Introduction

A formal context is a triple $`(G, M, I)`$ formed by two sets $`G`$ (of objects) and $`M`$ (of attributes), and a binary relation $`I`$ between them, i.e. $`I\subseteq G\times M`$. In formal contexts, attribute implications are used to extract information about the dependencies between attributes. Thus, an implication is a relation between two sets of attributes $`A`$ and $`B`$, denoted by $`A \rightarrow B`$, and is valid if, whenever an object has all attributes in $`A`$, then it also has all attributes in $`B`$. Implications have been the subject of several studies , notably those of Duquenne and Guigues , which, for a given formal context, led to the construction of the canonical base of implications. By incorporating the condition for which an object has an attribute, the notion of a formal context is extended . This has led to the development of Triadic Concept Analysis (TCA) as an extension of Formal Concept Analysis (FCA) . A triadic context is defined as a quadruple $`\mathbb{K}:=(G,M, \mathcal{C}, I)`$, where $`G`$ is a set of objects, $`M`$ is a set of attributes, $`\mathcal{C}`$ is a set of conditions, and $`I`$ is a relation between objects, attributes, and conditions ($`I \subseteq G{\times}M{\times}\mathcal{C}`$). In this article, we focus on implications of triadic contexts, which are specific connections between subsets of $`M`$ and $`\mathcal{C}`$ ; they were introduced in the triadic framework by Biedermann . Ganter and Obiedkov extended this work by defining other types of implications. Implications in triadic contexts fall into two categories, namely Biedermann, and Ganter & Obiedkov ones.

Those defined by Biedermann are the following:

$`\star`$
Biedermann’s conditional attributes implications (or BCAI for short), denoted by $`(A_1 \rightarrow A_2)_C`$, where $`A_1\subseteq M`$ is the premise, $`A_2 \subseteq M`$ is the conclusion, and $`C \subseteq \mathcal{C}`$ is the set of conditions (constraint). They can be interpreted as ’if an object of $`G`$ has all attributes in $`A_1`$ under all conditions in $`C`$, then it also has all attributes in $`A_2`$ under the same conditions’. They are seen as knowledge from the point of view of attributes.

$`\star`$
Biedermann’s attributional conditions implications (or BACI for short), denoted by $`(C_1 \rightarrow C_2)_A`$ , where $`C_1\subseteq \mathcal{C}`$ is the premise, $`C_2 \subseteq \mathcal{C}`$ is the conclusion and $`A \subseteq M`$ is the constraint. They can be interpreted as ’if an object of $`G`$ has all attributes in $`A`$ under all conditions in $`C_1`$, then it also has all attributes in $`A`$ under all conditions in $`C_2`$’. They are seen as knowledge from the point of view of conditions.

The ones defined by Ganter and Obiedkov are:

$`\star`$
Attribute$`\times`$condition implications (or A$`\times`$CI for short), are of the form $`E\rightarrow F`$, where $`E`$ (premise) and $`F`$ (conclusion) are subsets of $`M{\times}\mathcal{C}`$, and interpreted as: “any object $`g \in G`$ in relation with all attribute-condition pairs in $`E`$ is also in relation with all attribute-condition pairs in $`F`$”.

$`\star`$
Conditional attribute implications (CAI for short), denoted by $`A_{1}\overset{C}{\rightarrow}A_{2}`$, with $`A_{1}, A_{2} \subseteq M`$ and $`C\subseteq \mathcal{C}`$, are interpreted as: “if an object $`g \in G`$ has all attributes in $`A_{1}`$ under all set of conditions $`X \subseteq C`$, then $`g`$ also has all attributes in $`A_{2}`$ under $`X`$”.

$`\star`$
Attributional condition implications (ACI for short), denoted by $`C_{1} \overset{A}{\rightarrow}C_{2}`$, with $`C_{1}, C_{2} \subseteq \mathcal{C}`$ and $`A \subseteq M`$, are interpreted as: “whenever an object $`g \in G`$ has under the conditions in $`C_{1}`$ all attributes in $`X \subseteq A`$, then $`g`$ also has under the conditions in $`C_{2}`$ all attributes in $`X`$”, for all $`X\subseteq A`$.

This article is an extension of the work carried out in . It focuses on CAI and ACI because they are more compact and convey richer semantics than BACI and BCAI. Here, we present three key notions: feature, quasi-feature and pseudo-feature and we show that pseudo-features correspond to the smallest family likely to generate a minimal and optimal basis of BCAI and BACI . We then introduce the notion of unit pseudo-feature and show that it corresponds to the smallest family of elements generating a minimal and optimal basis of CAI and ACI. Finally, we propose an algorithm for performing these constructions, followed by a theoretical study of its complexity.

The rest of this document is organized as follows. Section 2 introduces some basic notions on FCA and TCA. In Section 3, we show how triadic context augmentation can contribute to the construction of quasi-features. Then we construct in Section 4 complete bases and minimal bases of BCAI, BACI, CAI and ACI respectively, using quasi-features, pseudo-features and unit pseudo-feature, and provide a construction of their optimal bases. Section 5 provides an algorithm for constructing those bases and studies its complexity. The paper ends with a conclusion.

Basic notions

FCA was introduced by Wille in , based on the understanding that a concept is constituted by its extent and intent. Indeed, to formalize the notion of concept, a universe of discourse or dyadic formal context is set by a triple $`(G, M, I)`$ consisting of two sets $`G`$ (of objects) and $`M`$ (of attributes) and a binary relation $`I\subseteq G\times M`$. A concept of $`(G,M,I)`$ is a pair $`(A,B)`$ such that $`A \subseteq G`$, $`B \subseteq M`$, $`A'=B`$ and $`B'=A`$, where $`A'`$ (the set of all attributes common to all objects in $`A`$) and $`B'`$ (the set of all objects sharing all attributes in $`B`$) are computed using the derivation operator $`'`$ defined as follows:

A' :=\{m \in M; (a,m) \in I, \ \forall a \in A\} \text{ and } B':=\{g \in G; (g,b) \in I, \ \forall b \in B\}.

For a dyadic concept $`(A,B)`$, the set $`A`$ is called the extent and $`B`$ the intent of $`(A,B)`$. The set of all concepts is ordered by the relation:

{(A_1,B_1) \leq (A_2,B_2) :\iff  A_1 \subseteq A_2\quad (:\iff  B_2 \subseteq B_1)}

and forms a complete lattice called the concept lattice of $`(G, M, I)`$, and denoted by $`\underline{\mathfrak{B}}(G,M,I)`$.

From Example 1, we can add a third dimension named Suppliers and study the relation between Clients, Products, and Suppliers. This representation has motivated Lehmann and Wille in 1995 to extend FCA to TCA.

A triadic context is a triple denoted $`\mathbb{K}:=(G, M, \mathcal{C}, I)`$, where $`I\subseteq G{\times}M{\times}\mathcal{C}`$ is a relation between objects in $`G`$, and the attributes in $`M`$ under conditions in $`\mathcal{C}`$.

The conditions are understood in as valuations, modalities, meanings, purposes, and reasons concerning connections between objects and attributes. A triadic context can be represented by a table. The two tables in Fig. 3 actually represent the same example. taken from as an adaptation of a table in .

