Partial formal contexts are trivalued contexts that, besides allowing to establish whether a property is satisfied or not, allow to represent situations in which there is ignorance about whether a property is satisfied. This can be useful, not only for the cases in which the modeled phenomenon has intrinsically unknown information, but also when summarizing information from a formal context by grouping similar rows. In this paper, we prospect for its extension including degrees of knowledge.
topic can be found in [3]. In this paper we extend the research line in [4]. We present a graded
extension of the algebraic framework presented. With this extension, we defined the graded
partial formal contexts that are given by a triple (, , ) where and are the sets with the
objects and the attributes (as usually) and ∶ × → [
Given a 3-set , we call support of to the set Spp() = { ∈ ∣ () ≠ ∘} mappings Neg, Pos, Unk ∶ 3 → 2 where, for each ∈ 3 , . We define also the Neg() = { ∈ ∣ () = −} = Pos() = { ∈ ∣ () = +} = Unk() = { ∈ ∣ () = ∘} = −1(−) −1(+) −1(∘) An equivalent formalisation of 3-sets can be found in [3] where only the known information (Positive or Negative) is given, and is represented as the so-called orthopairs by using sets ( , ) with ∩ = ∅ . When the support of a 3-set is finite, we express it as a sequence of elements (with no delimiters). It can be seen as an ordered re-writing of the concatenation of the elements of the orthopair, where the atoms in are overlined. For instance, = 1 5̄ 7 means that ( 1) = ( 7) = +, ( 5) = −, and () = ∘ otherwise. In particular, when Spp() = ∅ , we denote = (i.e. the empty sequence).
As mentioned above, we interpret a 3-valued set as the knowledge that we have about the properties of certain object. Thus, we have a conjunctive interpretation on these sets and, consequently, when we join two diferent 3-sets, we might find inconsistencies, that is, a property can be found positive in one of the sets and negative in the other set. Then, in the joined set, we have an inconsistent element. The sets 3 and 3 can be extended to model this situation by introducing a fourth element, denoted , which represents inconsistent or contradictory information. In order to obtain a lattice with the order of information, this new element will play the role of the (missing) maximum element of 3. This lattice of four elements + ↖ ↗ − ∘ (a) ∧-semilattice 3 ↗ ↖ ∘ + ↖ ↗ − ↗ (1, 1)
↖ (0, 0) (1, 0) ↖ ↗ (0, 1) (b) lattice (4, ≤) (c) Boolean Algebra 2 × 2 is denoted (4, ≤) and is shown in Fig. 1b; it is isomorphic to the Boolean algebra 2 × 2 (see
In addition, 4 denotes the set of mappings ∶ → 4 or 4-sets, and we assume that 3 is a subset of 4 . In the same way that we did for 3 , the order of 4 is componentwise extended to 4 . Note that the infimum of (4 , ≤) is (the constant mapping to ∘) and the supremum is the
The 4 -sets can be seen as paraconsistent orthopairs [6] but, since in our framework ex contradictione quodlibet still holds, all the orthopairs containing a contradiction are identified and ̇is used as the canonical representative of this class. To formalise it, we define the following closure operator: ∶ 4 → 4 being () = { ̇ if ∈
3 , otherwise.
