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Extended representation of the conceptual element in temporal context and the diachronism of the knowledge system
Knowledge-Based Systems 33 (2012) 136–144
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Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys
Extended representation of the conceptual element in temporal context and the diachronism of the knowledge system Jian Zhao a,b, Lei Liu a, Liang Hu a,⇑ a b
College of Computer Science and Technology, Jilin University, Changchun 130012, PR China College of Software, Changchun Institute of Technology, Changchun 130012, PR China
a r t i c l e
i n f o
Article history: Received 27 July 2011 Received in revised form 8 March 2012 Accepted 8 March 2012 Available online 15 March 2012 Keywords: Granular computing Formal concept analysis Rough set Diachronism Knowledge system
a b s t r a c t Based on granular computing, the conceptual element was established as the conceptual category of knowledge, with formal concept analysis as a basic tool, and the indiscernibility relation was established for rough set theory. The conceptual element in temporal context was then extended to the phenomenon element yielding temporal granular representation. Through the introduction of partially ordered timeseries, the diachronism of the knowledge system was studied. An isomorphic mapping was conducted between the sequence structure and concept lattice structure, consisting of a phenomenon element and other time-series materials, providing visualization of the movement of the phenomenon element over time. Dynamic modeling was applied to knowledge granules in the open problem domain of context and time, providing a theoretical basis for reference. 2012 Published by Elsevier B.V.
1. Introduction In general, knowledge is considered to be formed by information gathering, processing and interpretation, from the vast and complex information environment. The source of knowledge is usually taken as conceptual descriptions of the visual experience of objective targets, made by observers in the real world, such as descriptions of observed objects like celestial bodies or atoms, their structure and laws of motion. Such sources and characteristics of human knowledge are described such that time and space are the transcendental forms of human consciousness, with their effects on the corresponding objects of sensation forming the phenomenon while the phenomenon is comprehensively judged to form the knowledge. In other words, although the materials for the formation of a phenomenon are acquired, the forms still exist firmly in the consciousness, due to the a priori effect of time and space on the phenomenon. For example, the knowledge formed by general observers making judgments of objects is fixed, and essentially consistent. However, the sources of knowledge also contain descriptions of symbols or conceptual structures which cannot be obtained by visual experience of the language system or ideology. Fixed and consistent knowledge of such phenomena cannot be obtained, and ambiguity and distortion occur due to differences in human ⇑ Corresponding author. E-mail addresses: [email protected] (J. Zhao), [email protected] (L. Liu), [email protected] (L. Hu). 0950-7051/$ - see front matter 2012 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.knosys.2012.03.006
subjective factors or changes in perspective. In order to obtain understandable knowledge from such phenomena in a given field, various phenomena must be correlated with differences and generalities removed to form a knowledge community. Thus, the entire field context can be understood as a system, defined by Ervin Laszlo as: ‘‘the system is an organic whole with structures and specific functions composed of a number of interrelated and interactive elements’’. This system consists of interrelated intellectual phenomena, and the interaction can be understood as correlated phenomena brought into the knowledge community with boundaries, out of which non-correlated phenomena are abandoned. Based on the above precepts, the basic element constituting the knowledge system was named a phenomenon element in this study, and the system was called a phenomenon system, where the phenomenon is taken as material for the formation of knowledge. The states of the phenomena in this system will subsequently be changed and re-integrated to form a new knowledge community, when changes occur in the boundary or the nature of the field context. This activity reflects that the system is in constant motion, showing the historical state at a certain point in time. The phenomena are flowing through time, and in this study, this nature of the phenomenon system is called the diachronism of the phenomenon system. The formal representations were carried out on the concept and time categories of the phenomenon system, as well as on their diachronism. First, accuracy and continuity could not be achieved due to limitations of the knowledge acquisition in the phenomenon system, such that all materials for the formation of knowledge
