Algorithm 5 – Lattice
3.1 Formal concept analysis in the fuzzy setting
3 Fuzzy Formal Concept Analysis
We now introduce the reader to some developments on fuzzy extensions of FCA, here referred to as Fuzzy Formal Concept Analysis. In Sec.3.1we shall presented the ideas and results by using standard fuzzy sets (i. e., with L✏ t0,1✉). The contents in this section have been developed independently by the authors of this text1 and later found to fall within the scope of a broader theory, which we shall briefly discuss in Sec. 3.2.
true, so must be Ψ, but in case Φ is false we have no interest in Ψ3. Hence we may say that a PO✝ iff oPO ÝÑoIa.
By using an analogous argument for A❫ we see that the expressions forO✝ and A❫ can be translated respectively as
χO✝♣˜aq ✏ ❅o PO♣o PO ÝÑoI˜aq , χA❫♣˜oq ✏ ❅aP A♣aPAÝÑ ˜oIaq .
Now recall that in Section 2.3.2 we argued that the universal quantifier can be extended by taking the infimum. As a result, we may define the following.
Definition 3.1.2. Let Cf ✏ ①O,A, If② be a fuzzy formal context and let ñ be a fuzzy implication. Then given fuzzy subsets O ❸O and A❸A we define fuzzy subsets O✝ ❸A and A❫❸O respectively by their membership functions:
µO✝♣aq ✏ inf
oPOrµO♣oq ñ µIf♣o, aqs , (3.1.3) µA❫♣oq ✏ inf
aPArµA♣aq ñµIf♣o, aqs . (3.1.4) Afuzzy formal concept is a pair①O, A② PO✂A such thatO✝ ✏AandA❫✏O.
In order to extend results achieved working out the classical theory, we need to find out what fuzzy connectives preserve the classical properties we want to keep. For instance, remember that item 2. of Theorem 1.1.10 stated that given a formal context C✏ ①O,A, I② and given O ❸O, we have
O ❸O✝❫ .
Considering how the fuzzy sets O✝ and A❫ are defined, we need to do some investigation concerning fuzzy implications. We start by asking ourselves: does continuity preserve the properties we want?
Example 3.1.5. Let Cf ✏ ①O,A, If② be the fuzzy context defined so that O ✏ to1, o2✉, A ✏ ta1, a2✉and If is expressed by
a1 a2
o1 0.2 0.7 o2 0.4 0.6
3 We say that the classical implication isvacuously true, that is, an expression ΦÝÑΨ is true whenever Φ is false. It is also true, of course, when both Φ and Ψ are true, and false only in case Φ is true and Ψ is false.
Let ♣x ñ yq :✏ ♣1✁xq ❴y for each x, y P r0,1s (this is the Kleene-Dienes implication). Consider the fuzzy set O✏ 0.5
o1
0.8 o2
. Then
µO✝♣a1q ✏ inf
oPOrµO♣oq ñµIf♣o, a1qs
✏ r♣1✁0.5q ❴0.2s➞
r♣1✁0.8q ❴0.4s
✏0.5❫0.4✏0.4 and
µO✝♣a2q ✏ inf
oPOrµO♣oq ñµIf♣o, a2qs
✏ r♣1✁0.5q ❴0.7s➞
r♣1✁0.8q ❴0.6s
✏0.7❫0.6✏0.6 , so that O✝ ✏ 0.4
a1
0.6 a2
. Now, we have
µO✝❫♣o2q ✏ inf
aPArµO✝♣aq ñµIf♣o2, aqs
✏ r♣1✁0.4q ❴0.4s➞
r♣1✁0.6q ❴0.6s
✏0.6❫0.6✏0.6 .
Since µO✝❫♣o2q ✏ 0.6➔0.8✏µO♣o2q, we conclude that O ❺O✝❫.
The implication used in this example is continuous as it is the composition of continuous functions (maximum and difference). Thus continuity is not a sufficient condition on a fuzzy implication for us to extend Theorem1.1.10. As we shall see, continuity is not a necessary condition as well. Since item 4. of the aforementioned theorem uses a Cartesian product we are led to think that t-norms may be involved and that we must use their residua.
However, not any residuum solves the problem we are currently tackling. For example, remember that the item 4. of Theorem 1.1.10states that if C✏ ①O,A, I② is a formal context, then given O ❸O and A❸A have
A❸O✝ iff O✂A❸I .