Example 2. If the suppliers are Peter, Nelson, Rick, Kevin, and Simon, we can set $`\mathcal{C} := \{P,N,R,K, S\}`$ as our set of conditions. Note that the elements of $`\mathcal{C}`$ are capital letters, as the initials of proper nouns. The relation $`I`$ is then defined by $`(x, y, z) \in I`$ if and only if the client $`x`$ orders the product $`y`$ from the supplier $`z`$. The value¹ $`ac`$ in row $`1`$ and column $`R`$ in the table in Figure 3 (left) means that the client $`1`$ ordered the products $`a`$ and $`c`$ from the supplier $`R`$. In the table shown in Fig. 3 (right), $`PNRKS`$ in Line $`1`$ and Column $`a`$ means that the client $`1`$ ordered the product $`a`$ from all suppliers.

P N R K S 1 abd abd ac ab a 2 ad bcd abd ad d 3 abd d ab ab a 4 abd bd ab ab d 5 ad ad abd abc a

a b c d 1 PNRKS PNK R PN 2 RPK NR N PNRKS 3 PRKS RPK PN 4 PRK PNRK PNS 5 PNRKS KR K PNR

From a triadic context $`\mathbb{K}:=(G, M, \mathcal{C}, I)`$ we can extract the following dyadic contexts: $`\mathbb{K}^{(1)}:=(G, M{\times}\mathcal{C}, I^{(1)})`$, $`\mathbb{K}^{(2)}:=(M, G{\times}\mathcal{C}, I^{(2)})`$, and $`\mathbb{K}^{(3)}:=(\mathcal{C}, G{\times}M, I^{(3)})`$, where

(o, (a, c)) \in I^{(1)} \iff (a, (o, c)) \in I^{(2)} \iff (c, (o, a)) \in I^{(3)} \iff (o, a, c) \in I.

Their derivations are called $`i`$-derivation, $`i \in \{1,2,3\}`$. Additionally, we can extract the following dyadic contexts: $`(G, M, I_{C}^{12})`$ with $`C \subseteq \mathcal{C}`$, $`(G, \mathcal{C}, I_{A}^{13})`$ with $`A \subseteq M`$ and $`(M, \mathcal{C}, I_{O}^{23})`$ with $`O \subseteq G`$, where

\begin{eqnarray*}
     (o, a) \in I_{C}^{12} :\iff (o, a, c) \in I, \ \ \text{for all} \quad c \in C;  \\
     (o, c) \in I_{A}^{13} :\iff (o, a, c) \in I, \ \ \text{for all} \quad a \in A;  \\
     (a, c) \in I_{O}^{23} :\iff (o, a, c) \in I, \ \ \text{for all} \quad o \in O.
\end{eqnarray*}

Their derivations are: $`(1, 2, C)`$-derivation for $`I_{C}^{12}`$ with $`C\subseteq \mathcal{C}`$, $`(1, 3, A)`$-derivation for $`I_{A}^{13}`$ with $`A\subseteq M`$, $`(2, 3, O)`$-derivation for $`I_{O}^{23}`$ with $`O\subseteq G`$. For example, if $`O\subseteq G`$,

O^{(1)}=\{(a,c)\in M{\times}\mathcal{C}; (o,a,c) \in I, \  \forall o \in O \}\quad 
 \text{ and }

O^{(1,2,C)} =\{a \in M; (o,a,c) \in I, \  \forall (o,c) \in O{\times}C \}.

For any $`X, Y`$ subsets of $`G`$, $`M`$ or $`\mathcal{C}`$ and $`i \in \{1,2,3\}`$, we have $`X\subseteq X^{(i)(i)}`$, and if $`X\subseteq Y`$, then $`Y^{(i)} \subseteq X^{(i)}`$. If $`X_1\subseteq Y_1`$ and $`X_2 \subseteq Y_2`$, then $`(Y_1{\times}Y_2)^{(i)} \subseteq (X_1{\times}X_2)^{(i)}`$. These relations are valid if we replace the i-derivation by the $`(1, 2, C)`$-, $`(1, 3, A)`$- or $`(2, 3, O)`$-derivation.

A triadic concept of $`\mathbb{K}`$ is a 3-tuple $`(O, A, C) \in 2^{G}{\times}2^{M}{\times}2^{\mathcal{C}}`$ such that $`O = (A{\times}C)^{(1)}`$, $`A = (O{\times}C)^{(2)}`$, $`C = (O{\times}A)^{(3)}`$. We call $`O`$, $`A`$, $`C`$ and $`A{\times}C`$ respectively extent, intent, modus and feature of the concept $`(O,A,C)`$.

\begin{align*}
        \{1,2,3,4,5\}&={(\{d\}{\times}\{P,N\})}^{(1)} \\
        \{d\}&={(\{1,2,3,4,5\}{\times}\{P,N\})}^{(2)} \\
        \{P,N\}&={(\{1,2,3,4,5\}{\times}\{d\})}^{(3)}
\end{align*}

For any concept $`\mathit{c}`$, we denote by $`\mathrm{ext}(\mathit{c})`$, $`\mathrm{int}(\mathit{c})`$, $`\mathrm{modus}(\mathit{c})`$, $`\mathrm{feat}(\mathit{c})`$ respectively its extent, intent, modus and feature. $`\mathfrak{T}(\mathbb{K})`$ denotes the set of all concepts and $`\mathfrak{F}(\mathbb{K})`$ the set of all features of $`\mathbb{K}`$. Note that each feature defines a unique concept.

We recall that in a finite dyadic context $`(G, M, I)`$, a set $`P\subseteq M`$ is a pseudo-intent if it is not closed, but contains the closure of any other pseudo-intent it contains. The set $`\{P \to P'' \mid P\subseteq M`$ is pseudo-intent$`\}`$ forms an implication base of $`(G,M,I)`$, called the stem base ; it is also a base with the smallest cardinality. The recursive definition of pseudo-intent makes it computationally expensive to directly check whether a set is a pseudo-intent. Sebastian Rudolph provided in an optimized algorithm for the pseudo-intent verification. Indeed, he introduced the notion of incrementor and used it to provide a non-recursive characterization of pseudo-intents. To achieve this, for a given subset of attributes $`P \subseteq M`$, he added a new object $`o_P`$ such that $`o_P'=P`$, and therefore turned $`P`$ into an intent in the augmented context; this context is said to be augmented by $`P`$. He called incrementor any set of attributes that produces by augmentation just one new concept. He observed that any pseudo-intent is an incrementor and an incrementor $`P`$ is a pseudo-intent if for every incrementor $`Q \subseteq P`$, there is an intent $`R`$ such that $`Q \subseteq R \subseteq P`$.

In we provide an extension of this construction and a characterization for BCAI and BACI implications. However, no such construction has been proposed for CAI and ACI implications; yet, we know that the latter form allows a compact representation of implications, i.e. a representation of implications of the form $`X \overset{B}{\rightarrow} Y`$ (which correspond respectively to a family of implications described by $`\{(X \rightarrow Y)_{\{b\}} : b \in B\}`$). It is therefore important for us to propose such a construction for CAI and ACI implications.

In addition, these constructions require the process of context augmentation, which plays several roles:

$`\star`$
It highlights any augmentation to the context;

$`\star`$
It allows the construction of pseudo-features which are the counterparts of pseudo-intents in dyadic contexts.

