We enrich concept-forming operators in L-rough Concept Analysis with linguistic hedges which model semantics of logical connectives 'very' and 'slightly'. Using hedges as parameters for the concept-forming operators we are allowed to modify our uncertainty when forming concepts. As a consequence, by selection of these hedges we can control the size of concept lattice.
In [
First, we work with uncertain information. To do this we extend formal fuzzy contexts to contain two truth-degrees for each object-attribute pair. The two truth-degrees represent necessity and possibility of the fact that an object has an attribute. The interval between these degrees represents the uncertainty presented in a given data.
Second, we parametrize the concept-forming operators used in the framework by unary operators called truth-stressing and truth-depressing linguistic hedges. Their intended use is to model semantics of statements ‘it is very sure that this attribute belongs to a fuzzy set (intent)’ and ‘it is slightly possible that an attribute belongs a fuzzy set (intent)’, respectively. In the paper, we demonstrate how the hedges influence the size of concept lattice.
In this section we summarize the basic notions used in the paper.
We use complete residuated lattices as basic structures of truth-degrees.
A complete residuated lattice [
Elements of L are called truth degrees. Operations b (multiplication) and Ñ (residuum) play the role of (truth functions of) “fuzzy conjunction” and “fuzzy implication”. Furthermore, we define the complement of a P L as a “ a Ñ 0.
An L-set (or fuzzy set) A in a universe set X is a mapping assigning to each x P X some truth degree Apxq P L. The set of all L-sets in a universe X is denoted LX.
The operations with L-sets are defined componentwise. For instance, the intersection of L-sets A, B P LX is an L-set AXB in X such that pAXBqpxq “ Apxq^ Bpxq for each x P X. An L-set A P LX is also denoted tApxq{x | x P Xu. If for all y P X distinct from x1, . . . , xn we have Apyq “ 0, we also write tApx1q{x1, . . . , Apxnq{xnu.
An L-set A P LX is called normal if there is x P X such that Apxq “ 1. An L-set A P LX is called crisp if Apxq P t0, 1u for each x P X. Crisp L-sets can be identified with ordinary sets. For a crispA, we also write x P A for Apxq “ 1 and x < A for Apxq “ 0.
For A, B P LX we define the degree of inclusion of A in B by SpA, Bq “ ŹxPX Apxq Ñ Bpxq. Graded inclusion generalizes the classical inclusion relation. Described verbally, SpA, Bq represents a degree to which A is a subset of B. In particular, we write A Ď B iff SpA, Bq “ 1. As a consequence, we have A Ď B iff Apxq ď Bpxq for each x P X.
By L´1 we denote L with dual lattice order. An L-rough set A in a universe X is a pair of L-sets A “ xA, Ay P pL ˆ L´1qU. The A is called an lower approximation of A and the A is called a upper approximation of A.1
The operations with L-rough sets are again defined componentwise, i.e. čxA, Ay “ xč A, č´1Ay “ xč A, ď Ay, iPI iPI iPI iPI iPI ďxA, Ay “ xď A, ď´1Ay “ xď A, č Ay. iPI iPI iPI iPI iPI Similarly, the graded subsethood is then applied componentwise
SpxA, Ay, xB, Byq “ SpA, Bq ^ S´1pA, Bq “ SpA, Bq ^ SpB, Aq 1 In our setting we consider intents to be L-rough sets; the lower and upper approximation are interpreted as necessary intent and possible intent, respectively. and the crisp subsethood is then defined using the graded subsethood: xA, Ay Ď xB, By iff SpxA, Ay, xB, Byq “ 1, iff A Ď B and B Ď A.
An L-rough set xA, Ay is called natural if A Ď A.
Binary L-relations (binary fuzzy relations) between X and Y can be thought of as L-sets in the universe X ˆ Y. That is, a binary L-relation I P LXˆY between a set X and a set Y is a mapping assigning to each x P X and each y P Y a truth degree Ipx, yq P L (a degree to which x and y are related by I). L-rough relations are then pL ˆ L´1q-sets in X ˆ Y. For L-relation I P LXˆY we define its inverse I´1 P LYˆX as I´1py, xq “ Ipx, yq for all x P X, y P Y.
An L-context is a triplet xX, Y, Iy where X and Y are (ordinary) sets and I P LXˆY is an L-relation between X and Y. Elements of X are called objects, elements of Y are called attributes, I is called an incidence relation. Ipx, yq “ a is read: “The object x has the attribute y to degree a.”