Example 3.1.6. Let Cf ✏ ①to✉,ta✉, If② be a fuzzy formal context in which µIf♣o, aq ✏ 0.4.
Let O ✏0.5④o and let A✏1④a. Let △ be the drastic product t-norm4 and ñthe residuum
4 As introduced in Example 2.3.2,
x△Dy✏
✩✬
✫
✬✪
0, if ♣x❴yq ➔1 x, ify✏1 y, ifx✏1 .
of △. Then
♣µO♣oq ñµIf♣o, aqq ✏ ♣0.5ñ0.4q
✏suptz P r0,1s: 0.5△z ↕0.4✉
✏1
as 0.5△z ✏0 for all z P r0,1r. Thus, O✝ ✏1④a, and we have A ❸O✝. Nevertheless, µO♣oq△µA♣aq ✏ 0.5△1
✏0.5
→0.4✏µIf♣o, aq . Therefore,O✂△A❺If.
We see that O✂△A❺If results from the discontinuity of △. Nonetheless, it turns out that, for many of the results we wish to prove, something weaker than continuity of the t-norm is required: all we need are lower semicontinuous t-norms5.
Theorem 3.1.7. Let C✏ ①O,A, If② be a fuzzy context. Let O, O1, O2 be fuzzy subsets of O and A, A1, A2 be fuzzy subsets of A. Let △ be a lower semicontinuous t-norm, and let ñ be the residuum of △. Then:
1. If O1 ❸O2 then O2✝ ❸O✝1 1’. If A1 ❸A2 then A❫2 ❸A❫1
2. O❸O✝❫ 2’. A❸A❫✝
3. O✝ ✏O✝❫✝ 3’. A❫ ✏A❫✝❫
4. O❸A❫ iff A❸O✝ iff O✂△A❸If
Proof. We shall prove items 1., 2., 3. and 4. Items with a prime can be proved analogously.
1. Suppose thatO1 ❸O2. Let a P A. By hypothesis, µO1♣oq ↕ µO2♣oq for each o P O. Since ñis decreasing in its first component we have
♣µO2♣oq ñµIf♣o, aqq ↕ ♣µO1♣oq ñ µIf♣o, aqq .
Taking the infimum over o on both sides, we have (by definition of O✝1 and O✝2) µO✝
2♣aq ↕ µO✝
1♣aq. Buta PA is arbitrary. Thus O✝2 ❸O1✝.
2. Leto PO. Remember that ñis decreasing in its first component. Thus
µO♣oq ↕ r♣µO♣oq ñµIf♣o, aqq ñ µIf♣o, aqs Lemma 2.4.14
↕✑ inf˜
oPO♣µO♣o˜q ñµIf♣˜o, aqq ñµIf♣o, aq✙
Monotonicity of ñ
✏ rµO✝♣aq ñµIf♣o, aqs .
5 Lower semicontinuous functions were introduced in Def. 2.4.8.
Taking the infimum over a on the right side, we get µO♣oq ↕µO✝❫♣oq. Since oPO is arbitrary, O ❸O✝❫.
3. From item 2., we have O ❸O✝❫. Thus, using 1., O✝❫✝ ❸ O✝. On the other hand, using A✏O✝ in 2’., we have O✝ ❸O✝❫✝. Hence,O✝ ✏O✝❫✝.
4. We first prove thatO ❸ A❫ iff A ❸O✝. Suppose that O ❸ A❫. By 1., A❫✝ ❸ O✝. Using 2’. and transitivity of ❸, we have A ❸ O✝. The proof that A ❸ O✝ implies O ❸A❫ is analogous.
Now we prove thatO ❸A❫ iff O✂△A ❸If. In fact,O ❸A❫ iff for alloP O and for alla PA
µO♣oq ↕µA❫♣oq
✏ inf
˜
aPArµA♣˜aq ñµIf♣o,˜aqqs
↕ rµA♣aq ñ µIf♣o, aqs ,
By Corollary 2.4.12 (and applying commutativity of△) this holds iff µO♣oq△µA♣aq ↕µIf♣o, aq .
Thus O ❸A❫ iff O✂△A❸If.
Notice that, as in the classical case, ♣✝,❫q is a Galois connection between
①F♣Oq,❸② and ①F♣Aq,❸②, so that ✝❫ and ❫✝ are closure operators.