Since each augmentation adds new relations to the context, all concepts and implications of the augmented context must be re-computed. In this paper, we will focus on building implications of the augmented context from implications of the initial context.

We start with the augmentation process for triadic contexts.

Augmentation of a triadic context

Unless otherwise stated, we assume that $`\mathbb{K}:=(G,M, \mathcal{C}, I)`$ is a finite triadic context. A context can be augmented by an attribute, an object, a condition or several of these elements simultaneously. However, the process and properties associated with augmentation remain the same, as exchanging the positions of objects, attributes and conditions does not change the relation of the context. In this section, we illustrate this process by considering the augmentation by a new object. It should be recognized that an augmentation by several elements (attributes, objects or conditions) corresponds to a sequence of augmentations of an element.

In what follows, we investigate on the link between the derivation in $`\mathbb{K}`$ and $`\mathbb{K}[Z]`$ (Proposition 1) and the link between their features (Proposition 2). We will denote by $`^{(i_{_Z})}`$ the i-derivation in $`\mathbb{K}[Z]`$ to distinguish it from the i-derivation in $`\mathbb{K}`$.

As the name suggests, augmentation generates new information. However, it is essential to note that not all augmentations generate the same amount of information. As Sebastian Rudolf shows in , incrementors can characterize pseudo-closed sets in a dyadic context. Can we expect similar results in triadic contexts?

Observe that, if $`P`$ is a quasi-feature of $`\mathbb{K}`$, then $`|\mathfrak{F}(\mathbb{K}[P])| = 1 + |\mathfrak{F}(\mathbb{K})|`$. Hence, $`P`$ is a quasi-feature of $`\mathbb{K}`$ if and only if $`P \notin \mathfrak{F}(\mathbb{K})`$ and for any $`Z \in \mathfrak{F}(\mathbb{K}[P]), \ Z = P`$ or $`Z \in \mathfrak{F}(\mathbb{K})`$.

In the following, we will highlight the link between quasi-features and implications.

quasi-features and implications in triadic contexts

quasi-features are important for constructing implications. We use them here to construct bases of triadic implications of the forms BCAI, BACI, CAI and ACI. To facilitate understanding, we begin with a few reminders of these triadic implications.

\begin{eqnarray*}
        (A_{1}{\times}C)^{(1)} \subseteq (A_{2}{\times}C)^{(1)} \quad (\Longleftrightarrow A_{2} \subseteq A_{1}^{(1,2,C)(1,2,C)})
\end{eqnarray*}

\begin{eqnarray*}
        (A{\times}C_{1})^{(1)} \subseteq (A{\times}C_{2})^{(1)} \quad (\Longleftrightarrow C_{2} \subseteq C_{1}^{(1,3,A)(1,3,A)})
\end{eqnarray*}

\begin{eqnarray*}
       (A_{1}{\times}X)^{(1)} \subseteq (A_{2}{\times}X)^{(1)} \quad \text{for all} \ X \subseteq C \quad (\Longleftrightarrow A_{1} \subseteq A_{2}^{(1,2,X)(1,2,X)}  \quad \text{for all} \ X \subseteq C).
\end{eqnarray*}

\begin{eqnarray*}
      (X{\times}C_{1})^{(1)} \subseteq (X{\times}C_{2})^{(1)}  \quad \text{for all} \ X \subseteq A  \quad (\Longleftrightarrow C_{2} \subseteq C_{1}^{(1,3,X)(1,3,X)} \quad \text{for all} \ X \subseteq A).
\end{eqnarray*}

We recall that $`\mathbb{K}\models \sigma`$ means that $`\sigma`$ is valid in $`\mathbb{K}`$. If $`\Sigma`$ is a set of implications verifying $`\mathbb{K}\models \sigma`$, for all $`\sigma`$ in $`\Sigma`$, then we can write $`\mathbb{K}\models \Sigma`$. Finally, $`\Sigma`$ semantically follows from $`\sigma`$ (or $`\Sigma \models \sigma`$ for short) if and only if $`\sigma`$ is valid in every context, in which all implications of $`\Sigma`$ are valid.
In dyadic contexts, for two distinct pseudo-closed sets $`P_1`$ and $`P_2`$ such that $`P_1 \subset P_2`$, there exists a closed set $`F`$ between them ($`P_1 \subset F \subset P_2`$). This makes it possible to obtain two distinct implications ($`P_1\to P_1'' \subseteq F`$ and $`P_2 \rightarrow P_2^{''}`$) from $`P_1`$ and $`P_2`$. This analogy is interesting when implementing in triadic contexts.

\text{BCAI} \quad (X \rightarrow X^{(1,2,Y)(1,2,Y)} )_{_Y} \qquad  \textbf{and} \qquad  \text{BACI} \quad (Y \rightarrow Y^{(1,3,X)(1,3,X)} )_{_X}

The product $`X{\times}Y`$ will essentially be considered as a quasi-feature in what follows.

Next, we characterize quasi-features that generate non-trivial implications, i.e. implications that provide meaningful information; to be more precise, implications whose conclusion is not a subset of the premise, or whose condition is not empty.

In all what follows, $`\mathbb{P}_{2}(\mathbb{K})`$ will be the set of all relevant quasi-features of $`\mathbb{K}`$ with respect to $`M`$ and $`\mathbb{P}_{3}(\mathbb{K})`$ those with respect to $`\mathcal{C}`$.

The following outlines an interesting fact about quasi-features with one empty component.

In what follows, we recall the axiomatic system for triadic implications . Throughout the rest of this document, we will use the notation $`\Sigma \vdash \sigma`$ to mean that ’an implication $`\sigma`$ is a syntactic consequence of a set of implications $`\Sigma`$’; in other words, ’$`\sigma`$ can be deduced from $`\Sigma`$’. The following principles describe this deduction.

For $`X, Y, W, Z \subseteq M`$ and $`C, C_1, C_2 \subseteq \mathcal{C}`$ the logic for BCAI relies on two axioms:
[Non-constraint] $`\vdash (\emptyset \rightarrow M)_{\emptyset}`$
[Reflexivity] $`\vdash (X \rightarrow X)_{\mathcal{C}}`$
And three inference rules, which are :
[Augmentation] $`(X \rightarrow Y)_{C} \vdash (X \cup Z \rightarrow Y \cup Z)_{C}`$
[Transitivity] $`\{ (X \rightarrow Y)_{C} ; \ (Y \rightarrow Z)_{C} \} \vdash (X \rightarrow Z)_{C}`$
[Conditional composition] $`\{ (X \rightarrow Y)_{C_{1}} ; \ (Z \rightarrow W)_{C_{2}} \} \vdash (X \cup Z \rightarrow Y \cap W)_{C_{1} \cup C_{2}}`$

Since we are dealing with Biedermann’s implications, it follows that [conditional decomposition]:$`(X \rightarrow Y)_{C_1 \cup C_2} \vdash (X \rightarrow Y)_{C_1}`$ cannot be possible. The soundness and completeness of the above logic for Biedermann’s implications derive from the study of conditional attributes implicational logic made in . In the following, we recall other rules that derive from the above logic (see for details).