Consider the following pairs of operators induced by an L-context xX, Y, Iy. First, the pair xÒ, Óy of operators Ò : LX Ñ LY and Ó : LY Ñ LX is defined by AÒpyq “ ľ Apxq Ñ Ipx, yq and BÓpxq “ ľ Bpyq Ñ Ipx, yq.
xPX yPY Second, the pair xX, Yy of operators X : LX Ñ LY and Y : LY Ñ LX is defined by AXpyq “ ł Apxq b Ipx, yq and BYpxq “ ľ Ipx, yq Ñ Bpyq.
xPX yPY
To emphasize that the operators are induced by I, we also denote the operators by xÒI, ÓIy and xXI, YIy.
Fixpoints of these operators are called formal concepts. The set of all formal concepts (along with set inclusion) forms a complete lattice, called L-concept lattice. We denote the sets of all concepts (as well as the corresponding L-concept lattice) by BÒÓpX, Y, Iq and BXYpX, Y, Iq, i.e.
BÒÓpX, Y, Iq “ txA, By P LX ˆ LY | AÒ “ B, BÓ “ Au,
BXYpX, Y, Iq “ txA, By P LX ˆ LY | AX “ B, BY “ Au.
For an L-concept lattice BpX, Y, Iq, where B is either BÒÓ or BXY, denote the corresponding sets of extents and intents by ExtpX, Y, Iq and IntpX, Y, Iq. That is, ExtpX, Y, Iq “ tA P LX | xA, By P BpX, Y, Iq for some Bu,
IntpX, Y, Iq “ tB P LY | xA, By P BpX, Y, Iq for some Au.
An pL1, L2q-Galois connection between the sets X and Y is a pair x f, gy of mappings f : L1X Ñ L2Y, g : L2Y Ñ L1X, satisfying
SpA, gpBqq “ SpB, f pAqq for every A P L1X, B P LY.
2
One can easily observe that the couple xÒ, Óy forms an pL, Lq-Galois connection between X and Y, while xX, Yy forms an pL, L´1q-Galois connection between X and Y.
An L-rough context is a quadruple xX, Y, I, Iy, where X and Y are (crisp) sets of objects and attributes, respectively, and the xI, Iy is a L-rough relation. The meaning of xI, Iy is as follows: Ipx, yq (resp. Ipx, yq) is the truth degree to which the object x surely (resp. possibly) has the attribute y. The quadruple xX, Y, I, Iy is called a L-rough context.
The L-rough context induces two operators defined as follows. LetxX, Y, I, Iy be an L-rough context. Define L-rough concept-forming operators as
AM “ xAÒI , AXI y, xB, ByO “ BÓI X BYI (1) for A P LX, B, B P LY. Fixed points of xM, Oy, i.e. tuples xA, xB, Byy P LX ˆpLˆL´1qY such that AM “ xB, By and xB, ByO “ A, are called L-rough concepts. The B and B are called lower intent approximation and upper intent approximation, respectively.
In [
Truth-stressing hedges were studied from the point of fuzzy logic as logical
connectives ‘very true’, see [
On every complete residuated lattice L, there are two important truthstressing hedges: (i) identity, i.e. a˚ “ a pa P Lq; (ii) globalization, i.e.
a˚ “ " 1, if a “ 1,
0, otherwise.
A truth-depressing hedge is a mapping : L Ñ L such that following conditions are satisfied a ď b implies a ď b , a
“ a
for each a, b P L. A truth-depressing hedge is a (truth function of) logical
connective ‘slightly true’, see [
On every complete residuated lattice L, there are two important truthdepressing hedges: (i) identity, i.e. a “ a pa P Lq; (ii) antiglobalization, i.e.
a “ " 0, if a “ 0,
1, otherwise . ˚G ˚1 ˚2 ˚3 ˚4 ˚5 ˚6 G 1 2 3 4 5 6 id id 1 0.75 0.5 0.25 0 1 0.75 0.5 0.25 0 Fig. 1. Truth-stressing hedges (top) and truth-depressing hedges (bottom) on 5-element chain with Łukasiewicz operations L “ xt0, 0.25, 0.5, 0.75, 1u, min, max, b, Ñ, 0, 1y. The leftmost truth-stressing hedge ˚G is the globalization, leftmost truth-depressing hedge G is the antiglobalization. The rightmost hedges denoted by id are the identities.
For truth-stressing/truth-depressing hedge ˚ we denote by fixp˚q set of its idempotent elements in L; i.e. fixp˚q “ ta P L | a˚ “ au.