J. Zhao et al. / Knowledge-Based Systems 33 (2012) 136–144
were represented as granular [18], so the knowledge formed was called a knowledge granule. In this study the focus was on the intensive and extensive attributes of the knowledge granule. When the granule is taken as a whole, its intensive attribute is considered to be determined by the attributes of all sub-granules. When the granule is taken as a part, its extensive attribute is considered to determine how the granule will be sensed and how it will be correlated with other granules. A detailed discussion on the association can be found by referring to this author’s articles [11]. Formal concept analysis (FCA) was used as a basic tool to describe the knowledge granule in this study. The basic unit is called a concept to describe the knowledge object in the FCA, and the structure of concept lattice was adopted to describe the sequence relations between concepts. However, the knowledge granule has its own structure and compositions, making it the sub-system under the phenomenon system, which precisely reflects the relationship of generalization and specialization between the internal components of the knowledge granule. In addition, the attributes of the knowledge granule may have multiple characteristics. The color brown for example, is not only different from red, blue and yellow but also from such characteristics as triangle, wooden and large. Thus the formation of knowledge is largely ambiguous, with fuzzy boundaries, necessitating that the overall system of knowledge granules has the capacity to roughly identify, classify, and organize the various knowledge granules. Therefore, it is necessary to classify the represented characteristics of knowledge granules by equivalence class. Concepts including indiscernibility relation in rough set theory are used to facilitate such classifications. Rough set theory and scale theory of FCA are combined to construct concept elements as the basis for formation of the phenomenon, and thus the knowledge system is more accurately represented. Meanwhile, in order to study the motion in the knowledge system, we must describe its state changes during the passage of time. In this study, the phenomenon change is understood to occur in a specific, discrete time slice, and the formal representation of the time slice and time sequence are given based on FCA. The phenomenon element is constructed as a basic element of the knowledge system,using the time elements embedded in the concept element. The remainder of this study was organized within Sections 2–5, as described below. Section 2 contains comparisons of the strengths and limitations between this study’s theory and those of related works, within this field. The basic structure of the knowledge granule was given in Section 3, as well as the definition of the conceptual element with granular structure. The basic form of granular time was presented in Section 4, where the conceptual element was extended to the phenomenon element in temporal context, and the formal definition of the phenomenon system was given. Additionally, related definitions of the state, phase and flow of phenomenon knowledge in the evolutionary process were also provided. Based on the partial order of concept lattice, time series was constructed, so as to study the diachronism of the phenomenon system. Visualization of the movement of the phenomenon element was provided by descriptive an example in Section 5. 2. Relevant work Several influential and effective methods and theories have emerged in areas like knowledge representation, processing and description and the solution of structured problems in the objective world. 2.1. Ideas regarding granular computing and rough set Granular computing is an emerging discipline in problem solution methods and information processing areas, according to which