[Decomposition] $`(X \rightarrow Y \cup Z)_{C} \vdash (X \rightarrow Y)_{C}`$

[Pseudotransitivity] $`\{ (X \rightarrow Y)_{C} ; \ (Y \cup Z \rightarrow W)_{C} \} \vdash (X \cup Z \rightarrow W)_{C}`$

[Additivity] $`\{ (X \rightarrow Y)_{C} ; \ (X \rightarrow Z)_{C} \} \vdash (X \rightarrow Y \cup Z)_{C}`$

[Accumulation] $`\{ (X \rightarrow Y \cup Z)_{C} ; \ (Z \rightarrow W)_{C} \} \vdash (X \rightarrow Y \cup Z \cup W)_{C}`$

We are interested in finding a family that can generate (with respect to the above logic) all valid implications of a context, i.e. a family of implications capable of generating any implication valid in the same context.

\sigma \ \text{is valid in}  \ \mathbb{K}\quad \text{if and only if} \quad \mathcal{B} \models \sigma.

From , we know that an implication $`\sigma`$ semantically follows from a set of implications $`\mathcal{B}`$ if $`\sigma`$ can be derived syntactically from $`\mathcal{B}`$ using the axiomatic system described above. Therefore, $`\mathcal{B}`$ is complete in a context $`\mathbb{K}`$ if for all $`\sigma`$ valid in $`\mathbb{K}`$, we have $`\mathcal{B} \vdash \sigma`$.

We are going to recall two important propositions to better understand the construction of a complete family of implications in triadic contexts . First, we need to extend the notion of closure operator known in formal contexts .

\begin{eqnarray*}
 (.)_{\Sigma, C} \ : \  2^{M}   &  \to     &   2^{M}  \\
                             A     & \mapsto  & (A)_{\Sigma, C} := \{ a \in M : \Sigma \vdash (A \rightarrow a)_{C} \}
\end{eqnarray*}

This closure operator enables us to simplify the derivation process of implications as is shown below.

\begin{eqnarray*}
      \Sigma \vdash (A_{1} \rightarrow A_{2})_{C} \qquad \text{and} \qquad  A_{2} \subseteq (A_{1})_{\Sigma, C}
\end{eqnarray*}
```*

</div>

We have already shown how to extract some valid implications from a
triadic context (Lemma 
<a href="#lemme valid implication" data-reference-type="ref"
data-reference="lemme valid implication">1</a>). It would be interesting
to build a complete subfamily of these implications.

<div id="lemme complete set of implications" class="lemma">

*Lemma 2*. The following sets of implications are complete in
$`\mathbb{K}`$.
``` math
\begin{eqnarray*}
 \mathcal{B}_{BCAI} = \{ (X \rightarrow X^{(1,2,Y)(1,2,Y)} )_{_Y} : X{\times}Y \in \mathbb{P}_{2}(\mathbb{K}) \}  \\
 \mathcal{B}_{BACI} = \{ (Y \rightarrow Y^{(1,3,X)(1,3,X)} )_{_X} : X{\times}Y \in \mathbb{P}_{3}(\mathbb{K}) \}
\end{eqnarray*}

\mathbb{K}\ \vDash \ (A_{1} \rightarrow A_{2})_{C}   \ \Rightarrow \     \mathcal{B}_{BCAI} \ \vdash \ (A_{1} \rightarrow A_{2})_{C}

\mathcal{B}_{BCAI} \ \nvdash \ (A_{1} \rightarrow A_{2})_{C}  \ \Rightarrow \      \mathbb{K}\ \nvDash \ (A_{1} \rightarrow A_{2})_{C}

\mathcal{B}_{BCAI} \vDash  (A_{1} \rightarrow A_{2})_{C} \quad \Longrightarrow \quad \mathcal{B}_{BCAI} \vdash  (A_{1} \rightarrow A_{2})_{C}

\begin{align*}
     \mathcal{B}_{BCAI} & = \{(a \rightarrow ad)_{P} \ ; \ (a \rightarrow ad)_{N} \ ;  \ (b \rightarrow abd)_{P} \ ; \ (b \rightarrow bd)_{N} \ ;  \ (b \rightarrow ab)_{R} \ ; \ (b \rightarrow ab)_{K} ; \\ 
      & \phantom{==}   (b \rightarrow abcd)_{S} \ ; \ (c \rightarrow abcd)_{P} \ ; \ (c \rightarrow bcd)_{N} \ ;  \ (c \rightarrow ac)_{R} \ ; \ (c \rightarrow abc)_{K} \ ; \\
      & \phantom{==}   (c \rightarrow abcd)_{S} \ ; \ (c \rightarrow abcd)_{PS} \ ; \ (d \rightarrow ad)_{P} \ ; \ (d \rightarrow abd)_{R} \ ; \  (d \rightarrow ad)_{K} \ ; \\
      & \phantom{==}   (bc \rightarrow abcd)_{S} \ ; \ (\emptyset \rightarrow ad)_{P} \ ; \ (\emptyset \rightarrow d)_{N} \ ; \ (\emptyset \rightarrow a)_{R} \ ; \ (\emptyset \rightarrow a)_{K} \ ; \ (\emptyset \rightarrow d)_{PN} \ ; \\
      & \phantom{==}   (\emptyset \rightarrow a)_{PR} \ ; \ (\emptyset \rightarrow a)_{PK} \ ; \ (\emptyset \rightarrow a)_{RK} \ ; \ (\emptyset \rightarrow a)_{RPK} \ ; \  (acd \rightarrow abcd)_{P} \ ; \\
      & \phantom{==} (abd \rightarrow abcd)_{RPK} \ ; \ (abc \rightarrow abcd)_{RPK} \ ; \ (ad \rightarrow abcd)_{PNRKS} \ ; \\  
      & \phantom{==}   (ab \rightarrow abcd)_{RPKS} \ ; \ (abc \rightarrow abcd)_{RPKS} \ ; \ (abd \rightarrow abcd)_{PNRKS}\}
\end{align*}

\begin{align*}
       \mathcal{B}_{BACI} &= \{(P \rightarrow KPR)_{a} \ ; \ (N \rightarrow KPNRS)_{a} \ ; \ (R \rightarrow KPR)_{a} \ ; \ (K \rightarrow KPR)_{a} \ ; \\
                          & \phantom{==} (S \rightarrow KRPS)_{a} \ ; \ (PR \rightarrow KPR)_{a} \ ; \ (PK \rightarrow KPR)_{a} \ ; \ (RK \rightarrow KPR)_{a} \ ; \\
                          & \phantom{==} (P \rightarrow KP)_{b} \ ; \ (S \rightarrow KPNRS)_{b} \ ; \ (P \rightarrow KPNRS)_{c} \ ; \ (S \rightarrow KPNRS)_{c} \ ; \\
                          & \phantom{==} (PS \rightarrow KPNRS)_{c} \ ; \ (P \rightarrow NP)_{d} \ ; \ (N \rightarrow NP)_{d} \ ; \ (R \rightarrow NPR)_{d} \ ; \\
                          & \phantom{==} (K \rightarrow KPNRS)_{d} \ ; \ (S \rightarrow NPS)_{d} \ ; \  (S \rightarrow KPNRS)_{bc} \ ; \ (\emptyset \rightarrow RPK)_{a} \ ; \\
                          & \phantom{==}  (\emptyset \rightarrow PN)_{d} \ ; \ (\emptyset \rightarrow P)_{ad} \ ; \ (P \rightarrow PNRKS)_{acd} \ ; \ (RPK \rightarrow PNRKS)_{abd} \ ; \\  
                          & \phantom{==} (RPK \rightarrow PNRKS)_{abc} \ ; \ (RPKS \rightarrow PNRKS)_{ab} \ ; \ (PNK \rightarrow PNRKS)_{d} \ ; \\ 
                          & \phantom{==} (PNRK \rightarrow PNRKS)_{a} \ ; \ (PNRK \rightarrow PNRKS)_{d} \ ; \ (PNRS \rightarrow PNRKS)_{d} \ ; \\
                          & \phantom{==} (PNKS \rightarrow PNRKS)_{d} \ ; \ (RPKS \rightarrow PNRKS)_{abc} \ ; \ (P \rightarrow PNRKS)_{abcd} \ ; \\
                          & \phantom{==}  (PN \rightarrow PNRKS)_{abcd} \ ; \ (RPK \rightarrow PNRKS)_{abcd}\}
\end{align*}