Let ˚1, ˚2 be truth-stressing hedges on L such that fixp˚1q Ď fixp˚2q; then for each a P A, a˚1˚2 “ a˚1 holds. The same holds true for ˚1, ˚2 being truthdepressing hedges.
We naturally extend application of truth-stressing/truth-depressing hedges to L-sets: A˚pxq “ Apxq˚ for all x P U. 3
The L-rough concept-forming operator M gives for each L-set of objects two L-sets of attributes. The first one represents a necessity of having the attributes and second one a possibility of having the attributes. We add linguistic hedges to the concept-forming operators to control shape of the two L-sets.
Since the L-rough concept-forming operators are defined viaxÒ, Óy and xX, Yy,
we first recall the parametrization of these operators as described in [
Let xX, Y, Iy be an L-context and let r, q be truth-stressing hedges on L. The antitone concept-forming operators parametrized by r and q induced by I are defined as for all A P LX, B P LY.
Let r and ♠ be truth-stressing hedge and truth-depressing hedge on L, respectively. The isotone concept-forming operators parametrized by r and ♠ induced by I are defined as
AÒr pyq “ BÓq pxq “ ľ Apxqr Ñ Ipx, yq, xPX ľ Bpyqq Ñ Ipx, yq yPY AXr pyq “ BY♠ pxq “ ł Apxqr b Ipx, yq, xPX ľ Ipx, yq Ñ Bpyq♠ yPY for all A P LX, B P LY.
Properties of the hedges in the setting of multi-adjoint concept lattices with
heterogeneous conjunctors were studied in [
Let r, q be truth-stressing hedges on L and let ♠ be a truth-depressing hedge on L. We parametrize the L-rough concept-forming operators as
AN “ xAÒr , AXr y and xB, ByH “ BÓq X BY♠ (3) for A P LX, B, B P LY.
Remark 1. When the all three hedges are identities the pair xN, Hy is equivalent to xM, Oy; so it is an pL, L ˆ L´1q-Galois connection. For arbitrary hedges this does not hold.
The following theorem describes properties of xN, Hy.
Theorem 1. The pair xN, Hy of L-rough concept-forming operators parametrized by hedges has the following properties. (a) AN “ ArM “ ArN and xB, ByH “ xBq, B♠yO “ xBq, B♠yH (b) AM Ď AN and xB, ByO Ď xB, ByH (c) SpA1r, A2rq ď SpA2N, A1Nq and SpxB1, B1y, xB2, B2yq ď SpxB2, B2yH, xB1, B1yHq (d) Ar Ď ANH and xBq, B♠y Ď xB, ByHN; (e) A1 Ď A2 implies AN
2 Ď A1N and xB1, B1y Ď xB2, B2y implies xB2, B2yH Ď xB1, B1yH (f) SpAr, xB, ByHq “ SpxBq, B♠y, ANq (g) pŤiPI AirqN “ ŞiPI AiN and pxŤiPI Biq, ŞiPI Bi♠yqH “ ŞiPIxBi, BiyH (h) ANH “ ANHNH and xB, ByHN “ xB, ByHNHN.
Proof. (a) Follows immediately from definition of N and H and idempotency of hedges.
(b) From (2) we have Ar Ď A; by properties of Galois connections the inclusion implies AM Ď ArM, which is by (a) equivalent to AM Ď AN. Proof of the second statement in (b) is similar.
(c) Follows from (a) and properties of Galois connections.
(d) By [2, Corollary 1(a)] we have Ar Ď ArMO. Using (a) we get Ar Ď ANO and from (b) we have ANO Ď ANH, so Ar Ď ANH. Similarly for the second claim.
(e) Follows directly from [2, Corollary 1(c)] and properties of Galois connections.
(f) Since xM, Oy forms pL, L ˆ L´1q-Galois connection and using (a) we have SpAr, xB, ByHq “ SpAr, xBq, B♠yOq “ SpxBq, B♠y, ArMq “ SpxBq, B♠y, ANq.
(g) We can easily get and pď AirqN “ xpď AirqÒr , pď AirqXr y “ xč AiÒr , ď AiXr y iPI iPI iPI iPI iPI “ čxAiÒr , AiXr y “ č AiN, iPI
iPI pxď Biq, č Bi♠yqH “ pď BiqqÓq X pč Bi♠qY♠ “ iPI iPI iPI iPI č BiÓq X iPI č BiY♠ (h) Using (a), (d) and (e) twice, we have ANH Ď ANHNH. Using (d) for xB, By “ AN we have ANr Ď ANrHN “ ANHN. Then applying (e) we get ANHNH Ď ANH proving the first claim. The second claim can be proved analogically.