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[2,12,18,19], human intelligence is provided with global analysis capacity, or the ability to process problems in different granular worlds and observe and analyze objective phenomena with different granularities. The work of Zadeh [12] and Yao [17–19] have provided the basic representation of granular structure, utilizing tools such as fuzzy set, neighboring system, or rough set, depending on which relevant approximation theory and knowledge representation method was developed. Rough set theory [1,8,15,20,21], put forward by Polish Scholar Z. Pawlak, views knowledge as the division of given universe U. In other words, it granulizes irrelevant knowledge through equivalence class and utilizes the granular knowledge as the main processing object. Definition 2.1. Set a universe U and a cluster of equivalence relation A on U, a 2-tuple (U, A) is called a knowledge base with respect to the universe U. If there exists an equivalence relation B # A(B – U), then a \B is called an indiscernibility relation of B, denoted by IND(B) T and "u 2 U, [u]IND(B) = "R2B[u]IND(R). U/IND(B) = {[u]IND(B)j"u 2 U} indicates the knowledge related to IND(B) and is called a B-baseset (or B-knowledge) of the knowledge base (U, A) with respect to the universe U. This theory is an expansion of the classic set theory. It inserts knowledge used for classification into the set: the degree of a given object belonging to set X can be judged according to upper approximation and lower approximation. Definition 2.2. Upper and lower approximationsSet a knowledge base (U, A). So "G # U and an equivalence relation B A; the lower and upper approximations of subset G with respect to the equivalence relation B are respectively defined [21]:
BðGÞ :¼ fg 2 Uj½gB # Gg ¼ [fF # Gj8F 2 U=Bg BðGÞ :¼ fg 2 Uj½gB \ G – Ug ¼ [fF 2 UjF \ G – Ug Theories like granular computing have certain advantages in modeling approximate representations of knowledge semantics and structured problems in the objective world. However, the dynamic contact among knowledge granules, as well as the research on the overall structure of the knowledge space constituted by all knowledge granules and its evolution in time has been ignored, and therefore, it has certain limitations in processing open problem domain. Based granular computing, a study was carried out on the overall structure of a knowledge system consisting of knowledge granules, as well as its evolution over time, so as to provide a better understanding of open knowledge granulation and evolution in context and time. 2.2. Formal concept analysis The Classical FCA theory usually begins with the basic concepts of some particular formal context, such as a single-value context, which focus on the object sets, attribute sets, and their relationships. Definition 2.3 [3]. A single-value context is a 3-tuple K :¼ (G, M, I), where G and M are sets, and the binary relation I # G M and elements of G and M are called the objects and attributes accordingly. (g, m) 2 I or gIm indicates the object g has the attribute m. Definition 2.4 [3]. Suppose A is a subset of object set G, and define f(A) :¼ {m 2 Mj"g 2 A, gIm} (the set of common attributes of objects in A). Accordingly, suppose B is a subset of attribute set M, and define. g(B) :¼ {g 2 Gj"m 2 B,gIm} (the set of the objects of all attributes in B).
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Definition 2.5 [3]. A formal concept of the formal context (G, M, I) is a 2-tuple (A, B), where A # G, B # M and f(A) = B, g(B) = A. A is called the extent of the concept (A, B), while B is the intent thereof. B(G, M, I) is used to indicate the set of all concepts on the context (G, M, I). Let c(g) = (g(f(g)), f(g)) is an object concept; l(m) = (g(m), f(g(m))) is an attribute concept. Concepts form a ‘‘concept lattice’’ structure via ‘‘hierarchical sequence’’, which represents the relation between generalization and specialization among the concepts. FCA highlights the formalized and structured representation of knowledge. However, it has certain limitations in processing fuzzy and uncertain knowledge, in terms of semantic representation of knowledge. In order to deal with fuzzy or uncertain knowledge, it is necessary to use multivalued context in formal concept analysis for combining two theories in the form [9,10,13,14,17]. The definition of the multivalued context is first given as follows.
defined sufficiently based on RS. Thus, redefinition and a better understanding were provided for time, process, flow, state and other concepts in the event calculus and other theories.
Definition 2.6 [4]. Let (G, M, W, I) be a multi-valued context, with a ternary relation (I # G M W), while Sm(m 2 M) is called the conceptual scales. If m(G) # Gm, then the objects in the scale are the values of the attribute m, while the attributes in the scale are those with single-value context. The single-value context (G, N, J) is derived, where N ¼ [m2M M m and gJ(m, n) , m(g) = w and w Imn. The formal context of the multi-valued attributes can depict the formal context of the exact relationship between the objects and the attributes, and more accurately reflect the connectivity and transmissibility between the concepts, with the condition that the knowledge structure is basically constant. Two theories were combined in this study, where a rough formal concept was introduced for a semantic representation of phenomenon elements. Specific propositions will be given in Section 3.
