Formal L-concepts with Rough Intents‹ Eduard Bartl and Jan Konecny Data Analysis and Modeling Lab Dept. Computer Science, Palacky University, Olomouc 17. listopadu 12, CZ-77146 Olomouc Czech Republic Abstract. We provide a new approach to synthesis of Formal Concept Analysis and Rough Set Theory. In this approach, the formal concept is considered to be a collection of objects accompanied with two collections of attributes—those which are shared by all the objects and those which are possessed by at least one of the objects. We define concept-forming operators for these concepts and describe their properties. Furthermore, we deal with reduction of the data by rough approximation by given equivalence. The results are elaborated in a fuzzy setting. 1 Introduction Formal concept analysis (FCA) [12] is a method of relational data analysis iden- tifying interesting clusters (formal concepts) in a collection of objects and their attributes (formal context), and organizing them into a structure called concept lattice. Numerous generalizations of FCA, which allow to work with graded data, were provided; see [19] and references therein. In a graded (fuzzy) setting, two main kinds of concept forming-operators— antitone and isotone one—were studied [2, 13, 20, 21], compared [7, 8] and even covered under a unifying framework [4, 18]. We describe concept-forming oper- ators combining both isotone and antitone operators in such a way that each formal (fuzzy) concept is given by two sets of attributes. The first one is a lower intent approximation, containing attributes shared by all objects of the concept; the second one is an upper intent approximation, containing those at- tributes which are possessed by at least one object of the concept. Thus, one can consider the two intents to be a lower and upper approximation of attributes possessed by an object. Several authors dealing with synthesis of FCA and Rough Set Theory have noticed that intents formed by isotone and antitone operators (in both, crisp and fuzzy setting) correspond to upper and lower approximations, respectively (see e.g. [15, 16, 24]). To the best of our knowledge, no one has studied concept- forming operators which would provide both approximations being present in one concept lattice. In this papers we present such concept-forming operators, structure of their concepts, and reduction of the data by means of rough approximations by equiv- alences. Due to page limitation we omit proofs of some theorems. ‹ Supported by grant no. P202/14-11585S of the Czech Science Foundation. 2 Preliminaries In this section we summarize the basic notions used in the paper. Residuated Lattices and Fuzzy Sets We use complete residuated lattices as basic structures of truth-degrees. A complete residuated lattice [1, 14, 23] is a struc- ture L “ xL, ^, _, b, Ñ, 0, 1y such that xL, ^, _, 0, 1y is a complete lattice, i.e. a partially ordered set in which arbitrary infima and suprema exist; xL, b, 1y is a commutative monoid, i.e. b is a binary operation which is commutative, asso- ciative, and a b 1 “ a for each a P L; b and Ñ satisfy adjointness, i.e. a b b ď c iff a ď b Ñ c. 0 and 1 denote the least and greatest elements. The partial order of L is denoted by ď. Throughout this work, L denotes an arbitrary complete residuated lattice. Elements a 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 , or LX if the structure of L is to be emphasized. The operations with L-sets are defined componentwise. For instance, the intersection of L-sets A, B P LX is an L-set A X B in X such that pA X Bqpxq “ 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 tApx1 q{x1 , . . . , Apxn q{xn u. An L-set A P LX is called normal if there is x P X such that Apxq “ 1, and it is called crisp if Apxq P t0, 1u for each x P X. Crisp L-sets can be identified with ordinary sets. For a crisp A, we also write x P A for Apxq “ 1 and x R A for Apxq “ 0. 