The set of fixed points of xN, Hy endowed with partial order ď given by xA1, B1, B1y ď xA2, B2, B2y iff A1 Ď A2 iff xB1, B1y Ď xB2, B2y is denoted by BrNH,q,♠pX, Y, I, Iq.
Remark 2. Note that from (4) it is clear that if a concept has non-natural L-rough intent then all its subconcepts have non-natural intent. If such concepts are not desired, one can simply ignore them and work with the iceberg lattice of concepts with natural L-rough intents.
The next theorem shows a crisp representation of BrNH,q,♠pX, Y, I, Iq.
NH Theorem 2. Br,q,♠pX, Y, I, Iq is isomorphic to ordinary concept lattice BÒÓpXˆfixprq, Yˆ fixpqq ˆ fixp♠q, Iˆq where
xxx, ay, xy, b, byy P Iˆ iff a b b ď Ipx, yq and a Ñ b ě Ipx, yq.
Proof. This proof can be done by following the same steps as in [
The following theorem explains the structure of BrNH,q,♠pX, Y, I, Iq.
NH Theorem 3. Br,q,♠pX, Y, I, Iq is a complete lattice with suprema and infima defined as
AN “ xpAÒqq, pAXq♠y and xB, ByH “ pBÓ X BYqr AN “ xpAÒr qq, pAXr q♠y and xB, ByH “ xB, ByO
AN “ AM and xB, ByH “ pBÓq X BY♠ qr we obtain an isomorphic concept lattice. In addition (5) and (6) produce the same concept lattice.
This part provides analogous results on reduction with truth-stressing and
truthdepressing hedges as [
For the next theorem we need the following lemma.
Lemma 1. Let r, ♥, q, ♦ be truth-stressing hedges on L such that fixprq Ď fixp♥q, fixpqq Ď fixp♦q; let ♠, s be truth-depressing hedges on L such that and fixp♠q Ď fixpsq. We have
AN♥ Ď ANr and xB, ByH♦,s Ď xB, ByHq,♠ .
Proof. We have Ar♥ Ď A♥ from (2). From the assumption fixprq Ď fixp♥q we get Ar♥ “ Ar; whence we have Ar Ď A♥. Theorem 1(e) implies A♥N Ď ArN which is by the claim (a) of this theorem equivalent to AN♥ Ď ANr . The second claim can be proved similarly. \[ Theorem 4. Let r, ♥, q, ♦ be truth-stressing hedges on L such that fixprq Ď fixp♥q, fixpqq Ď fixp♦q; let ♠, s be truth-depressing hedges on L s.t. and fixp♠q Ď fixpsq, for all L-rough contexts xX, Y, I, Iy.
In addition, if r “ ♥ “ id, we have Similarly, if q “ ♦ “ ♠ “ s “ id, we have
ExtrNH,q,♠pX, Y, I, Iq Ď Ext♥NH,♦,spX, Y, I, Iq.
IntrNH,q,♠pX, Y, I, Iq Ď Int♥NH,♦,spX, Y, I, Iq.
Proof. (4) follows directly from Theorem 2 and results on subcontexts in [
Now, we show (4). Note that each A P ExtrNH,q,♠pX, Y, I, Iq we have
A “ ANrHq,♠ “ AN♥Hq,♠ Ě AN♥H♦,s Ě A. Example 1. Consider the truth-stressing hedges ˚G, ˚1, ˚2, id and truth-depressing hedges G, 1, 2, id from Figure 1. One can easily observe that fixp˚Gq Ď fixp˚1q Ď fixp˚2q Ď fixpidq fixp Gq Ď fixp 1q Ď fixp 2q Ď fixpidq.
Consider the L-context of books and their graded properties in Fig. 2 with L being 5-element Łukasiewicz chain. Using various combinations of the hedges we obtain a smooth transition in size of the associated fuzzy rough concept lattice going from 10 concepts up to 498 (see Tab. 1). When the 5-element G o¨del chain is used instead, we again get a transition going from 10 concepts up to 298 (see Tab. 2). 1 2 3 4 5 6
We have shown that the L-rough concept-forming operators can be parameterized by truth-stressing and truth-depressing hedges similarly as the antitone and isotone fuzzy concept-forming operators.
Our future research includes a study of attribute implications using whose
semantics is related to the present setting. That will combine results on fuzzy
attribute implications [
Supported by grant No. 15-17899S, “Decompositions of Matrices with Boolean and Ordinal Data: Theory and Algorithms”, of the Czech Science Foundation.