In general, granular structure can be established in a special domain based on specific context. A certain constraint is defined in this domain, where granular connotation is expressed by constraint rules, reflecting the general characteristics of all elements in the domain, while granular extension is the element set in this domain satisfying the constraint, or the set of elements covered in granular structure. However, context relation represents the external attribute of interaction among granular structures. The formalized representation of the conceptual category of knowledge is carried out through granular structure as described below. First, considering that the conceptual categories of most phenomena are ambiguous, containing indistinct boundaries, a knowledge domain is required with the capacity for rough identification, classification and organization of scattered grains. To that end, the indiscernibility relation of rough set theory was used to process incomplete and inadequate information, and then the knowledge domain was divided into different equivalence classes, so the phenomena are classified according to different attributes or characteristics. Meanwhile, the granular structure forms of the conceptual category in phenomena are given by combining the scale theory of FCA. The isomorphism between both theories is first given by the following propositions, and a detailed proof of the proposition can be found by referring to the author’s article [11].
2.3. Ontology engineering Meanwhile, a number of theories and methods have been proposed on the research of timing materials in the knowledge representation field, such as Temporal Logic [5–7] and Event Calculus [16]. According to situation calculus, it is stated that low is a function and predicated to change from one scene to another. The event calculus is designed to allow reasoning in the time interval, including talking time and time interval capability. For example, Initiates(e, f, t) indicates that the event e occurs in the time t, leading to the flow f is true. Terminates(w, f, t) indicates that the flow f is no longer true. Happens(e, t) indicates that the event e occurs at the time t. Clipped(f, t, t2) indicates that f is terminated by a certain event in a moment from t to t2. The following facts were provided by an event calculus axiom: a flow will be true in a moment, if this flow is induced by an event in a certain past moment and not terminated by a certain event. The axiom formalization is shown as follows:
EVENT CALCULUS AXIOM Tðf ; t 2 Þ () 9e; t
3. Knowledge granule and concept element In granular structure, there are three basic attributes that must be satisfied, including: (1) context attributes indicating the existence of a granule in special environment; (2) internal attributes reflecting elemental interaction in a granule, (3) external attributes reflecting the interaction between a granule and others. Based on the above attributes, the basic form of granular structure is given below. Definition 3.1. Granular structureA granular structure is described with a triple (EG, IG, R), where EG and IG are the sets called as granular extension and granular connotation respectively. Binary relation R # EG IG is called as context relation.
Proposition 3.1. Any knowledge base (U, A) is given, where U is the universe and A :¼ {Bmjm 2 M}. S(U, A) :¼ ((U, A, W, I), (SBjB 2 A)) is known, where SB is the nominal scale with its derived context (U, N, J). c is set to represent the mapping of object concept in the context (U, N, J), and thus, if (u, v) 2 IND(P)((u, v) 2 U2, P # A), then c(u) = c(v). It is known by Proposition 3.1 that if a knowledge base (U, A) is given, then a many-valued context can be generated under the effect of scale operator, and the derived context of a many-valued context is
Happensðe; tÞ ^ Initiatesðe; f ; tÞ
^ ðt < t2 Þ ^ :Clippedðf ; t; t2 Þ Clippedðf ; t; t 2 Þ () 9e; t 1
Happensðe; t 1 Þ ^ Terminatesðe; f ; t 1 Þ
^ ðt < t 1 Þ ^ ðt 1 < t2 Þ Thus, equation Happens(Rise(Moon), 11:00) can be used to illustrate that the moon rises at 11:00. First-order logic is commonly used as a research tool in the majority of these theories. On this basis, direct descriptions of an object and its category affiliation, scenarios, processes, etc. can be obtained with easy maneuverability. However, limitations exist in the semantic representation of objects, the studies of object evolution caused by changes in the semantic environment, as well as the representation of time order. The time-series was defined on the basis of FCA lattice structure. Meanwhile, knowledge was
U; fðB; ½uB ÞjB 2 A; ½uB 2 U=Bg; JÞ; i.e., the knowledge base (U, A) can be expressed with the derived context of a many-valued context of a formal concept. Then, the consistency between basic concepts of both theories is a basis for establishing granular structure. Based on this proposition, the formalized description of a conceptual category is given and called a concept element. Definition 3.2 (Concept element). A knowledge base (U, A) is given as well as a many-valued context (G, M, W, I), of which, M # A, G # U/M,W :¼ {[u]Bju 2 G, B 2 M} and I # G M W. The concept element is the derived context of a nominal scale as (G,{(B, [u]B)jB 2 M, [u]B 2 U/B}, J). uJ(B, [v]B) , [u]B = [v]B is satisfied, where "u, v 2 U and u – v. It is abbreviated as v :¼ (G, N, J), where N :¼ {(B, [u]B) jB 2 M, [u]B 2 U/B}.