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). For L-relation I P LXˆY we define its transpose I T P LY ˆX as I T py, xq “ Ipx, yq for all x P X, y P Y . The composition operators are defined by ł pI ˝ Jqpx, zq “ Ipx, yq b Jpy, zq, yPY ľ pI Ž Jqpx, zq “ Ipx, yq Ñ Jpy, zq, yPY ľ pI Ż Jqpx, zq “ Jpy, zq Ñ Ipx, yq yPY for every I P LXˆY and J P LY ˆZ . A binary L-relation E is called an L-equivalence if it satisfies IdX Ď E (reflexivity), E “ E T (symmetry), E ˝ E Ď E (transitivity). An L-set B P LY is compatible w.r.t. L-equivalence E P LY ˆY if Bpy1 q b Epy1 , y2 q ď Bpy2 q. for any y1 , y2 P Y . Formal Concept Analysis in the Fuzzy Setting 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.” An L-context may be described as a table with the objects corresponding to the rows of the table, the attributes corresponding to the columns of the table and Ipx, yq written in cells of the table (for an example see Fig. 1). α β γ δ A 0.5 0 1 0 B 1 0.5 1 0.5 C 0 0.5 0.5 0.5 D 0.5 0.5 1 0.5 Fig. 1. Example of L-context with objects A,B,C,D and attributes α, β, γ, δ. 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 ľ ľ AÒ pyq “ Apxq Ñ Ipx, yq, B Ó pxq “ Bpyq Ñ Ipx, yq. (1) xPX yPY Second, the pair xX, Yy of operators X : LX Ñ LY and Y : LY Ñ LX is defined by ł ľ AX pyq “ Apxq b Ipx, yq, B Y pxq “ Ipx, yq Ñ Bpyq. (2) xPX yPY To emphasize that the operators are induced by I, we also denote the op- erators by xÒI , ÓI y and xXI , YI y. Fixpoints of these operators are called formal concepts. The set of all formal concepts (along with set inclusion) forms a com- plete lattice, called L-concept lattice. We denote the sets of all concepts (as well as the corresponding L-concept lattice) by B ÒÓ pX, Y, Iq and B XY pX, Y, Iq, i.e. B ÒÓ pX, Y, Iq “ txA, By P LX ˆ LY | AÒ “ B, B Ó “ Au, (3) B XY pX, Y, Iq “ txA, By P LX ˆ LY | AX “ B, B Y “ Au. For an L-concept lattice BpX, Y, Iq, where B is either B ÒÓ or B XY , 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, (4) IntpX, Y, Iq “ tB P LY | xA, By P BpX, Y, Iq for some Au. When displaying L-concept lattices, we use labeled Hasse diagrams to include all the information on extents and intents. In B ÒÓ pX, Y, Iq, for any x P X, y P Y and formal L-concept xA, By we have Apxq ě a and Bpyq ě b if and only if there is a formal concept xA1 , B1 y ď xA, By, labeled by a{x and a formal concept xA2 , B2 y ě xA, By, labeled by b{y. We use labels x resp. y instead of 1{x resp. 1 {y and omit redundant labels (i.e., if a concept has both the labels a{x and b{x then we keep only that with the greater degree; dually for attributes). The whole structure of B ÒÓ pX, Y, Iq can be determined from the labeled diagram using the results from [2] (see also [1]). In B XY pX, Y, Iq, for any x P X, y P Y and formal L-concept xA, By we have Apxq ě a and Bpyq ď b if and only if there is a formal concept xA1 , B1 y ď xA, By, labeled by a{x and a formal concept xA2 , B2 y ě xA, By, labeled by b{y (see examples depicted in Fig. 2). B, 0.5{β, 0.5{δ 0.5 ‚ {γ ‚ D, 0.5{α ‚ A, 0.5{α, γ ‚ ‚ C, 0.5{β, 0.5{δ A, 0 {β, 0.5{δ ‚ ‚ C, 0.5{β, 0.5{γ D ‚ 0.5 {B ‚ ‚ C, 0 {α 0.5 {C, β, δ ‚ ‚ 0.5{A, B, α 0.5 {A, 0.5{D ‚ ‚ 0.5 {B, 0.5{D ‚ 0 {γ Fig. 2. Concept lattice BÒÓ pX, Y, Iq (left) and BXY pX, Y, Iq (right) of the L-context in Fig. 1. 