The absence of a relation between some attributes-conditions-objects in a context can be considered as information that can be described by implications. In what follows, we highlight a few remarkable facts about them. The third assertion describes a rule almost similar to [Conditional decomposition].

In the examples below, we deduce new implications from $`\mathcal{B}_{BCAI}`$.

With a complete set of implications $`\Sigma`$ of $`\mathbb{K}`$, we are sure that all implications valid in $`\mathbb{K}`$ can be derived from $`\Sigma`$. However, we cannot confirm whether there could exist some implications $`\sigma_{i}`$ in $`\Sigma`$ that could be derived from $`\Sigma \setminus \{\sigma_{i}\}`$. Thus, it is important to search for a minimum set of valid implications.

To avoid redundancy, Sebastian Rudolph proved that an incrementor $`P`$ is a pseudo-intent if, for any incrementor $`Q`$, there exists a feature $`R`$ such that $`Q \subseteq R \subseteq P`$ . In triadic contexts, this can be adapted (with respect to simplification logic) as follows: for any relevant quasi-feature $`A{\times}C`$,

• [conditional composition]: if there are some other quasi-features $`A_i{\times}C_i`$, strict subsets of $`A \times C`$, such that $`\underset{i}{\cup} C_i = C`$ and $`A^{(1,2,C)(1,2,C)} = \underset{i}{\cap} A_i^{(1,2,C_i)(1,2,C_i)}`$, then $`\{ (A_i \rightarrow A_i^{(1,2,C_i)(1,2,C_i)} )_{_{C_i}} : \ i \in \{1, \dots, n\} \} \vdash (A \rightarrow A^{(1,2,C)(1,2,C)} )_{_C}.`$

• [Augmentation]: if there is another quasi-feature $`A_1{\times}C_1`$, strict subsets of $`A \times C`$, such that $`C_1 = C`$ and $`A^{(1,2,C)(1,2,C)} = A_1^{(2,1,C_1)(1,2,C_1)}`$, then $`(A_1 \rightarrow A_1^{(1,2,C_1)(1,2,C_1)} )_{_{C_1}} \vdash (A \rightarrow A^{(1,2,C)(1,2,C)} )_{_C}.`$

• [Transitivity]: if there are some other quasi-features $`A_i{\times}C_i`$, such that $`C_i = C, \ A_1=A, \ A_{n-1}^{(1,2,C_{n-1})(1,2,C_{n-1})}=A_n`$ and $`A^{(1,2,C)(1,2,C)} = A_n^{(1,2,C_n)(1,2,C_n)}`$, then $`\{ (A_i \rightarrow A_i^{(1,2,C_i)(1,2,C_i)} )_{_{C_i}} : \ i \in \{1, \dots, n\}\} \vdash (A \rightarrow A^{(1,2,C)(1,2,C)} )_{_C}.`$

To simplify the notation, we can write $`\{A_i{\times}C_i:\ i \in \mathbb{N} \} \vdash A{\times}C`$ instead of

\{ (A_i \rightarrow A_i^{(1,2,C_i)(1,2,C_i)} )_{_{C_i}} : \ i \in \{1, \dots, n\}\} \vdash (A \rightarrow A^{(1,2,C)(1,2,C)} )_{_C}.

\begin{align*}
      & \star \ \mathcal{P}_i(\mathbb{K}) \vdash A{\times}C, \ \text{for all quasi-feature} \ A{\times}C \\
      & \star \star \ \mathcal{P}_i(\mathbb{K})\backslash \{A{\times}C\} \nvdash A{\times}C
\end{align*}

In Example 10, we have

\begin{align*}
        & \emptyset {\times}P \vdash \{a{\times}P; \ b{\times}P; \ d{\times}P\}; \\
        & \emptyset {\times}N \vdash \{a{\times}N; \ b{\times}N\}; \\
        & \emptyset {\times}R \vdash \{b{\times}R; \ c{\times}R\}; \\
        & \emptyset {\times}K \vdash \{b{\times}K; \ d{\times}K\}; \\
        & \{\emptyset {\times}P; \ \emptyset {\times}N\} \vdash \emptyset {\times}PN; \\
        & \{\emptyset {\times}P; \ \emptyset {\times}R; \ \emptyset {\times}K\} \vdash \{ \emptyset {\times}PR; \ \emptyset {\times}PK; \ \emptyset {\times}KR; \ \emptyset {\times}RPK\}; \\
        & \{b{\times}S; \ c{\times}S\} \vdash  bc{\times}S; \\
        & \{c{\times}P; \ c{\times}S\} \vdash  c{\times}PS;  \\  
        & \{\emptyset {\times}P; \ c{\times}P\} \vdash acd{\times}P ; \\
        & \{ab{\times}RPKS; \ c{\times}S\} \vdash abc{\times}RPKS; \\
        &  ad{\times}PNRKS   \vdash  abd{\times}PNRKS.
\end{align*}

Thus,

\begin{align*}
\mathcal{P}_2(\mathbb{K})=&\{\emptyset {\times}P \ ; \ \emptyset {\times}N \ ; \ \emptyset {\times}R \ ; \ \emptyset {\times}K \ ; \ c{\times}P \ ; \ d{\times}R \ ; \ c{\times}K \ ; \ c{\times}N \ ; \ b{\times}S \ ; \ c{\times}S \ ; \\
                   & abd{\times}RPK \ ; \ abc{\times}RPK \ ; \ ad{\times}PNRKS \ ; \ ab{\times}RPKS\}
 % \ ; \ ad{\times}S \ ; \ ad{\times}NR \ ; \\
                   % & \phantom{==}  ad{\times}NK \ ; \ ad{\times}PKS \ ; \ ad{\times}NRK \ ; \ ab{\times}PKS
\end{align*}

