, the attributes that play the roles of the bounds of general intents are equipped with particular features:

[TABLE]

which is a direct consequence of Proposition 3.3. In what follows, by matching any property of a given object with a GCL one can determine its class belongingness in the GCL structure.

Proposition 4.1**.**

Given a GCL subject to the formal context $F(G,M)$, $\forall\mu\in M^{\ast}\ (\mu^{R},[\mu\cdot 1_{\eta},\mu+0_{\rho}])$ is a general concept, cf. Proposition 2.1 and Eq. (4), every general concept of the GCL can be unambiguously represented by $(\mu^{R},[\mu^{R}]_{F})$ with some $\mu\in M^{\ast}$.

Proof.

Equation (11) implies that $\forall\mu$, $\mu\cdot 1_{\eta}$ is a lower bound of some general intent, i.e. $\eta(X)$ with some $X\in E_{F}$, because $\mu\cdot 1_{\eta}\leq 1_{\eta}$. Likewise, $\mu+0_{\rho}$ can be identified as the upper bound $\rho(X^{\prime})$ with some $X^{\prime}\in E_{F}$. Moreover, $X^{\prime}=(\mu+0_{\rho})^{R}=\mu^{R}=(\mu\cdot 1_{\eta})^{R}=X$. Therefore, $(\mu\cdot 1_{\eta})=\eta(\mu^{R})$ and $(\mu+0_{\rho})=\rho(\mu^{R})$ implies that $(\mu^{R},[\mu\cdot 1_{\eta},\mu+0_{\rho}])$ is always a general concept. On the other hand, for any given $X\in E_{F}$, any $\mu\in[X]_{F}$ will render $(\mu^{R},[\mu^{R}]_{F})$ a general concept, as is demonstrated above. This is not ambiguous since general intents are disjoint, namely, $\forall X_{i}\forall X_{j}\in E_{F}\ [X_{i}]_{F}\cap[X_{j}]_{F}=\emptyset\iff X_{i}\neq X_{j}$ (Proposition 2.1).∎

Corollary 4.2**.**

Given a formal context $F(G,M)$, $\forall X\in E_{F}$,

• •

$\rho(X)=\eta(X)+0_{\rho}$* and $\eta(X)=\rho(X)\cdot 1_{\eta}$,*

• •

$\rho(X)\cdot\neg\eta(X)=0_{\rho}$* and $\neg\rho(X)+\eta(X)=1_{\eta}$.*

Proof.

• •

According to Proposition 4.1, substituting $\mu=\eta(X)$ and $\mu=\rho(X)$ into the expression $(\mu^{R},[\mu\cdot 1_{\eta},\mu+0_{\rho}])$ must lead to the same general concept since $\rho(X)^{R}=\eta(X)^{R}=X$. Consequently, $(X,[\eta(X),\eta(X)+0_{\rho}])=(X,[\rho(X)\cdot 1_{\eta},\rho(X)])$ implies that $\eta(X)+0_{\rho}=\rho(X)$ and $\eta(X)=\rho(X)\cdot 1_{\eta}$.

• •

$\eta(X)$ and $\rho(X)$ can be respectively identified as $\mu\cdot 1_{\eta}$ and $\mu+0_{\rho}$ for some $\mu\in M^{\ast}$. Therefore, $\rho(X)\cdot\neg\eta(X)=(\mu+0_{\rho})\cdot\neg(\mu\cdot 1_{\eta})=(\mu+0_{\rho})\cdot(\neg\mu+0_{\rho})=0_{\rho}$. On the other hand, $\neg\rho(X)+\eta(X)=\neg(\rho(X)\cdot\neg\eta(X))=1_{\eta}$.

∎

Thus, $0_{\rho}$ and $1_{\eta}$ can furnish the connection between the upper and lower bounds for any $[X]_{F}$. In addition, $\rho(X)\cdot\neg\eta(X)$ and $\neg\rho(X)+\eta(X)$ are in fact constants over the full GCL.

Corollary 4.3**.**

The general intent $[X]_{F}$ is the equivalent class of attributes generated from the pair ($\rho(X),\eta(X)$):

[TABLE]

Proof.

For all $\mu_{1}\in\{\mu\in M^{\ast}\mid\mu+0_{\rho}=\rho(X),\ \mu\cdot 1_{\eta}=\eta(X)\}$ one has the general concept $(\mu_{1}^{R},\rho(\mu_{1}^{R}),\eta(\mu_{1}^{R}))$ in which $\mu_{1}^{R}=X$, i.e. $\mu_{1}\in[X]_{F}$ (Proposition 4.1). Thus, $\mu_{1}\in\{\mu\in M^{\ast}\mid\mu+0_{\rho}=\rho(X),\ \mu\cdot 1_{\eta}=\eta(X)\}\implies\mu_{1}\in[X]_{F}$. On the other hand, if $\mu_{1}\in\{\mu\in M^{\ast}\mid\mu+0_{\rho}=\rho(X),\mu\cdot 1_{\eta}=\eta(X)\}$ then $\mu_{1}^{R}=X$.