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It is known from granular structure of concept element that the constraints among all elements with similarity in the domain U can be reflected in granular connotation {(B, [u]B)jB 2 M, [u]B 2 U/B} through the indiscernibility relationship. While granular extension G is a set of elements satisfying this indiscernibility relationship, and specific characterization of constraints are given by J. Thus, conceptual category is a granular structure satisfying granular attributes.
4. Expansion of the concept element and diachronism of a phenomenon system 4.1. Formal representations of basic elements in a phenomenon system In order to ensure the individual continuity and uniqueness of a certain phenomenon unit in a phenomenon system, which will avoid the occurrence of multiple replications, it was necessary to clarify the definition of its state and the changes occurring in the time dimension. For example, in order to determine if a fresh or rotten apple is the same object, it is required to give reconfirmation according to the time and state. Thus, the phenomenon unit was a necessary part of the inner, embedding time sequence. First, we considered how to define the state of a phenomenon unit. The knowledge object studied is essentially a conceptual description of the visual experience of objective targets, by observers in the real world, which exists as a phenomenon within the structure of a granule surrounding the knowledge core. However, the knowledge core is essentially accumulated and formed by the knowledge in the core, from correlated concept elements. When the current content of specific knowledge is investigated, the new knowledge restructuring will be carried out by changing perspectives and intercepting different equivalence class slices. Thus, the state of knowledge in flux can be considered as the set of equivalence class slices with different perspectives. However, if such change is placed in the field of a complex system for observation, it won’t involve human factors, but will proceed with a self-organization approach, so the state of knowledge is required to be described as the set (knowledge core) of knowledge slices shared in different concept elements of the granule at different points in time. Then, a further study of the composition of knowledge granule is required to determine how to embed the time element in the concept element. Time is continuous in the real world, while discrete in this context set. For example, the maturation process of an apple occurs in continuous time, while knowledge such as: The nature of the Soviet Union will change from one state to another due to changes in politics and world trends. Therefore, the motion of knowledge exists in a time slice rather than the uniform time of the physical world. Thus, the occurrence of a general event in the knowledge system is considered to consist of state slices and time slices of knowledge. In order to discuss discrete time, the time must be represented by granular structure. The granular time is defined as a time element, and it is represented formally as a multivalue context, for example: s :¼ ((GT, MT, WT, IT), (Stjt 2 MT)). The formal context of time elements is shown in Table 1. Time is the element of time domain Gs, representing the time slice, which is known as the time object. Gs indicates the set of all time points in an interval. Its elements can be expressed by ordered numbers (1, 2, 3, . . .), or by strings such as ‘‘851914 (representing 14 o’clock on May 19, 2008)’’ which can also be used to indicate time points with specific meanings. Other time attributes are the specific descriptions of time objects. In this multi-valued context, various scales can be used according to different granularities. The nominal scales can be applied to accurately indicate the granulation of time, and other scales (such as ordinal scales) may be used to indicate time in a less precise, more fuzzy manner.