Therefore,

\begin{align*}
     \mathcal{B}_{BCAI} & = \{(\emptyset \rightarrow ad)_{P} \ ; \ (\emptyset \rightarrow d)_{N} \ ; \ (\emptyset \rightarrow a)_{R} \ ; \ (\emptyset \rightarrow a)_{K} \ ; \ (c \rightarrow abcd)_{P} \ ; \ (d \rightarrow abd)_{R} \ ; \\ 
                       & \phantom{==} (c \rightarrow abc)_{K} \ ;  \ (c \rightarrow bcd)_{N} \ ;  \ (b \rightarrow abcd)_{S} \ ; \ (c \rightarrow abcd)_{S} \ ; \ (abd \rightarrow abcd)_{RPK} \ ; \\
                       & \phantom{==} (abc \rightarrow abcd)_{RPK} \ ; \ (ad \rightarrow abcd)_{PNRKS} \ ; \ (ab \rightarrow abcd)_{RPKS}\}
                       % \ ; \ (ad \rightarrow abcd)_{S} \ ; (ad \rightarrow abcd)_{NR} \ ; \\
                       % & \phantom{==} (ad \rightarrow abcd)_{NK} \ ; \ (ad \rightarrow abcd)_{PKS} \ ; \ (ad \rightarrow abcd)_{NRK} \ ; \
                     % (ab \rightarrow abcd)_{PKS}
\end{align*}

Similarly, from the relevant quasi-features with respect to $`C`$ in Example 10, we have

\begin{align*}
        & a{\times} \emptyset \vdash \{a{\times}P; \ a{\times}R; \ a{\times}K; \ a{\times}S; \  a{\times}PR; \  a{\times}PK; \  a{\times}RK \}; \\
        & d{\times} \emptyset \vdash \{d{\times}P; \ d{\times}N; \ d{\times}R; \ d{\times}S \}; \\
        & \{a{\times} \emptyset; \ d{\times} \emptyset\}  \vdash \{ad{\times} \emptyset \}; \\
        & \{c{\times}P; \ c{\times}S\} \vdash  c{\times}PS; \\
        & \{b{\times}S; \ c{\times}S\} \vdash  bc{\times}S;  \\
        & a{\times}N  \vdash a{\times}PNRK; \\
        &  d{\times}K \vdash \{ d{\times}PNK; \ d{\times}PNRK ; \ d{\times}PNKS\} \\
        & abcd{\times}P  \vdash  \{abcd{\times}PN; \ abcd{\times}RPK\}        \\
        &  abc{\times}RPK   \vdash  abc{\times}RPKS     \\
\end{align*}

Therefore,

\begin{align*}
  \mathcal{P}_3(\mathbb{K}) &= \{ a{\times}N \ ; \ b{\times}P \ ;\ b{\times}S \ ; \ c{\times}P \ ; c{\times}S \ ; \ d{\times}K \ ; \ a{\times} \emptyset \ ; \ d{\times} \emptyset \ ; \ acd{\times}P \ ; \ abd{\times}PRK \ ; \\
                    & \phantom{==} abc{\times}PRK \ ; \ ab{\times}RPKS \ ; \ d{\times}PNRS \ ; \ abcd{\times}P \}.
\end{align*}

That is,

\begin{align*}
       \mathcal{B}_{BACI} &= \{ (N \rightarrow KPNRS)_{a} \ ; \ (P \rightarrow KP)_{b} \ ; \ (S \rightarrow KPNRS)_{b} \ ; \  (P \rightarrow KPNRS)_{c} \ ; \\
                          & \phantom{==}   (S \rightarrow KPNRS)_{c} \ ; \ (K \rightarrow KPNRS)_{d} \ ; \ (\emptyset \rightarrow RPK)_{a} \ ; \ (\emptyset \rightarrow PN)_{d} \ ; \\ 
                          & \phantom{==}  (P \rightarrow PNRKS)_{acd} \ ; \ (RPK \rightarrow PNRKS)_{abd} \ ; \ (RPK \rightarrow PNRKS)_{abc} \ ; \\
                          & \phantom{==} (RPKS \rightarrow PNRKS)_{ab} \ ; \ (PNRS \rightarrow PNRKS)_{d} \ ; \ (P \rightarrow PNRKS)_{abcd}  \}
\end{align*}

We then examine these bases of implications, paying particular attention to the minimal criterion.

\begin{eqnarray*}
 \mathcal{B}_{BCAI} = \{ (A \rightarrow A^{(1,2,C)(1,2,C)} )_{_C} : A{\times}C \in \mathcal{P}_{2}(\mathbb{K}) \}  \\
 \mathcal{B}_{BACI} = \{ (C \rightarrow C^{(1,3,A)(1,3,A)} )_{_A} : A{\times}C \in \mathcal{P}_{3}(\mathbb{K}) \}
\end{eqnarray*}
```*

</div>

<div class="proof">

*Proof.* According to
Lemma <a href="#lemme valid implication" data-reference-type="ref"
data-reference="lemme valid implication">1</a>, all implications in
$`\mathcal{B}_{BCAI}`$ are valid in the context $`\mathbb{K}`$.
Therefore,
Lemma <a href="#lemme complete set of implications" data-reference-type="ref"
data-reference="lemme complete set of implications">2</a> reassures us
that $`\mathcal{B}_{BCAI}`$ is complete. The non-redundancy comes from
the definition of $`\mathcal{P}_{2}(\mathbb{K})`$. The minimality of
$`\mathcal{B}_{BCAI}`$ follows from the definition of
$`\mathcal{P}_{2}(\mathbb{K})`$ and
Lemma <a href="#lemme minimal base" data-reference-type="ref"
data-reference="lemme minimal base">3</a>. Thus, $`\mathcal{B}_{BCAI}`$
is a minimal base of implications.

The proof is similar for BACI. ◻

</div>

In the following paragraph, we want to construct a base for CAI and ACI.
To do this, we begin by recalling the interplay between an implication
of the form $`A_1 \overset{C}{\rightarrow} A_2`$ and that of the form
$`(A_1 \rightarrow A_2)_{C}`$ as described below:
``` math
\begin{equation}
\label{equivalence CAI et BCAI}
    A_1 \overset{C}{\rightarrow} A_2 \quad \text{if and only if} \quad (A_1 \rightarrow A_2)_{\{c\}}, \quad \forall c \in C
\end{equation}

\begin{eqnarray*}
 \mathcal{B}_{CAI} = \{A \overset{c}{\rightarrow} A^{(1,2,c)(1,2,c)}: \ A{\times}c \in \mathbb{UP}_{2}(\mathbb{K}) \}  \\
 \mathcal{B}_{ACI} = \{C \overset{a}{\rightarrow} C^{(1,3,a)(1,3,a)}: \ C{\times}a \in \mathbb{UP}_{3}(\mathbb{K}) \}
\end{eqnarray*}
```*

</div>

<div class="proof">

*Proof.* The validity and relevance of each implication follow from
Lemma <a href="#lemme valid implication" data-reference-type="ref"
data-reference="lemme valid implication">1</a> and the fact that these
unit quasi-features are relevant. The completeness of the constructed
sets is a corollary of
Remark <a href="#base of CAI" data-reference-type="ref"
data-reference="base of CAI">4</a>. ◻

</div>

<div class="remark">

*Remark 5*. The bases $`\mathcal{B}_{CAI}`$ and $`\mathcal{B}_{ACI}`$
can be made minimal using the simplification logic. We present an
illustration of this statement for CAI in the following example.

</div>

<div id="exple base" class="example">

*Example 11*. From
Example <a href="#exple complete base" data-reference-type="ref"
data-reference="exple complete base">10</a>,
``` math
\begin{align*}
   \mathbb{UP}_2(\mathbb{K})= & \{a{\times}P \ ; \ a{\times}N \ ; \ b{\times}P \ ;\ b{\times}N \ ; \  b{\times}R \ ; \  b{\times}K \ ; \ b{\times}S \ ; \ c{\times}P \ ;\ c{\times}N \ ; \  c{\times}R \ ; \\
                    & \phantom{==} c{\times}K \ ; \ c{\times}S \ ; \ d{\times}P \ ; \ d{\times}R \ ; \ d{\times}K \ ; \ bc{\times}S \ ; \  \emptyset {\times} P \ ; \  \emptyset {\times} N \ ; \\
                    & \phantom{==} \emptyset {\times} R \ ; \  \emptyset {\times} K \}
\end{align*}