Another interesting matter is that not only the set Bf♣Cfqof fuzzy concepts on Cf is a complete lattice (under an order similar to that of classical concepts): the formulae for finding infima and suprema on this lattice are similar to those of the classical case.
Proposition 3.1.8. Let Cf ✏ ①O,A, If② be a fuzzy context. Let △ be a lower semicontin-uous t-norm, and let ñ be its residuum. Let J be an index set and, for each α P J, let Oα ❸O and Aα ❸A. Then
↔
αPJ
O✝α ✏
✄↕
αPJ
Oα
☛✝
, (3.1.9)
↔
αPJ
A❫α ✏
✄↕
αPJ
Aα
☛❫
. (3.1.10)
Proof. We prove (3.1.9). The proof of (3.1.10) is analogous. LetaP A. For eachα0 PJ we
have
µ♣❨αPJOαq✝♣aq ✏ inf
oPOrµ❨αPJOα♣oq ñµIf♣o, aqs
✏ inf
oPO
✒✂
sup
αPJ
µOα♣oq
✡
ñµIf♣o, aq
✚
↕
✒✂
sup
αPJ
µOα♣oq
✡
ñµIf♣o, aq
✚
↕✏
µOα0♣oq ñµIf♣o, aq✘ ,
as ñis decreasing in the first component. Applying the infimum over oP O and then the infimum over αPJ on the right-hand side yields
µ♣❨αPJOαq✝♣aq ↕µ❳αPJO✝α♣aq . Since aP A is arbitrary, ✄
↕
αPJ
Oα
☛✝
❸ ↔
αPJ
O✝α . Conversely, we want to prove
↔
αPJ
Oα✝ ❸
✄↕
αPJ
Oα
☛✝
. (3.1.11)
Notice that if for all oP O and for all aPA we have µ❳αPJO✝α♣aq ↕✏
µ➈αPJOα♣oq✟
ñµIf♣o, aq✘
, (3.1.12)
then taking the infimum over o P O on the right-hand side of (3.1.12) yields (3.1.11).
Hence it is sufficient to prove (3.1.12).
Suppose for the sake of contradiction that (3.1.12) does not hold, that is, for some o PO and aPA,
κ:✏ inf
αPJ
✑inf
˜
oPO♣µOα♣o˜q ñµIf♣˜o, aqq✙
→
✒✂
sup
αPJ
µOα♣oq
✡
ñµIf♣o, aq
✚ . Define x0 ✏sup
αPJ
µOα♣oq. Let ♣αnq be a sequence on J such that µOαn♣oq✟
converges to x0
and, for each n→0, define xn ✏µOαn♣oq. By hypothesis, for each n→0,
♣x0 ñµIf♣o, aqq ➔ inf
αPJ
✑inf
˜
oPO♣µOα♣˜oq ñµIf♣˜o, aqq✙
♣✏ κq
↕ inf
˜
oPO µOαn♣˜oq ñ µIf♣˜o, aq✟
↕ µOαn♣oq ñµIf♣o, aq✟
✏ ♣xnñµIf♣o, aqq . Thus,
♣x0 ñµIf♣o, aqq ➔κ↕lim sup
nÑ✽ ♣xn ñµIf♣o, aqq ,
and so xÞÑ ♣xñµIf♣o, aqq is not upper semicontinuous, contradicting Proposition2.4.17.
Hence, (3.1.12) holds.
Theorem 3.1.13. Let Cf ✏ ①O,A, If② be a fuzzy formal context. Using a lower semicon-tinuous t-norm and its residuum to define the maps ✝ and ❫, define the order ↕ on the set Bf♣Cfq of all the fuzzy formal concepts of Cf by
①O1, A1② ↕ ①O2, A2② iff O1 ❸O2 ♣iff A2 ❸A1q . (3.1.14) Then LCf :✏ ①Bf♣Cfq,↕② is a complete lattice, called the fuzzy concept lattice of Cf.
If K is an index set and Cκ ✏ ①Oκ, Aκ② PB♣Cq for each κP K then
κinfPKCκ ✏
❈↔
κPK
Oκ,
✄↕
κPK
Aκ
☛❫✝●
, (3.1.15)
sup
κPK
Cκ ✏
❈✄↕
κPK
Oκ
☛✝❫
,↔
κPK
Aκ
●
. (3.1.16)
Proof. Similar to that of the Basic Theorem (Theorem 1.1.18).