[TABLE]

Therefore, $\forall\mu_{1}\in M^{\ast},\ \mu_{1}\in[X]_{F}\iff\mu_{1}\in\{\mu\in M^{\ast}\mid\mu+0_{\rho}=\rho(X),\ \mu\cdot 1_{\eta}=\eta(X)\}$.∎

It is also notworthy that by the $n_{F}$ general intents one in fact exhausts the whole generalised attribute set $M^{\ast}$ in view of $2^{(\log_{2}^{|M^{\ast}|}-n_{F})}\times 2^{n_{F}}=|M^{\ast}|$.

Corollary 4.4**.**

Given a collection of general intents, say $\left\{[X_{1}]_{F},[X_{2}]_{F},\ldots[X_{n}]_{F}\right\}$,

• •

if $\mu_{i}\in[X_{i}]_{F}$ for $1\leq i\leq n$ then $\sum_{i=1}^{n}\mu_{i}\in\left[\bigcup_{i=1}^{n}X_{i}\right]_{F}$ as well as $\prod_{i=1}^{n}\mu_{i}\in\left[\bigcap_{i=1}^{n}X_{i}\right]_{F}$,

• •

all the general intents for the general concept lattice are of the same cardinality: $|[X]_{F}|=2^{(\log_{2}^{|M^{\ast}|}-n_{F})}\ \forall X\in E_{F}$ subject to the formal context $F(G,M)$.

Proof.

• •

Since $\mu_{i}\in[X_{i}]_{F}$, one has $\mu_{i}=\eta(X_{i})+\tau_{i}$ with some $\tau_{i}\in[{\bf 0},0_{\rho}]$ by Proposition 2.1. Therefore,

[TABLE]

by Proposition 3.3 since $\tau=\sum_{i=1}^{n}\tau_{i}\in[{\bf 0},0_{\rho}]$ and $\tau^{\prime}=\prod_{i=1}^{n}\tau_{i}\in[{\bf 0},0_{\rho}]$.

• •

$\rho(X)=\eta(X)+0_{\rho}\ \forall X\in E_{F}$ (Corollary 4.2). Hence, any $\mu\in[X]_{F}\equiv[\eta(X),\eta(X)+0_{\rho}]$ can be written as $\mu=\eta(X)+\tau$ with ${\bf 0}\leq\tau\leq 0_{\rho}$. Upon employing the convention of Eq. (8) and (9), one may identify $0_{\rho}=\sum_{k=n_{F}+1}^{\log_{2}^{|M^{\ast}|}}e_{k}$ since $0_{\rho}\cdot 1_{\eta}={\bf 0}$ and $0_{\rho}+1_{\eta}={\bf 1}$. Subsequently, any $\tau\in[{\bf 0},0_{\rho}]$ can be written as $\tau=\sum_{k\in Q}e_{k}$ for some $Q\subseteq K^{\ast}\backslash K_{0}=\{n_{F}+1,\ldots,\log_{2}^{|M^{\ast}|}\}$. Therefore, $|[X]_{F}|=|[{\bf 0},0_{\rho}]|=|2^{K^{\ast}\backslash K_{0}}|=2^{(\log_{2}^{|M^{\ast}|}-n_{F})}$.

∎

According to Proposition 4.1, an object set $X=\mu^{R}$ for which $\mu\in[X]_{F}$ can be acquired by $\mu+0_{\rho}=[B_{X}]_{\rho}$ or $\mu\cdot 1_{\eta}=[B_{X}]_{\eta}$, see Eq. (10). For instance, consider $\mu=a\cdot\neg e+c$ in Fig. 3. Then,

[TABLE]

both imply $B_{X}=$“11010”, hence $X=\{1,2,5\}$, which could also be deduced from $\mu^{R}=a^{R}\cap e^{Rc}\cup c^{R}=\{1,2,5\}$. This is the logic foundation provided by inspecting $\mu$ for its object partial ordering from the object-attribute relationship embedded in the GCL structure. However, the conventional interest of rules of implication are attribute based, where the object reference is implicit. Thus, one’s primary concern here is the so-called implication informative above the GCL framework, which was discussed originally in the traditional FCL framework GD86 .

Note that for FCL an implication relation $A\rightarrow B$ is considered between the attribute sets $A\subseteq M$ and $B\subseteq M$. Since the FCL in effect deals with the conjunctions of simple attributes in $M$, see Lemma 3.1 in Ref. LLJD12-1 , every FCL rule receives a corresponding rule in the GCL:

[TABLE]

Therefore, one is interested in the implication $\mu_{1}\rightarrow\mu_{2}$ which relates between the generalised attributes $\mu_{1}\in M^{\ast}$ and $\mu_{2}\in M^{\ast}$ in the GCL theory. In particular, according to Ref. GD86 , $A\stackrel{{\scriptstyle fcl}}{{\rightarrow}}B$ is a tautology (i.e. not informative) if $A\supseteq B$, which means $\mu_{1}=\prod A\leq\prod B=\mu_{2}$. Hence, if $\mu_{1}\not\leq\mu_{2}$ then the GCL rule $\mu_{1}\rightarrow\mu_{2}$ is informative.