\begin{align*}
     \mathcal{B}_{CAI} & = \{a \overset{P}{\rightarrow} ad \ ; \ a \overset{N}{\rightarrow} ad \ ;  \ b \overset{P}{\rightarrow} abd \ ; \ b \overset{N}{\rightarrow} bd \ ;  \ b \overset{R}{\rightarrow} ab \ ; \ b \overset{K}{\rightarrow} ab ; \  b \overset{S}{\rightarrow} abcd \ ; \\ 
      & \phantom{==}    c \overset{P}{\rightarrow} abcd \ ; \ c \overset{N}{\rightarrow} bcd \ ;  \ c \overset{R}{\rightarrow} ac \ ; \ c \overset{K}{\rightarrow} abc \ ; \ c \overset{S}{\rightarrow} abcd \ ; \ d \overset{P}{\rightarrow} ad \ ; \ d \overset{R}{\rightarrow} abd \ ; \\
      & \phantom{==}   d \overset{K}{\rightarrow} ad \ ; \ bc \overset{S}{\rightarrow} abcd \ ; \ \emptyset \overset{P}{\rightarrow} ad \ ; \ \emptyset \overset{N}{\rightarrow} d \ ; \ \emptyset \overset{R}{\rightarrow} a \ ; \ \emptyset \overset{K}{\rightarrow} a \}
\end{align*}

\begin{align*}
     \mathcal{B}_{CAI} & = \{b \overset{S}{\rightarrow} abcd \ ; \ c \overset{P}{\rightarrow} abcd \ ; \ c \overset{N}{\rightarrow} bcd \ ; \ c \overset{K}{\rightarrow} abc \ ; \ c \overset{S}{\rightarrow} abcd \ ; \ d \overset{R}{\rightarrow} abd \ ; \\
      & \phantom{==}    bc \overset{S}{\rightarrow} abcd \ ; \ \emptyset \overset{P}{\rightarrow} ad \ ; \ \emptyset \overset{N}{\rightarrow} d \ ; \ \emptyset \overset{R}{\rightarrow} a \ ; \ \emptyset \overset{K}{\rightarrow} a \}
\end{align*}

\begin{align*}
     \mathcal{B}_{CAI} & = \{b \overset{S}{\rightarrow} abcd \ ; \ c \overset{P}{\rightarrow} abcd \ ; \ c \overset{N}{\rightarrow} bcd \ ; \ c \overset{K}{\rightarrow} abc \ ; \ c \overset{S}{\rightarrow} abcd \ ; \ d \overset{R}{\rightarrow} abd \ ; \\
      & \phantom{==}  \emptyset \overset{P}{\rightarrow} ad \ ; \ \emptyset \overset{N}{\rightarrow} d \ ; \ \emptyset \overset{R}{\rightarrow} a \ ; \ \emptyset \overset{K}{\rightarrow} a \}
\end{align*}

\begin{align*}
  \mathbb{UP}_3(\mathbb{K})= & \{a{\times}P \ ; \ a{\times}N \ ; \ a{\times}R \ ; \ a{\times}K \ ; \ a{\times}S \ ; \  a{\times}PR \ ; \  a{\times}PK \ ; \  a{\times}RK \ ; \ b{\times}P \ ; \\
                      & \phantom{==} b{\times}S \ ; \ c{\times}P \ ; \ c{\times}S \ ; \ c{\times}PS \ ; \ d{\times}P \ ; \ d{\times}N \ ; \ d{\times}R \ ; \ d{\times}K \ ; \ d{\times}S \ ; \ a{\times} \emptyset \ ; \\
                      & \phantom{==} d{\times} \emptyset \ ; \ d{\times}PNK \ ;  \ a{\times}PNRK \ ; \ d{\times}PNRK \ ; \ d{\times}PNRS \ ; \ d{\times}PNKS.\}
\end{align*}

\begin{align*}
       \mathcal{B}_{ACI} &= \{P \overset{a}{\rightarrow} KPR \ ; \ N \overset{a}{\rightarrow} KPNRS \ ; \ R \overset{a}{\rightarrow} KPR \ ; \ K \overset{a}{\rightarrow} KPR \ ; \\
                          & \phantom{==} S \overset{a}{\rightarrow} KRPS \ ; \ PR \overset{a}{\rightarrow} KPR \ ; \ PK \overset{a}{\rightarrow} KPR \ ; \ RK \overset{a}{\rightarrow} KPR \ ; \\
                          & \phantom{==} P \overset{b}{\rightarrow} KP \ ; \ S \overset{b}{\rightarrow} KPNRS \ ; \ P \overset{c}{\rightarrow} KPNRS \ ; \ S \overset{c}{\rightarrow} KPNRS \ ; \\
                          & \phantom{==} PS \overset{c}{\rightarrow} KPNRS \ ; \ P \overset{d}{\rightarrow} NP \ ; \ N \overset{d}{\rightarrow} NP \ ; \ R \overset{d}{\rightarrow} NPR \ ; \\
                          & \phantom{==} K \overset{d}{\rightarrow} KPNRS \ ; \ S \overset{d}{\rightarrow} NPS \ ; \ \emptyset \overset{a}{\rightarrow} RPK \ ; \ \emptyset \overset{d}{\rightarrow} PN \ ; \\
                          & \phantom{==} PNK \overset{d}{\rightarrow} PNRKS \ ; \ PNRK \overset{a}{\rightarrow} PNRKS \ ; \ PNRK \overset{d}{\rightarrow} PNRKS \ ; \\
                          & \phantom{==}  PNRS \overset{d}{\rightarrow} PNRKS \ ; \ PNKS \overset{d}{\rightarrow} PNRKS\}
\end{align*}

\begin{align*}
     \mathcal{B}^{op}_{BCAI} & = \{(\emptyset \rightarrow ad)_{P} \ ; \ (\emptyset \rightarrow d)_{N} \ ; \ (\emptyset \rightarrow a)_{R} \ ; \ (\emptyset \rightarrow a)_{K} \ ; \ (c \rightarrow b)_{P} \ ; \ (d \rightarrow b)_{R} \ ; \\ 
                       & \phantom{==} (c \rightarrow ab)_{K} \ ;  \ (c \rightarrow b)_{N} \ ;  \ (b \rightarrow acd)_{S} \ ; \ (c \rightarrow abd)_{S} \}
\end{align*}