Space–time in the physical world generally consists of continuous four-dimensional space–time, where an event’s occurrence is constituted by all aspects of a specific space–time slice. For example, the Cold War is an event occurring at a number of space–time points, of which, sub-events such as the arms race, the space race, or the disintegration of the Soviet Union occur in a certain phase within this space–time slice. Each phase can also be continuously decomposed into increasingly more specific phases or sub-events. Each event has attributes in both time and space, while the space–time attributes also exist with boundaries. For example, the Cold War Period has a limited time boundary of 1947–1991, and the largest space boundary. At a certain spatial location, the Soviet Union has the largest time boundary, while the space boundary changes over time. The descriptions and hierarchical relations of the event can be analogous to the concept lattice structures of FCA, while the overall event and its space–time specifications can also be indicated by formal context. The major environment of discussion is the phenomenon system. For instance, in describing the Cold War event, the main focus is on the knowledge level world, like politics, economy, culture, ideology and so forth. Therefore, its space–time slice is different from the continuous space–time environment in the physical world, and its time is granular and discrete, while the space dimension is actually the state of a concept element at a certain point of time. In order to found a basis for descriptions on the motion characteristics of the phenomenon system, an analogy is carried out of the physical system, using related concepts of the physical system, such as state, process and phase. Above all, the concept of state is discussed. In the real-world system, it is very difficult to define the state. For example, the state of a moving person can be described in many aspects, such as physical, biological and psychological, etc. Generally, the ‘‘idealized’’ approach is adopted in order to simplify the study. In the mechanical motion system of physics, for example, a motional element can be idealized to a particle, while the state of a particle can be simply described with location and momentum. In the phenomenon system, the extended forms of concept element are first given with the time dimension included, so as to determine the state of concept element under the time dimension. Definition 4.1 (Time–concept element and its state). A knowledge base (U, A) and a time domain Gs are given as well as the concept element v :¼ ðGv ; Mv ; W v ; Iv Þ. Then, time–concept element is defined as v :¼ ðG; M v ; W v ; Iv Þ, where G = Gv Gs. The derived context is as K v :¼ ðG; fðB; ½uB Þ j B 2 M v ; ½uB 2 U=Bg; JÞ, the state of concept element v is defined as the connotation of object concept cv(g) of formal context Kv. In Definition 4.1, time is understood as being granular: a time granule is a time slice (time domain Gs), in which a number of time points are contained. The time is included in the inner of layer of the general concept element, i.e. G = Gv Gs is used to indicate the object domain of time–concept element. This form can represent the element changing with time points in the domain Gv, such as the description on objects in the time context like the dawn moon. The internal instant state of a concept element occurs in the current time, a specific time point in the time domain Gs. While discussing the situation of two states of the concept element are the same, it is understood that the concept elements at two different time points have equal values. Under the framework of formal Table 1 Formal context representation of time elements. Time
Year
Month
Day
Daytime
1 2 3
2008 2010 2010
May January April
19 12 14
14 16 7
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concept analysis, the value of a concept element is expressed by its scale Sm :¼ (Wm, Mm, Im)(m 2 Mv), i.e., if "m, n 2 Wm are equal, if and only if m, n have the same object concepts in Sm. Thus, the two states of concept element v are the same at the two different time points of g1, g2, or the corresponding object concepts of cv(g1) and cv(g2) have the same intents. Therefore, the state of the concept element here is expressed by the intent of the object concept cv(g), i.e., representing the set of different attributes contained in the object at a specific point in time. Example. The lunar phase is taken as a simple example. The knowledge can be described as the state (lunisolar longitude difference) changes of the moon during the period of a specific time slice such as (from the night of the second day of a lunar month to the daylight of the seventh day of a lunar month. By isolating a part of the knowledge, the time and event contexts are represented in Tables 2 and 3, where the element of time domain Gs, like ‘‘221’’, represents the time of 21 o’clock on the second day of a lunar month. By using the form of the time–concept element, it is indicated as Table 4. The