Definition 4.5**.**

Consider the implication statement $\mu_{1}\rightarrow\mu_{2}$ ($\mu_{1}$ implies $\mu_{2}$), where $\mu_{1}\in M^{\ast}$ and $\mu_{2}\in M^{\ast}$.

• •

If $\mu_{1}>\mu_{2}$, $\mu_{1}\rightarrow\mu_{2}$ is referred to as a rule of purely informative implication (RPII).

• •

If $\mu_{1}\not\leq\mu_{2}$, $\mu_{1}\rightarrow\mu_{2}$ is referred to as a rule of informative implication (RII).

• •

If $\mu_{1}\rightarrow\mu_{2}$ is manifestly true ($\mu_{1}\leq\mu_{2}$) then it is referred to as a tautology (TT), in the sense that the implication tells nothing new, and is re-denoted as $\mu_{1}\implies\mu_{2}$.

Note that RPII is a particular class of RII’s in which no TT is involved.

Lemma 4.6**.**

For the GCL subject to the formal context $F(G,M)$, the rules of implication between two attributes in $M^{\ast}$ are well defined. Explicitly, $\forall\mu_{1}\forall\mu_{2}\in M^{\ast}$, $\mu_{1}\rightarrow\mu_{2}$ can be identified as $\mu_{1}^{R}\subseteq\mu_{2}^{R}$: $\mu_{1}\leftrightarrow\mu_{2}$ ($\mu_{1}^{R}=\mu_{2}^{R}$) is referred to as the T1 rule; $\mu_{1}\rightarrow\mu_{2}$ ($\mu_{1}^{R}\subset\mu_{2}^{R}$) is referred to as the T2 rule.

• •

One may deduce T2 from T1 by TT.

• •

Knowing all the RPII’s suffices the full characterisation of RII’s (Definition 4.5).

Proof.

Similar to the idea of $A\rightarrow A^{II}$ in Ref. GD86 , the rule of implication $\mu_{1}\rightarrow\mu_{2}$ means that the object class possessing $\mu_{1}$ must also be equipped with $\mu_{2}$. In other words, the object class possessing $\mu_{1}$ is included in the object class possessing $\mu_{2}$, hence, $\mu_{1}^{R}\subseteq\mu_{2}^{R}$. Note that the T1 rule is a bi-implication. With $\mu_{1}^{R}=\mu_{2}^{R}:=X$, both $\mu_{1}$ and $\mu_{2}$ are in $[X]_{F}$, they belong to the common property of the same object class and are thus regarded as equivalent. For T2, let $X_{1}=\mu_{1}^{R}$ and $X_{2}=\mu_{2}^{R}$, where $X_{1}\subset X_{2}$. Thus, $(\mu_{1}\cdot\mu_{2})^{R}=X_{1}\cap X_{2}=X_{1}$, and $\mu_{1}\cdot\mu_{2}\in[X_{1}]_{F}$. Accordingly, $gR\mu_{1}\implies gR(\mu_{1}\cdot\mu_{2})$ entails that the object possessing $\mu_{1}$ must also possess $\mu_{2}$.

• •

This is to show that $\mu_{1}^{R}\subset\mu_{2}^{R}$ implies $\mu_{1}\rightarrow\mu_{2}$ based on T1.

Since $\mu^{R}\in E_{F}\ \forall\mu\in M^{\ast}$, if $\mu_{1}^{R}\subset\mu_{2}^{R}$ then $\eta(\mu_{1}^{R})<\eta(\mu_{2}^{R})$ by Proposition 2.1 (also cf. Lemma 3.14 of Ref. LLJD12-1 ), thus, $\eta(\mu_{1}^{R})\implies\eta(\mu_{2}^{R})$. In addition, with $\mu_{1}^{R}\subset\mu_{2}^{R}$ two T1 rules state that $\left\{\begin{smallmatrix}\mu_{1}\leftrightarrow\eta(\mu_{1}^{R})\\ \mu_{2}\leftrightarrow\eta(\mu_{2}^{R})\end{smallmatrix}\right.$. Therefore, based on T1, $\mu_{1}^{R}\subset\mu_{2}^{R}$ implies $\mu_{1}\rightarrow\mu_{2}$ (T2).

• •

Any rule $\mu_{1}\rightarrow\mu_{2}$ where $\mu_{2}$ and $\mu_{1}$ are not ordered is a RII. However, $\mu_{1}\rightarrow\mu_{2}$ is logically equivalent to $\mu_{1}\rightarrow\mu_{1}\cdot\mu_{2}$ because one may invoke the tautology $\mu_{1}\cdot\mu_{2}\implies\mu_{2}$ so as to recover $\mu_{1}\rightarrow\mu_{2}$. Apparently, $\mu_{1}\rightarrow\mu_{1}\cdot\mu_{2}$ is an RPII since $\mu_{1}\cdot\mu_{2}<\mu_{1}$. Therefore, knowing all the RPII’s suffices the full characterisation of RII’s.