\begin{eqnarray*}
 \mathcal{B}_{CAI}^{op} = \{ A \overset{c}{\rightarrow} A^{(1,2,c)(1,2,c)}\backslash A : A{\times}c \in \mathcal{UP}_{2}(\mathbb{K}) \}  \\
 \mathcal{B}_{ACI}^{op} = \{ C \overset{a}{\rightarrow} C^{(1,3,a)(1,3,a)}\backslash C : a{\times}C \in \mathcal{UP}_{3}(\mathbb{K})\}
\end{eqnarray*}
```*

</div>

<div class="proof">

*Proof.* From
Theorem <a href="#theorem base of CAI" data-reference-type="ref"
data-reference="theorem base of CAI">2</a>, $`\mathcal{B}_{CAI}^{op}`$
is a base. Furthermore, $`\mathcal{B}_{CAI}^{op}`$ is built from unit
pseudo-features $`\mathcal{UP}_2(\mathbb{K})`$ with respect to $`M`$;
so, $`\mathcal{B}_{CAI}^{op}`$ is minimal. Moreover, the simplification
logic and the reduction due to the implications described in
Proposition <a href="#propo 100percent-support implication and optimality"
data-reference-type="ref"
data-reference="propo 100percent-support implication and optimality">3</a>
guarantee optimality.

The proof is similar for $`\mathcal{B}_{ACI}^{op}`$. ◻

</div>

In the following examples, we compute the optimal bases for CAI and ACI
in our running context. Achieving a $`70\%`$ reduction rate, we moved
from $`\mathcal{B}_{CAI}`$ and $`\mathcal{B}_{ACI}`$ base to their
respective optimal bases ($`\mathcal{B}_{CAI}^{op}`$ and
$`\mathcal{B}_{ACI}^{op}`$).

<div id="exple optimalbase CAI" class="example">

*Example 12*. In this example, we will compute the set
$`\mathcal{B}_{CAI}^{op}`$ of the context in
Fig. <a href="#tab 1" data-reference-type="ref" data-reference="tab 1">3</a>
(left). We already know that
``` math
\begin{align*}
\mathcal{P}_2(\mathbb{K})=&\{\emptyset {\times}P \ ; \ \emptyset {\times}N \ ; \ \emptyset {\times}R \ ; \ \emptyset {\times}K \ ; \ c{\times}P \ ; \ d{\times}R \ ; \ c{\times}K \ ; \ c{\times}N \ ; \ b{\times}S \ ; \ c{\times}S \ ; \\
                   & abd{\times}RPK \ ; \ abc{\times}RPK \ ; \ ad{\times}PNRKS \ ; \ ab{\times}RPKS\}
\end{align*}

\begin{align*}
\mathcal{UP}_2(\mathbb{K})=&\{\emptyset {\times}P \ ; \ \emptyset {\times}N \ ; \ \emptyset {\times}R \ ; \ \emptyset {\times}K \ ; \ c{\times}P \ ; \ d{\times}R \ ; \ c{\times}K \ ; \ c{\times}N \ ; \ b{\times}S \ ; \ c{\times}S \}.
\end{align*}

\begin{align*}
      \{\emptyset \overset{P}{\rightarrow} ad  ; \ \emptyset \overset{N}{\rightarrow} d  ; \ \emptyset \overset{R}{\rightarrow} a  ; \ \emptyset \overset{K}{\rightarrow} a  ; \ c \overset{P}{\rightarrow} b  ; \ d \overset{R}{\rightarrow} b  ; \ c \overset{K}{\rightarrow} b  ;  \ c \overset{N}{\rightarrow} b  ;  \ b \overset{S}{\rightarrow} acd  ; \  c \overset{S}{\rightarrow} abd\}
\end{align*}

\begin{align*}
      \{\emptyset \overset{P}{\rightarrow} ad \ ; \ \emptyset \overset{N}{\rightarrow} d \ ; \ \emptyset \overset{KR}{\rightarrow} a \ ; \ c \overset{KPN}{\longrightarrow} b \ ; \ d \overset{R}{\rightarrow} b \ ; \ b \overset{S}{\rightarrow} acd \ ; \  c \overset{S}{\rightarrow} abd\}
\end{align*}

\begin{align*}
     \mathcal{B}_{CAI}^{op} & = \{\emptyset \overset{P}{\rightarrow} ad \ ; \ \emptyset \overset{N}{\rightarrow} d \ ; \ \emptyset \overset{KR}{\rightarrow} a \ ; \  c \overset{KPN}{\longrightarrow} b \ ; \ d \overset{R}{\rightarrow} b \ ; \ b \overset{S}{\rightarrow} c \ ; \  c \overset{S}{\rightarrow} abd\}.
\end{align*}

\begin{align*}
  \mathcal{P}_3(\mathbb{K}) &= \{ a{\times}N \ ; \ b{\times}P \ ;\ b{\times}S \ ; \ c{\times}P \ ; c{\times}S \ ; \ d{\times}K \ ; \ a{\times} \emptyset \ ; \ d{\times} \emptyset \ ; \ acd{\times}P \ ; \ abd{\times}PRK \ ; \\
                    & \phantom{==} abc{\times}PRK \ ; \ ab{\times}RPKS \ ; \ d{\times}PNRS \ ; \ abcd{\times}P \}.
\end{align*}

\begin{align*}
  \mathcal{UP}_3(\mathbb{K})=& \{ a{\times}N \ ; \ b{\times}P \ ;\ b{\times}S \ ; \ c{\times}P \ ; c{\times}S \ ; \ d{\times}K \ ; \ a{\times} \emptyset \ ; \ d{\times} \emptyset \ ; \  d{\times}PNRS\}.
\end{align*}

\begin{align*}
       & \{ N \overset{a}{\rightarrow} KPRS \ ; \ P \overset{b}{\rightarrow} K \ ; \ S \overset{b}{\rightarrow} KPNR \ ; \ P \overset{c}{\rightarrow} KNRS \ ; \ S \overset{c}{\rightarrow} KPNR \ ; \\
       & \phantom{==}  K \overset{d}{\rightarrow} PNRS \ ; \ \emptyset \overset{a}{\rightarrow} RPK \ ; \ \emptyset \overset{d}{\rightarrow} PN \ ; \ PNRS \overset{d}{\rightarrow} K\}
\end{align*}

\begin{align*}
       & \{N \overset{a}{\rightarrow} S \ ; \ P \overset{b}{\rightarrow} K \ ; \ S \overset{b}{\rightarrow} KPNR \ ; \ P \overset{c}{\rightarrow} KNRS \ ; \ S \overset{c}{\rightarrow} KPNR \ ; \\
       & \phantom{==}  K \overset{d}{\rightarrow} SR \ ; \ \emptyset \overset{a}{\rightarrow} RPK \ ; \ \emptyset \overset{d}{\rightarrow} PN \ ; \ RS \overset{d}{\rightarrow} K\}
\end{align*}

\begin{align*}
       \mathcal{B}_{ACI}^{op} = & \{N \overset{a}{\rightarrow} S \ ; \ P \overset{b}{\rightarrow} K \ ; \ S \overset{bc}{\rightarrow} KPNR \ ; \ P \overset{c}{\rightarrow} KNRS \ ; \\
                                & \phantom{==} K \overset{d}{\rightarrow} SR \ ; \ \emptyset \overset{a}{\rightarrow} RPK \ ; \ \emptyset \overset{d}{\rightarrow} PN \ ; \ RS \overset{d}{\rightarrow} K\}
\end{align*}

premise \overset{condition}{\longrightarrow} conclusion

📊 논문 시각자료 (Figures)

A Note of Gratitude