object concept of element (221, 1) in the domain is ({(221, 1)}, {0, East 25}), so its state is represented as {0, East 25}. The time–concept element represents the process and attributes of an event, through which, the event is often manifested itself as a phenomenon. In order to describe the phenomenon in the time dimension under a more complex context, the time slice must be described under multiple dimensions (such as year, month, and date), as extending the time domain Gs to be the time element s, and the phenomenon is described as the form of time element and time–concept element. In the meantime, the state of the time–concept element will be extended to be a phase for describing the state of the phenomenon unit in the system at a specific point of time. A formal definition of a phenomenon unit in the phenomenon system is given in Definition 4.2. Definition 4.2 (Phenomenon element). A knowledge base (U, A) is given, and the time element is set as s :¼ ðGs ; M s ; W s ; Is Þ, and concept element as v :¼ ðGv ; Mv ; W v ; Iv Þ. The two-tuple P :¼ ððGs ; M s ; W s ; Is Þ; ðGv ; M v ; W v ; Iv ÞÞ is called a phenomenon element occurring in the knowledge base (U, A) in the time domain Gs, abbreviated to P :¼ (s, v). It can be seen from the above-mentioned definition of phenomenon element that its form is essentially the combination of formal contexts in two different domains. In order to study it in the
Table 5 Formal contexts of phenomenon element. Gs
221 307 412
Time element
Concept element
Day
Hour
Gv
Angle ()
Location
Second day of a lunar month Third day of a lunar month Fourth day of a lunar month
21 o’clock 7 o’clock 12 o’clock
1 2 3
0 15 45
East 25 East 28 West 13
Table 6 Formal contexts of extended phenomenon element. Gv Gs
(221, 1) (307, 2) (412, 3)
Time element
Concept element
Day
Hour
Angle ()
Location
Second day of a lunar month Third day of a lunar month Foruth day of a lunar month
21 o’clock 7 o’clock 12 o’clock
0 15 45
East 25 East 28 West 13
same domain, the domain of the time–concept element is used to merge Gv and Gs to be Gv Gs. Thus, the following definition is obtained: Definition 4.3 (Phenomenon and phenomenon space). The phenomenon element P :¼ (s, v) is extended to P :¼ ðGs ; Gv ; D; s; vÞ. Among them, D = Gs Gv. Thus, the derived context KP of the formal context of phenomenon has actually become the apposition context of Ks and Kv [3]. KP is called as the phenomenon context, and the object concept of the phenomenon context c(g)(g 2 Gs Gv) as the phenomenon at the time point g. The set of all phenomena under the phenomenon element P :¼ {c(g)jg 2 Gs Gv} is called as the phenomenon space. The above example of the lunar phase is expressed as the Tables 5 and 6 with the form of a phenomenon, of which, the time element represents the time part of phenomenon, while the concept element represents the event part of phenomenon: In the forms of a phenomenon element, a static description of the time is given, as the time element part. However, in order to describe the state changes of a concept element, it is necessary to give a description of the time fluidity or flow, the state is changed successively according to the context of time points. Therefore, it is necessary to include the formal representation of the time sequence to study the diachronism of a phenomenon system. 4.2. Diachronism of a phenomenon system
Table 2 Formal contexts of time. Gv
Angle ()
Location (apparent declination)
1 2 3
0 15 45
East 25 East 28 West 13
Table 3 Formal contexts of event. Gs
Day
Hour
221 307 412
Second day of a lunar month Third day of a lunar month Fourth day of a lunar month
21 o’clock 7 o’clock 12 o’clock
Table 4 Formal context of the time–concept element. G = Gv Gs
Angle ()
Location (apparent declination)
(221, 1) (307, 2) (412, 3)
0 15 45
East 25 East 28 West 13
The time domain here is considered as a poset ðGs ; 6s Þ, in which, 6s is the partial order defined by the time successive relation in Gs, which is called the time sequence here. Meanwhile, "g1, g2, g3 2 Gs is set to satisfy the following attributes: (1) g1 6s g1; (2) g1 6s g2, and g1 – g2, then g2 6s g1 is untenable; (3) g1 6s g2, and g2 6s g3, then g1 6s g3. Gs # R is taken here (R is the real number set), and Gs is set to indicate a closed interval [g, h] of R("g, h 2 Gs), so the time sequence as a partial order is the natural order of real numbers. Similarly, in the domain Gv of the time–concept element, a natural sequence is used as the index set, so the partial order relation 6v of the poset ðGv ; 6v Þ is also in accordance with the natural order of real numbers, where it is equated with the time sequence 6s. In order to discuss the diachronism of the phenomenon, it is necessary to extend the domain Gsto Gv Gs, so that the definition below is obtained. Definition 4.4. Two posets ðGs ; 6s Þ and ðGv ; 6v Þ exist, in which, the partial order is the time sequence. Then, the direct product ðGs Gv ; 6sv Þ of the two sets is also the poset. Among them, 6sv indicates (g1, g2) 6sv (h1, h2) , g1 6sv h1, and g2 6sv h2.