∎

While T1 works in two directions, it is not always the case that both implications are informative. For instance, if $\mu_{1}\rightarrow\mu_{2}$ is an RPII then $\mu_{1}\leftarrow\mu_{2}$ is a TT since $\mu_{1}>\mu_{2}$. Likewise, $\mu_{1}\rightarrow\mu_{2}$ of the type T2 needs not be a TT because it is not necessarily $\mu_{1}<\mu_{2}$ although $\mu_{1}^{R}\subset\mu_{2}^{R}$. Moreover, none of T2 rules can be RPII since tautology has been involved in deducing them from the T1 rules.

Proposition 4.7**.**

All the rules of implication can be determined in the following sense.

• •

$\forall\mu\in M^{\ast}$, any RII with respect to $\mu$ can be deduced from $\mu\rightarrow\mu\cdot 1_{\eta}$ by TT.

• •

$\forall\mu\in M^{\ast}\ \mu\rightarrow\mu\cdot 1_{\eta}$* is equivalent to $\forall\mu\in M^{\ast}\ \mu+0_{\rho}\rightarrow\mu$ when implementing the rules of implication.*

Proof.

Since knowing RPII suffices the full characterization of RII and all the rules of implication can be deduced from the type T1 (Lemma 4.6), one is only interested in the T1/RPII $\forall\mu\in M^{\ast}$ written as

$\mu\rightarrow\nu_{1}\ \mbox{for}\ \nu_{1}\in[\eta(\mu^{R}),\mu)$ and $\nu_{2}\rightarrow\mu\ \mbox{for}\ \nu_{2}\in(\mu,\rho(\mu^{R})]$.

• •

By Proposition 4.1, $\mu\rightarrow\mu\cdot 1_{\eta}$ is equivalently to $\mu\rightarrow\eta(\mu^{R})$. Since $\eta(\mu^{R})$ is the lower bound of the interval $[\eta(\mu^{R}),\mu)$, the consequence of the disjunction-introduction $\left(\mu\rightarrow\eta(\mu^{R})\right)\implies\left(\forall\tau\in M^{\ast}\ (\mu\rightarrow\eta(\mu^{R})+\tau)\right)$ can exhaust all the possibilities of $\mu\rightarrow\nu_{1}$ for $\nu_{1}\in[\eta(\mu^{R}),\mu)$. On the other hand, $\nu_{2}\cdot 1_{\eta}=\eta(\mu^{R})$ for $\nu_{2}\in(\mu,\rho(\mu^{R})]\subseteq[\mu^{R}]_{F}$, rendering $\nu_{2}\rightarrow\eta(\mu^{R})$. Therefore, $\nu_{2}\rightarrow\mu$ for $\nu_{2}\in(\mu,\rho(\mu^{R})]$ since $\eta(\mu^{R})\implies\mu$.

• •

$(\forall\mu\in M^{\ast}\ \mu+0_{\rho}\rightarrow\mu)\iff(\forall\neg\mu\in M^{\ast}\ \neg\mu\rightarrow\neg(\mu+0_{\rho}))\iff(\forall\neg\mu\in M^{\ast}\ \neg\mu\rightarrow\neg\mu\cdot 1_{\eta})$

$\iff(\forall\mu\in M^{\ast}\ \mu\rightarrow\mu\cdot 1_{\eta})$.

∎

Therefore, by applying the formula $\mu\rightarrow\mu\cdot 1_{\eta}$ one is able to deduce all the logic implications implemented by the GCL. Moreover, two additional points remain crucial in concluding that the result at hand in fact avoids the tractability problem for implementing the logic content GW99 ; GD86 ; Kso04 ; KO08 ; Sb09 ; Df10 ; DS11 ; BK13 in FCL. Firstly, the GCL-based deduction is sufficiently general to include implication rules deduced from FCL and RSL. Secondly, the unique formula $\mu\rightarrow\mu\cdot 1_{\eta}$ can furnish a systematic decision about whether any given rule is supported by the formal context.

• •

From the perspective of Ref. GD86 $A\stackrel{{\scriptstyle fcl}}{{\rightarrow}}B$ is furnished by $A^{I}\subseteq B^{I}$, while one may have its analogue in RSL by identifying $A\stackrel{{\scriptstyle rsl}}{{\rightarrow}}B$ with $A^{\Diamond}\subseteq B^{\Diamond}$. Moreover, there are two equivalence relations inspired by Lemma 3.1 of Ref. LLJD12-1 : $A\stackrel{{\scriptstyle fcl}}{{\rightarrow}}B$ is equivalent to $\prod A{\rightarrow}\prod B$; $A\stackrel{{\scriptstyle rsl}}{{\rightarrow}}B$ is equivalent to $\sum A{\rightarrow}\sum B$. Both are particular cases of $(\mu_{1}\rightarrow\mu_{2})\iff(\mu_{1}^{R}\subseteq\mu_{2}^{R})$ (Lemma 4.6) since

[TABLE]

where $(A^{\Diamond},B^{\Diamond})$ and $(A^{I},B^{I})$ are referred to the variant forms given in Eq. (9) of Ref. LLJD12-1 . For instance, let one inspect the content of $\rho(\{1,2\})>c>\eta(\{1,2\})$ in Fig. 3 by employing the fact $c^{R}=\{1,2\}$ is a common extent of FCL and RSL, where implications of both types can be easily exhibited.