3 L-rough concepts We consider concept-forming operators induced by L-context xX, Y, Iy defined as follows: Definition 1. Let xX, Y, Iy be an L-context. Define L-rough concept-forming operators as O Y AM “ xAÒ , AX y and xB, By “ B Ó X B for A P LX , B, B P LY . L-rough concept is then a fixed point of xM, Oy, i.e. a O pair xA, xB, Byy P LX ˆ pL ˆ LqY such that AM “ xB, By and xB, By “ A.1 AÒ and AX are called lower intent approximation and upper intent approximation, respectively. That means, M gives intents w.r.t. both xÒ, Óy and xX, Yy; O then gives inter- section of extents related to the corresponding intents. We denote the set of all fixed-points of xM, Oy, in correspondence with (3), as B MO pX, Y, Iq and call it L-rough concept lattice. Below, we present an analogy of the Main theorem on concept lattices for L-rough setting. Theorem 1 (Main theorem on L-rough concept lattices). (a) L-rough concept lattice B MO pX, Y, Iq is a complete lattice with suprema and infima defined as follows ľ č ď č OM xAi , B i , B i y “ x Ai , x B i , B i y y, i i i i ł ď č ď MO xAi , B i , B i y “ xp Ai q , Bi, B i y. i i i i (b) Moreover, a complete lattice V “ xV, ďy is isomorphic to B MO pX, Y, Iq iff there are mappings γ :X ˆLÑV and µ:Y ˆLˆLÑV such that γpX ˆLq is supremally dense in V, µpY ˆLˆLq is infimally dense in V, and a b b ď Ipx, yq and Ipx, yq ď a Ñ b is equivalent to γpx, aq ď µpy, b, bq for all x P X, y P Y, a, b, b P L. When drawing a concept lattice we label nodes as in B ÒÓ for lower intent approximations and B XY for upper intent approximations. We write a{y or a{y instead of just a{y to distinguish them. Fig. 3 (middle) shows an L-rough concept lattice for the L-context from Fig. 1. The following theorem explains that normal extents have natural intent ap- proximations; that is B Ď B. 1 In what follows, we naturally identify xA, xB, Byy with xA, B, By. Theorem 2. For normal A P LX , we have AÒ Ď AX , for crisp singleton A P LX , we have AÒ “ AX . Proof. Since A is normal, there is x1 P X such that Apx1 q “ 1. Then we have ľ AÒ pyq “ Apxq Ñ Ipx, yq ď Apx1 q Ñ Ipx1 , yq “ Ipx1 , yq “ xPX ł (5) “ Apx1 q b Ipx1 , yq ď Apxq b Ipx, yq “ AX pyq xPX for each y P Y . For A being a crisp singleton, one can show AÒ “ AX by changing all inequal- ities in (5) to equalities. \ [ Since xM, Oy is defined via xÒ, Óy and xX, Yy, one can expect that there is a strong relationship between the associated concept lattices. In the rest of this section, we summarize them. Theorem 3. For S Ď LX , let rSs denote an L-closure span of S, i.e. the small- est L-closure system containing S. We have rExtÒÓ pX, Y, Iq Y ExtXY pX, Y, Iqs “ ExtMO pX, Y, Iq. O Proof. “Ď”: Let A P ExtÒÓ pX, Y, Iq. Then A “ AXX “ xAÒ , Y y P ExtMO pX, Y, Iq. Similarly for A P ExtXY pX, Y, Iq. “Ě”: Let A P ExtMO pX, Y, Iq and let xB1 , B2 y “ AM . Then we have A “ B Ó X B Y P rExtÒÓ pX, Y, Iq Y ExtXY pX, Y, Iqs since B Ó P ExtÒÓ pX, Y, Iq and B Y P ExtXY pX, Y, Iq. From Theorem 3 one can observe that no extent from ExtÒÓ pX, Y, Iq and ExtXY pX, Y, Iq is lost. Corollary 1. ExtÒÓ pX, Y, Iq Ď ExtMO pX, Y, Iq and ExtXY pX, Y, Iq Ď ExtMO pX, Y, Iq. In addition, no concept is lost. Corollary 