will be denoted 3̇ ; it consists of the elements in 3 , the consistent orthopairs, together with the oxymoron .̇ It is a closure system in (4 , ≤) and, therefore, it is a ∧-subsemilattice of (4 , ≤) (but not a sublattice) and (3̇ , ⊑) is a complete lattice (see Fig. 2). Since the infimum operation coincides in both structures, they will be denoted by the same symbol ∧, however the supremum in (4 , ≤) will be denoted by ∨, whereas in (3̇ , ⊑) will be denoted by ⊔. Thus, for all { ∶ ∈ } ⊆
3̇ , we have that ⨆ = ( ⋁ ) ∈
∈ ⨆ ≠ ̇ ∈ implies ⨆ = ⋁ . ∈ ∈ (1) and, in particular, and ∩ = ∅
The maximal sets of 3 are called full sets, and the set of all of them is denoted by Full( ) . They are the super-atoms of (3̇ , ⊑) and coincide with those orthopairs ( , ) such that ∪ =
We extend the mappings Neg, Pos, Unk ∶ 3̇ → 2 by considering Pos()̇ = Neg()̇ = , Proposition 1 ([4]). For any , ∈
3̇ , we have that 1. ⊑
if and only if Neg() ⊆ Neg() and Pos() ⊆ Pos() . 1↑ 2↖ 1 ↖ ↗ 1 2 ↖ 2 ↖ ↗ 1 2↖ ↗ 2 ↗ 1↑ 2 ↗ 1 (a) The ∧-semilattice 3{ 1, 2} ↗ ̇ ↖ ↗ ↖ 1↑ 2↖ 1 ↖ ↗ 1 2 ↖ 2 ↖ ↗ 1 2↖ ↗ 2 ↗ 1↑ 2 ↗ 1 (b) The lattice 3̇{ 1, 2} 3. The mappings Pos and Neg restricted to Full( ) are bijections in 2 .
Finally, we define the operation ( ) ∶ 3̇ → 3̇ , which we call opposite, such that =̇ ̇ and, for () =
+ if () = − { − if () = + ∘ if () = ∘ Thus, Neg() = Pos() , Pos() =
Neg() , and Unk() =
Unk() . 2.2. Formal concept analysis for unknown information We start the extension of the FCA framework by presenting the notion of partial formal context, introduced by Ganter in [7], which is defined as a triple ℙ = (, , ) being and non-empty sets and ∶ × → 3. We call the elements of and objects and attributes respectively.
Given ∈ ; (, ) = − , the assignment (, ) = +
means that the object has the attribute means that the object has not the attribute ; and (, ) = ∘ means that we do not know whether the object has the attribute or not. As in the classical case, these contexts are shown as tables (see Figure 3, for instance).
ℙ 1 2 • ℙ+ = (, , +) where + = −1(+), that is + if (, ) = + .
Their derivation operators are denoted by the symbol ⊕, that is, for all ⊆ and ⊆ ⊕ = ⋂ +( ) = { ∈ ∣
∈ ⊕ = ⋂ ( ) + = { ∈ ∣ ∈ +, ∀ ∈ } +, ∀ ∈ } • ℙ− = (, , −) where − = −1(−) and their derivation operators are denoted by the symbol ⊖ and defined in a similar way.
We use these formal contexts to define the derivation operators in the partial formal context as follows. The classical derivation operator is generalised by defining ( )↑ ∶ 2 → 3̇ and ( )↓ ∶ 3̇ → 2 as ↑ = ⋀ (, ), ∈
and ↓ = Pos( ) ⊕ ∩ Neg( ) ⊖ A pair (, ) ∈ 2 × 3̇ is said to be a (formal) concept if ↑ = and ↓ = . As in the classical case, the concepts can be hierarchically ordered as
( 1, 1) ≤ ( 2, 2) if and only if 1 ⊆ 2 (or, equivalently, if 2 ⊑ 1).
Theorem 1 ([4]). Let ℙ = (, , ) be a partial formal context. The couple (↑, ↓) is a Galois connection between the lattices (2 , ⊆) and (3̇ , ⊑) and, therefore, concepts are fix-points of the Galois connection and, if ⋆(ℙ) is the set of all concepts, the pair ( ⋆(ℙ), ≤)is a complete lattice.
In [4] we showed that a classical formal context can be seen as a partial formal context, and partial formal contexts without any unknown information were called total formal context. We also proved the existence of a classical formal context whose concept lattice is isomorphic to that of our partial formal context. Conversely, given a formal context, we proved the existence of a partial formal context whose concept lattice is isomorphic to that of the classic formal context.
For more details for the algebraic background, we refer to [8, 9], and for Formal Concept
Analysis, we refer to [10, 11].