For $c\rightarrow\eta(\{1,2\})$, the rule $c\rightarrow a\neg bc(de+\neg d\neg e)$ is of the type T1/RPII but its equivalent statement $c\rightarrow a\neg b(de+\neg d\neg e)$ is simply a T2-rule, which is due to $c^{R}=\{1,2\}\subset\{1,2,5\}=(a\neg b(de+\neg d\neg e))^{R}$. On the other hand, $\left\{c\right\}\stackrel{{\scriptstyle fcl}}{{\rightarrow}}\left\{a\right\}$ is due to $\{c\}^{I}=\left\{1,2\right\}\subset\left\{1,2,5,6\right\}=\{a\}^{I}$ GD86 , in addition, $\{c\}\stackrel{{\scriptstyle fcl}}{{\rightarrow}}\{a,c\}$. Based on Eq. (12), $\left\{c\right\}\stackrel{{\scriptstyle fcl}}{{\rightarrow}}\left\{a\right\}$ becomes the T2-rule $c\rightarrow a$ and $\{c\}\stackrel{{\scriptstyle fcl}}{{\rightarrow}}\{a,c\}$ becomes the T1-rule $c\rightarrow ac$, both are the consequences of $(c\rightarrow a\neg bc(de+\neg d\neg e))\implies(c\rightarrow ac)$.

For $\rho(\{1,2\})\rightarrow c$, the rule $c+d+(a+\neg b+\neg e)(\neg a+b+e)(\neg a+\neg b+\neg e)\rightarrow c$ can imply $(d\rightarrow c)$ and $(c+d\rightarrow c)$. Discarding the disjunction symbol, it turns out that $\{d\}\stackrel{{\scriptstyle rsl}}{{\rightarrow}}\{c\}$ and $\{c,d\}\stackrel{{\scriptstyle rsl}}{{\rightarrow}}\{c\}$, which are what one could have achieved in RSL 111To our knowledge, the discussion about such relations has not yet been found in the literature.: $\{d\}\stackrel{{\scriptstyle rsl}}{{\rightarrow}}\{c\}$ is based on $\{d\}^{\Diamond}\subseteq\{c\}^{\Diamond}$; $\{c,d\}\stackrel{{\scriptstyle rsl}}{{\rightarrow}}\{c\}$ is based on $\{c,d\}^{\Diamond}\subseteq\{c\}^{\Diamond}$. In principle, identifying $A\stackrel{{\scriptstyle rsl}}{{\rightarrow}}B$ with $A^{\Diamond}\subseteq B^{\Diamond}$ is the reasonable analogue of $A\stackrel{{\scriptstyle fcl}}{{\rightarrow}}B$ with $A^{I}\subseteq B^{I}$.

• •

Note that it is not practical to list out all the possible RII’s. Instead, the GCL provides criteria to determine whether a logic implication is allowable by the formal context. Since $\mu_{1}\rightarrow\mu_{2}$ is defined by $\mu_{1}^{R}\subseteq\mu_{2}^{R}$, which entails $\eta(\mu_{1}^{R})\leq\eta(\mu_{2}^{R})$ (Proposition 3.14 in Ref. LLJD12-1 ), the $\mu_{1}\rightarrow\mu_{2}$ is an implication allowable by the formal context iff $\mu_{1}\cdot 1_{\eta}\implies\mu_{2}\cdot 1_{\eta}$. Since $1_{\eta}$ is logically true subject to the formal context, it always coexists with any other attribute. In other words, one requires $\mu_{1}\rightarrow\mu_{2}$ to be true under the condition $1_{\eta}$:

[TABLE]

Moreover, if $\mu_{1}\rightarrow\mu_{2}$ is allowable by $1_{\eta}$ then one has ${\mu_{1}\cdot 1_{\eta}\implies\mu_{2}\cdot 1_{\eta}},\ \mbox{i.e., }\mu_{1}\cdot 1_{\eta}\leq\mu_{2}\cdot 1_{\eta}$. For example, by employing Eq. (7) for the $1_{\eta}$ obtained in Fig. 3, one has

[TABLE]

Another interesting point is about the limit at which the contextual truth $1_{\eta}$ becomes the real logical truth ${\bf 1}$. Under consideration is thus the degenerate GCL which emerges from a degenerate formal context as follows.