2. For each xA, By P B ÒÓ pX, Y, Iq there is xA, B, AX y P B MO pX, Y, Iq. For each xA, By P B XY pX, Y, Iq there is xA, AÒ , By P B MO pX, Y, Iq. Remark 1. One can observe from Fig. 3 that in ExtMO pX, Y, Iq there exist ex- tents which are present neither in ExtÒÓ pX, Y, Iq nor in ExtXY pX, Y, Iq. On the other hand, lower intent approximations are exactly those from IntÒÓ pX, Y, Iq and upper intent approximations are exactly those from IntXY pX, Y, Iq. With results on mutual reducibility from [8] we can state the following the- orem on representation of B MO by B ÒÓ . 0.5 {γ, 0.5{β, 0.5{δ ‚ 0.5 {α, 1{γ 0.5 0.5 0.5 {β, {δ ‚ ‚ ‚ 0.5{α {γ ‚ B, 0.5{β, 0.5{δ ‚ ‚ ‚ ‚ A, 0.5{α, γ ‚ ‚ C, 0.5{β, 0.5{δ 0.5 {γ D, 0.5{α ‚ ‚ ‚ ‚ A, 0{β, 0.5{δ D D ‚ A, 0{β, 0.5{δ ‚ ‚ 0.5{γ 1 B, {α ‚ ‚ ‚ C, 0{α 0.5 {C, β, δ ‚ ‚ 0.5{A, B, α 1 0.5 {β, 1{δ {B ‚ ‚ C, 0{α ‚ ‚ ‚ ‚ 0.5 0.5 0.5 {A {A, 0.5{D ‚ {B, 0.5{D 0.5 {B ‚ ‚ ‚ ‚ 0.5 0.5 {D ‚ ‚ 0.5{C, 0{γ {C, 0{γ ‚ Fig. 3. BMO pX, Y, Iq (middle) and positions of original concepts in BÒÓ pX, Y, Iq (left) and BXY pX, Y, Iq (right) with L being a three-element Lukasiewicz chain Theorem 4. For a L-context xX, Y, Iy, consider the L-context xX, Y ˆ L, Jy where J is defined by # Ipx, yq if a “ 1, Jpx, xy, ayq “ Ipx, yq Ñ a otherwise. Then we have that B ÒÓ pX, Y ˆ L, Jq is isomorphic to B MO pX, Y, Iq as a lattice. In addition, ExtÒÓ pX, Y ˆ L, Jq “ ExtMO pX, Y, Iq. Proof (sketch). In [8] we show that for L-contexts xX, Y, Iy and xX, Y ˆ Lzt1u, Jy such that Jpx, xy, ayq “ Ipx, yq Ñ a it holds that ExtXY pX, Y, Iq “ ExtÒÓ pX, Y ˆ Lzt1u, Jq. Using this fact, one can check that mapping i defined as 1 ipxA, B, Byq ÞÑ xA, B 1 Y B y, 1 where B 1 P LY ˆt1u , B P LY ˆLzt1u with B 1 pxy, 1yq “ Bpyq, 1 B pxy, ayq “ Bpyq Ñ a, is the desired isomorphism from B MO pX, Y, Iq to B ÒÓ pX, Y ˆ L, Jq. Theorem 4 shows how we can obtain a concept lattice formed by xÒ, Óy which is isomorphic to L-rough concept lattice of given L-context. 4 Rough approximation of an L-context and L-concept lattice In [17] Pawlak introduced Rough Set Theory where uncertain elements are ap- proximated with respect to an equivalence relation representing indiscernibility. Formally, given Pawlak approximation space xU, Ey, where U is a non-empty set of objects (universe) and E is an equivalence relation on U , the rough approx- imation of a crisp set A Ď U by E is the pair xAóE , AòE y of sets in U defined by x P A óE iff for all y P U, xx, yy P E implies y P A, òE xPA iff there exists y P U such that xx, yy P E and y P A. AóE and AòE are called lower and upper approximation of the set A by E, respectively. In the fuzzy setting, one can generalize xAóE , AòE y as in [10, 11, 22], ľ AóE pxq “ pEpx, yq Ñ Apyqq, yPU ł òE A pxq “ pApyq b Epx, yqq yPU for L-equivalence E P LU ˆU and L-set A P LU . Considering L-context xU, U, Ey, we can easily see that óE is equivalent to YE ; and òE is equivalent to XET . Since E is symmetric, we can also write xóE , òE y “ xYE , XE y. (6) Note that for L-set A, AóE is its largest subset compatible with E and AòE is its smallest superset compatible with E. Below, we deal with situation where lower and upper intent approximations are further approximated using Pawlak’s approach. In other words, instead of lower intent approximation AÒ we consider the largest subset of AÒ compatible with a given indiscernibility relation E, and similarly, instead of upper intent approximation AX we consider its smallest superset