3. Graded extension of the algebraic framework
In this section, we extend the previous algebraic framework to a graded one following the idea
of intuitionistic logic. Hereinafter, we consider a residuated lattice I = ([
We generalise the lattice 4 by considering the residuated lattice I2 = I × I where the ordering and the operations are componentwise extended. Thus, for a universal set , the 4-sets are extended by using grades as I2-sets, i.e. mappings from to I2. Equivalently, a I2-set ∶ → I2 can be seen as a graded paraconsistent orthopair ( +, −) such that +, − ∶ → I and () = ( +(), −()) where +() means the degree in which we know that belongs to and −() means the degree in which we know that does not belong to .
The set of I2-sets will be denoted by (I2) . The operations can be extended from I2 to (I2) as usual: ( ⋆ )() = () ⋆ ()
for all ∈ and all ⋆ ∈ {∧, ∨, ⊗, →}.
We also consider the order relation defined as
1 ⊑ 2 if 1() ≤ 2() for all ∈ , or, equivalently,
1 ⊑ 2 if 1+() ≤ 2+() and 1−() ≤ 2−() for all ∈ . +() ⊗
−() > 0 .
It is easy to see that (I2) = ((I2) , ∧, ∨, ⊗, →, , )̇ is also a residuated lattice where and ̇ are the I2-sets such that () = (0, 0) and (̇ ) = (1, 1) for all ∈ . The I2-set means that we have absolutely no knowledge about it, whereas the set ̇ denotes total contradiction (we have knowledge that all elements belong to the set and, at the same time, do not belong to it).
This leads us to distinguish between consistent and inconsistent sets. We will say that a I2-set on is consistent if +() ⊗
−() = 0 for all ∈ , i.e. its range is contained in ℂ = {( 1, 2) ∈ I2 ∶ 1 ⊗ 2 = 0}. Contrariwise, is inconsistent if there exists ∈ such that
The notion of consistent set is a generalization of the well-known Atanassov intuitionistic
fuzzy set [12], which is a mapping ∶ → [
is equivalent to + ≤ 1 . ∧-semilattice that we will be denoted by ℂ .
Obviously, ℂ is a ∧-subsemilattice of I2 that generalises 3, in the same way that I2 generalises 4. The set of consistent sets on , denoted ℂ , with the componentwise ordering is also a
Following the same scheme as in the non-graded case, we consider equivalent all the inconsistent sets and we identify them with .̇ Thus, we define the closure operator ∶ ( I2) → (I2) where ( ) = { ̇ if ∈ ℂ , otherwise. and denote by ℂ̇ its range that is ℂ ∪ {}̇. Since this set is a closure system, we have that it is a ∧-subsemilattice of (I2) , and also a complete lattice that we denote by ℂ̇ . To distinguish the supremum of (I2) from that of ℂ̇ we will use the symbols ∨ and ⊔ respectively. Thus, given { ∶ ∈ } ⊆ ℂ̇ , ⨆ = ( ⋁ ) ∈ ∈ 4. Graded partial formal contexts We begin by extending the notion of partial formal context. A graded partial formal context is a triple ℙ = (, , )
where and are sets whose elements are called objects and attributes respectively, and is an ℂ-set on × , i.e. ∶ × → ℂ . Thus, for each (, ) ∈ × degree +(, ) is those in which we know that has the attribute and −(, ) , the is the degree in which we know that does not have the attribute . As usual, eventually we use a currifying process and, given ∈
, we consider the ℂ-sets (, .) ∈ ℂ and (., ) ∈ ℂ defined as: (, .)() = (, ) for all ∈ ; (., )() = (, ) for all ∈ .
A first approach to extend the results given for partial formal contexts is to consider the following Galois connection.
Theorem 2. Given a graded partial formal context ℙ = (, , ) , the derivation operators ( )↑ ∶ 2 → ℂ̇ and ( )↓ ∶ ℂ̇
→ 2 defined as ↑ = ⋀ (, ) ∈
and ↓ = { ∈ ∶ ⊑ (, )} form a Galois connection between the lattices 2 and ℂ̇ .