Proposition 4.8**.**

Subject to $F(G,M)$, the following statements are all equivalent, which defines the degenerate formal context: S1. $n_{F}=\log_{2}^{|M^{\ast}|}$. S2. $\forall X\in E_{F}\ \rho(X)=\eta(X)$. S3. $1_{\eta}={\bf 1}$ and $0_{\rho}={\bf 0}$. S4. $\forall\mu\in M^{\ast}\ [\mu^{R}]_{F}=\left\{\mu\right\}$.

Proof.

Consider ${\bf S1}\implies{\bf S2}\implies{\bf S3}\implies{\bf S4}\implies{\bf S1}$, which entails ${\bf S1}\iff{\bf S2}\iff{\bf S3}\iff{\bf S4}$, as follows.

S1$\implies$S2: If $n_{F}=\log_{2}^{|M^{\ast}|}$ then $|[X]_{F}|=1$ (Corollary 4.4). Moreover, $\rho(X)\geq\eta(X)$ and $\left\{\begin{smallmatrix}\eta(X)\in[X]_{F}\\ \rho(X)\in[X]_{F}\end{smallmatrix}\right.$, therefore, $\rho(X)=\eta(X)$.

S2$\implies$S3: $\rho(X)=\eta(X)$, thus, $\forall\mu\in M^{\ast}\ \mu\cdot 1_{\eta}=\mu+0_{\rho}$ (Proposition 4.1), which renders $\mu\geq\mu\cdot 1_{\eta}=\mu+0_{\rho}\geq\mu$ by Eq. (11). Therefore, $\forall\mu\ \mu\cdot 1_{\eta}=\mu=\mu+0_{\rho}$ which implies $1_{\eta}={\bf 1}$ and $0_{\rho}={\bf 0}$.

S3$\implies$S4: $\mu+0_{\rho}=\mu+{\bf 0}=\mu\cdot{\bf 1}=\mu\cdot 1_{\eta}=\mu\ \therefore[\mu^{R}]_{F}=[\mu+0_{\rho},\mu\cdot 1_{\eta}]=[\mu,\mu]=\left\{\mu\right\}$.

S4$\implies$S1: $\forall X_{i}\forall X_{j}\in E_{F}\ [X_{i}]_{F}\cap[X_{j}]_{F}=\emptyset$ iff $X_{i}\neq X_{j}$ (Proposition 2.1). Moreover, with S4, $\forall\mu\in M^{\ast}$, $\mu^{R}$ can be identified as some $X\in E_{F}$, which is given by $\left\{\mu^{R}\mid\mu\in M^{\ast}\right\}$. Therefore, $\forall X\in E_{F}\ |[X]_{F}|=1=2^{\log_{2}^{|M^{\ast}|}-n_{F}}$, which implies $\log_{2}^{|M^{\ast}|}=n_{F}$. ∎

Corollary 4.9**.**

Subject to a degenerate formal context, say $F_{\cal D}(G,M)$, $\forall\mu\forall\mu^{\prime}\in M^{\ast}$ $\mu\leq\mu^{\prime}$ iff $\mu^{R}{\subseteq}(\mu^{\prime})^{R}$.

Proof.

One now proceeds the proof in two parts.

Firstly, $\mu=\mu^{\prime}\stackrel{{\scriptstyle F_{\cal D}}}{{\iff}}\mu^{R}=(\mu^{\prime})^{R}$: It is clear that $\mu=\mu^{\prime}\implies\mu^{R}=(\mu^{\prime})^{R}$. Reversely, if $\mu^{R}=(\mu^{\prime})^{R}$ then $\left\{\mu\right\}=[\mu^{R}]_{F_{\cal D}}=[(\mu^{\prime})^{R}]_{F_{\cal D}}=\left\{\mu^{\prime}\right\}$, i.e. $\mu=\mu^{\prime}$. Therefore, $\mu=\mu^{\prime}\iff\mu^{R}{=}(\mu^{\prime})^{R}$.

Secondly, $\mu<\mu^{\prime}\stackrel{{\scriptstyle F_{\cal D}}}{{\iff}}\mu^{R}\subset(\mu^{\prime})^{R}$: With $\mu=\mu^{\prime}\iff\mu^{R}{=}(\mu^{\prime})^{R}$, the formula $\mu<\mu^{\prime}\implies\mu^{R}{\subseteq}(\mu^{\prime})^{R}$ (Lemma 2.10 in Ref. LLJD12-1 ) is reduced to $\mu<\mu^{\prime}\implies\mu^{R}{\subset}(\mu^{\prime})^{R}$. Reversely, assume $\mu^{R}{\subset}(\mu^{\prime})^{R}$ but $\mu\not<\mu^{\prime}$. Then, $\mu^{R}\cap(\mu^{\prime})^{Rc}=(\mu\cdot\neg\mu^{\prime})^{R}=\emptyset$ but $\mu\cdot\neg\mu^{\prime}\neq{\bf 0}$, which is contradictory since $\mu\cdot\neg\mu^{\prime}\in[\emptyset]_{F_{\cal D}}$ while $[\emptyset]_{F_{\cal D}}=\{{\bf 0}\}$. Therefore, $\mu^{R}{\subset}(\mu^{\prime})^{R}\iff\mu<\mu^{\prime}$. ∎