compatible with E. In The- orem 5 we show how to express this setting using L-rough concept forming operators. Definition 2. Let xX, Y, Iy be an L-context, E be an L-equivalence on Y . Define L-rough concept-forming operators as follows: AME “ xAÒóE , AXòE y, OE óE Y xB, By “ B òE Ó X B . Directly from (6) and results in [5] we have: Theorem 5. Let xX, Y, Iy be an L-context, E be an L-equivalence on Y . We have OE YI˝E AME “ xAÒIŻE , AXI˝E y and xB, By “ B ÓIŻE X B . Again, for normal extents we obtain natural upper and lower intent approx- imations. Theorem 6. For normal A P LX we have AÒIŻE Ď AXI˝E . In correspondence with (3) and (4), we denote set of the set of fixpoints of xME , OE y in L-context xX, Y, Iy by B MEOE pX, Y, Iq and set of its extents and intents by ExtMEOE pX, Y, Iq and IntMEOE pX, Y, Iq, respectively. The following theorem shows that a use of a rougher L-equivalence relation leads to a reduction of size of the L-rough concept lattices. Furthermore, this reduction is natural, i.e. it preserves extents. Theorem 7. Let xX, Y, Iy be an L-context, and E1 , E2 be L-equivalences on Y , such that E1 Ď E2 . Then ExtME2OE2 pX, Y, Iq Ď ExtME1OE1 pX, Y, Iq. Example 1. Fig. 4 shows L-rough concept lattice of the L-context in Fig. 1 and rough L-concept lattice approximated using the following L-equivalence relation on Y . α β γ δ α 1 0.5 0 0 β 0.5 1 0 0 γ 0 0 1 0.5 δ 0 0 0.5 1 To demonstrate Theorem 7, the concepts with the same extents in the two lattices are connected. 5 Conclusions and further research We proposed a novel approach to synthesis of RTS na FCA. It provides a lot of directions to be further explored. Our future research includes: Study of attribute implications using whose semantics is related to the present setting. That will combine results on fuzzy attribute implications [9] and at- tribute containment formulas [6]. Generalization of the current setting. Note that the operators Ò and X which compute the universal and the existential intent, need not be induced by the same relation to keep most of the described properties. Actually, this feature is used in Section 4. In our future research, we want to elaborate more on this. For instance, it can provide interesting solution of problem of missing values in a formal fuzzy context—the idea is to use Ò induced by the context with missing values substituted by 0, and X induced by the context with missing values substituted by 1. Reduction of L-rough concept lattice via linguistic hedges As two intents are considered in each L-rough concept, the size of concept lattice can grow very large. The RST approach to reduction of data, i.e. use of rougher L-relation, directly leads to reduction of L-rough concept lattice as we showed in Theorem 7. FFCA provides other ways to reduce the size, one of them is parametrization of concept-forming operators using hedges. 0.5 {γ, 0.5{β, 0.5{δ 0.5 {γ, 0.5{, 0.5{δ ‚ ‚ 0.5 {α, 1{γ A, 0.5{α, 0.5{β 0.5 0.5 {β, {δ ‚ ‚ ‚ 0.5 {α ‚ ‚ 0.5{δ B, 0.5{α, 0.5{β, γ ‚ ‚ ‚ ‚ ‚ 0.5 {γ C, 0.5{γ ‚ ‚ ‚ A, {β, 0 0.5 ‚ ‚ D {δ D 0 {δ 1 0 B, {α ‚ ‚ ‚ C, {α ‚ 1 {β, 1{δ 0.5 {A, 0{α, 0{β ‚ ‚ ‚ ‚ ‚ 1{δ 0.5 0.5 {A {B, 1{α, 1{β 0.5 {B ‚ ‚ ‚ ‚ ‚ 0.5 {C, 0{γ 0.5 {D ‚ ‚ 0.5 {C, {γ 0 ‚ ‚ 0.5{D ‚ ‚ Fig. 4. Rough L-concept lattices BMO pX, Y, Iq (left) and BMEOE pX, Y, Iq (right) with L being three-element Lukasiewicz chain. The corresponding extents are connected. References 1. Radim Belohlavek. 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