The concept lattice is defined in the usual way from the above Galois connection. Its minimum element is (∅, )̇ and, for any other formal concept ( , ) , if () = (, ) , the value is a degree in which it is known that all the objects in have the attribute , whereas is a degree in which it is known that all the objects in have not the attribute (rephrasing in more mathematical terms, and are, respectively, lower bounds of the degrees of each object of having, respectively not having, the attribute ).
To have a more general framework, a second attempt could be done by considering fuzzy sets of objects. To do so, we must introduce some additional notation.
In a natural way, from a graded partial formal context ℙ = (, , ) , two fuzzy formal contexts, ℙ+ = (, , +) and ℙ− = (, ,
−), can be induced. The derivation operators in ℙ+ and ℙ−
will be denoted by the symbols ⊕ and ⊖ respectively. Thus, given ∈ [
−(, ) ) ∈ ⊕() = ⋀ ( () →
+(, ) ) and ⊖() =
and, given ∈ [
We use these formal contexts to define the derivation operators from a partial formal context as follows.
Theorem 3. Given a graded partial formal context ℙ = (, , )
( ) ∶ [
→ [
From this theorem, which is the direct extension of Theorem 1, we can define formal concepts as fix pairs of the Galois connection and study the corresponding concept lattice. However, the main diference between this theorem and the previous ones is that could be inconsistent. , the derivation operators 5. Conclusions and further works Partial formal contexts are useful to work with formal contexts that have been obtained via a granularisation process from a (classical) formal context, as described in [7]. This framework for the management of formal contexts in which there is missing or unknown information has been extended by considering graded knowledge about whether a property holds. Thus, we consider the degree in which is known that an object has certain property and, on the other hand, the degree in which it is known that this object does not have the property. We conclude with a pair of theorems that establish Galois connections and from which we can generalise the concept lattice and all the machinery of FCA.
A first approach to this extension is provided by Theorem 2 where the induced concepts, leaving aside the case of the minimum concept, are couples ( , ) being a classical set and a consistent ℂ-set of attributes.
The Galois connection established in Theorem 3 induces concepts ( , ) where is fuzzy set of objects and is a I2-set of attribute. Nevertheless, contrariwise to the previous cases (those obtained from Theorems 1 and 2), the I2-set could be inconsistent. As a consequence, a further refinement can be made to identify any concepts inconsistent with the pair ( ↓̇, )̇ , in the same way that the closure operator does. The problem of finding alternative Galois connection between the lattices I and ℂ̇ that will allow to avoid the computation of inconsistent concepts is left as future work.
Further work in the short term is to situate the results obtained from Theorem 2 in the framework of the pattern structures introduced by Ganter and Kuznetsov [13], as well as the results arising from Theorem 3 with the works of Konecny [14] and Dubois et al [15].
The consistency issue is specially relevant when we work with attribute implications, where ex contradictione quodlibet principle must hold. In [4], not only the concept lattice of a partial formal context was introduced, but also the notion of attribute implication, and an axiomatic system proved to be sound and complete. Another interesting direction for future work will be to extend the previous study of implications to the framework defined from Theorem 2 by considering implications where premise and conclusion are ℂ̇ -sets of attributes.
Partially supported by the Spanish Ministry of Science, Innovation, and Universities (MCIU), State Agency of Research (AEI), Junta de Andalucía (JA), Universidad de Málaga (UMA) and European Regional Development Fund (FEDER) through the projects PGC2018-095869-B-I00 and TIN2017-89023-P (MCIU/AEI/FEDER), and UMA2018-FEDERJA-001 (JA/UMA/FEDER). [3] D. Ciucci, D. Dubois, J. Lawry, Borderline vs. unknown: comparing three-valued representations of imperfect information, Intl J of Approximate Reasoning 55 (2014) 1866–1889. [4] F. Pérez-Gámez, P. Cordero, M. Enciso, A. Mora, A new kind of implication to reason with unknown information, Lecture Notes in Computer Science 12733 (2021) 74–90. [5] N. D. Belnap, A useful four-valued logic, in: J. M. Dunn, G. Epstein (Eds.), Modern Uses of
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