Note that the occurrence of degenerate GCL is not as rare as one might have anticipated. For instance, by removing the attributes $a,b$ and $d$ from $F(G,M)$ in Table 1, where $M$ reduces to $\{c,e\}$, one will end up with $n_{F}=2^{|M|}=4$. As is depicted in Fig. 4, the resultant degenerate GCL then comprises $2^{4}$ nodes embedded in the original one, in which $1_{\eta}=ce+c\neg e+\neg ce+\neg c\neg e\equiv{\bf 1}$ and $0_{\rho}=(\neg c+\neg e)(\neg c+e)(c+\neg e)(c+e)\equiv{\bf 0}$. However, still more instructive is that to each formal context one may associate a degenerate formal context, which exhibits all the attribute freedom and can thus serve as the reference context for analysing the re-parametrisation for the general concept lattice.

5 discussion

One has demonstrated in this paper the merits of GCL with two insightful features, which are the generality and the tractability. For the generality feature, it is shown that the GCL incorporates the conventional FCL and RSL from both the perspectives of lattice structure and of logic content. From the lattice-structure perspective, the GCL furnishes a comprehensive categorisation for whatever distinctive object classes (general extents) based on $F(G,M)$, where every attribute in $M^{\ast}$ essentially pertains to a definite general intent, cf. Proposition 2.1. The GCL turns out to be the foundation of various generalised concept lattices (Proposition 2.2) such as the generalised versions of the FCL and RSL. In practice, all the nodes of the FCL and the RSL are identifiable on the GCL, as can be explicitly worked out in Fig. 2 and 3. From the logic-content perspective, the logic implication extracted from the GCL is concerned with the implication relations of the type $\mu_{1}\rightarrow\mu_{2}$ ($\mu_{1},\mu_{2}\in M^{\ast}$) where the FCL- and RSL- based implications emerge as particular cases due to Eq. (12).

For the tractability feature, both constructing the lattice structure and implementing the logic content are tractable.

To construct the lattice, the GCL develops a Hasse diagram of $2^{n_{F}}$ nodes, where each node is referred to as a general concept comprising a distinct 2-tuple given in terms of general extent and general intent. The $2^{n_{F}}$ general extents appear to be all possible unions of the smallest subsets discernible by the formal context, see Proposition 3.5 in Ref. LLJD12-1 , hence, no additional effort is needed for selecting them out. The general intents are $2^{n_{F}}$ disjoint closed sub-intervals of $M^{\ast}$ (Proposition 2.1, Corollary 4.4) with the constant cardinality $2^{rank(M)-n_{F}}$. The expression of a general concept is stated as $(X,[X]_{F})$, where $[X]_{F}=[\eta(X),\rho(X)]$, namely, $\eta(X)$ and $\rho(X)$ are respectively the lower and upper bounds of $[X]_{F}$, see Proposition 2.1. The construction of GCL is as tractable as listing out the formal context in an arbitray order since it is fully characterised by means of the $\eta$-representation ${\Upsilon}_{\eta}$ (or $\rho$-representation ${\Upsilon}_{\rho}$, see Definition 3.4), which can be completed by a single glance of the formal context, see Proposition 3.1. All the general concepts can then be read out on-demand from ${\Upsilon}_{\eta}$ (${\Upsilon}_{\rho}$). Note that based on Proposition 3.3 any of the components in the triplet $(X,\rho(X),\eta(X))$ will determine the other two, as can be illustrated by means of simple binary masks, see Eq. (10).

In determining the logic content, the GCL suggests to implement the implication relations via the entailment of the lower bound property. Note that any object set carrying a definite property should be categorised into a definite general extent because every attribute in $M^{\ast}$ essentially belongs to a definite general intent (Proposition 2.1). Such an implementation is tractable since the single formula $\forall\mu\in M^{\ast}\ \mu\ \rightarrow\mu\cdot 1_{\eta}$ (Proposition 4.7) suffices to present all the rules of informative implication where $1_{\eta}$ is the contextual truth obtained by summing all the components of the $\eta$-representation (Definition 3.4). Conjugately, the formula can be restated as $\forall\mu\in M^{\ast}\ \mu+0_{\rho}\ \rightarrow\mu$ with the contextual falsity $0_{\rho}\equiv\neg 1_{\eta}$, which turns out to implement the implication relations via the entailment of the upper bound property. Note that either of the formulas $\left\{\begin{smallmatrix}\mu\ \rightarrow\mu\cdot 1_{\eta}\\ \mu+0_{\rho}\ \rightarrow\mu\end{smallmatrix}\right.$ is capable of determining all the implication relations based on the formal context, including those which could be deduced from the FCL and RSL since both $A\stackrel{{\scriptstyle fcl}}{{\rightarrow}}B$ and $A\stackrel{{\scriptstyle rsl}}{{\rightarrow}}B$ can be interpreted as particular cases for $\mu_{1}\rightarrow\mu_{2}$ by Eq. (12). On the other hand, $\left\{\begin{smallmatrix}A\stackrel{{\scriptstyle fcl}}{{\rightarrow}}B\\ A\stackrel{{\scriptstyle rsl}}{{\rightarrow}}B\end{smallmatrix}\right.$ can by no means generate all the possible implications from the formal context. Obviously, an expression like “$\neg e\rightarrow a+\neg b\neg d$” in Eq. (14) can be identified neither with $A\stackrel{{\scriptstyle fcl}}{{\rightarrow}}B$ nor with $A\stackrel{{\scriptstyle rsl}}{{\rightarrow}}B$.

The logic reasoning based on the GCL is in fact rather intuitive. All the attributes $[X]_{F}$ possessed by the same object class $X$ are regarded as equivalent. For any $\mu\in M^{\ast}$, there is a bi-conditional equivalence $\forall\nu\in[\mu^{R}]_{F}\ \mu\leftrightarrow\nu$ that corresponds to the T1-rules (Lemma 4.6) from which one may determine all the rules by incorporating tautologies. While the explicit object reference is ignored here, logic relations only refer to the contextual Venn diagram ${\cal V}^{F}_{M}$, $\mu_{1}\rightarrow\mu_{2}$ gets its interpretation via $\mu_{1}^{R}\subseteq\mu_{2}^{R}$. In general, the set relation between $\mu_{1}^{R}$ and $\mu_{2}^{R}$ suggests an ordering on ${\cal V}^{F}_{M}$ that further determines whether $\mu_{1}\rightarrow\mu_{2}$ is an allowable implication, see Eq. (13) also cf. Proposition 3.14 in Ref. LLJD12-1 . Note that any attribute in effect serves as a logic statement that asserts property on a definite subject. In particular, the attribute $1_{\eta}=1_{\eta}^{F}=1_{\eta}({\cal V}^{F}_{M})$ exhibits a logic condition that governs the rules of implication according to ${\cal V}^{F}_{M}$. Hence, every single attribute enters as a logic statement that asserts property and then can be employed as a logic condition by means of $1_{\eta}({\cal V}^{lc}_{M})$. The idea to deal with the statements of pure attribute type then brings about a simplified algebraically manipulable reasoning called the primary deduction system LLJD12-3 where the logical OR, AND, NOT and implication among statements can be realised by the Boolean disjunction, conjunction and negation operators among attributes. Indeed, the primary deduction system readily suffices to provide an efficient reasoning tool that leads to non-trivial applications, e.g. solving certain well-known puzzles. Moreover, the rules of classical logic are found to be true in the primary deduction system since the Hilbert axioms in Ref. Hd27 all appear to be manifestly valid. It should however be remarked that the primary deduction system with pure attribute-type statements could not be satisfactory and is coined to be naïve as it contradicts one’s intuitions, Ref. LLJD12-4 thus strives to incorporate novel syntax in order to resolve such counterintuitive issue. Another point is that the primary deduction system is not expressive enough, as opposed to the conventional reasoning process, therefore, one has to look forward further to a comprehensive algebraically manipulable deduction LLJD12-5 .

In addition, the degenerate formal context (Proposition 4.8) describes a mathematical limit at which the number of object classes discernible by the formal context is exhausted. Notably, the degenerate formal context gives rise to a degenerate GCL in which the condition $\rho(X)=\eta(X)$ reduces every general intent into one sole member of $M^{\ast}$ (Corollary 4.4). The contextual Venn diagram for a degenerate formal context $F_{\cal D}$ in fact coincides with the conventional Venn diagram, say ${\cal V}^{F_{\cal D}}_{M}={\cal V}_{M}$, by means of $1_{\eta}^{F_{\cal D}}={\bf 1}$. Indeed, the implication formula (Proposition 4.7), when applying to the degenerate GCL, concludes that $\forall\mu\in M^{\ast}\ \mu\rightarrow\mu$ which proposes no interesting implications and thus becomes less appealing as a practical categorisation. Nevertheless, the degenerate formal context can serve as theoretical referential system. To each formal context, a referential context can be designed to provide a basis convention, by revealing the freedom of the generalised attribute system as was stated in Eq. (8). In practice, the referential context $F_{\cal D}(G\cup G_{f},M)$ is a degenerate formal context by appending $G_{f}$ to $F(G,M)$, a set of $rank(M)-n_{F}$ fictitious objects (hence, $n_{F_{\cal D}}=rank(M)$), as a means to expose the attribute freedom corresponding to the properties not carried by the existing objects. In Ref. LLJD12-3 , it will be shown the referential context is instructive to illustrate the extensive structure of GCL. Moreover, $F_{\cal D}(G\cup G_{f},M)$ also provides a very convenient framework above which one may study the equivalent classes of formal context.