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Proposition 4.1. Mapping of #:Gs ? Gs Gv exists between two posets ðGs ; 6s Þ and ðGs Gv ; 6sv Þ, so the two posets are order preserving. Proof. It is shown by Definition 4.4.
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Proposition 4.3. The posets ðGs Gv ; 6sv Þ; ðpðBðK s ÞÞpðBðK v ÞÞ; 6Þ exist, (of which, the partial order relation 6 is the set-inclusion relation ). If B(KP) is generated by ordinal scale, then these two posets are order preserving.
h
Proposition 4.2. The posets ðGs Gv ; 6sv Þ and (B(KP), 6) exist, of which, B(KP) is the concept lattice of the phenomenon context KP, 6 is the level sequence in B(KP). If B(KP) is generated by the ordinal scale, then the two posets are order isomorphic. Proof. "g1, g2 2 Gs is taken, and then g1 6s g2. The mapping is set as c:Gs ? B(Gs), i.e., c is the object concept mapping of g 2 Gs. The ordinal scale is taken as O :¼ (n, n, 6) to generate derived context, and epitaxial chains are formed in attribute values, so it is proven that: f(g1) f(g2), i.e. g(f(g1)) # g(f(g2)). Therefore, the object concept is shown as c(g1) :¼ (g(f(g1)), f(g1)) 6 c(g2) :¼ (g(f(g2)), f(g2)). That is, g1 6s g2 ) c(g1) 6 c(g2), and vice versa. Therefore, if g1 6s g2 , c(g1) 6 c(g2) exists, and c is known as bijection, then the two partially order sets ðGs ; 6s Þ and (B(KP),6) are order isomorphic under the mapping c. It is also known by Proposition 4.1 that ðGs ; 6s Þ and ðGs Gv ; 6sv Þ are order preserving, thus it is proven that ðGs Gv ; 6sv Þ and (B(KP), 6) are also order isomorphic. It is known from Proposition 4.2 that a structure like concept lattice can be used to represent the time sequence vividly, so that each object concept (i.e. phenomenon) in concept lattice can be considered in the passage of time. Meanwhile, due to the concept lattice generated by using ordinal scales, the partial order relationship can be considered as total ordering. All the following phenomenon context KP discussed will be derived by ordinal scale. The following will use the ordinal scale to obtain the derived context KP of the phenomenon P. The obtained phenomenon context KP is shown in Table 7. The lattice structure is generated by the phenomenon context KP as follows (see Fig. 1). The description of the concept elements discussed above specifies that in a certain time domain, phenomena are formed in concept elements, and divided into different stages by discrete time points. When the knowledge granule at each stage shows two states simultaneously, the state of the time dimension and knowledge are described. As previously discussed, at the time point g 2 Gs Gv, the knowledge state of concept element v is represented by the corresponding object concept cv(g) of Kv. However, the state of the time dimension is described by using the corresponding object concept cs(g) of Ks. Therefore, a combination of these two states can be used to represent the time sequenceknowledge state of concept element at a certain time ("g 2 Gs Gv), which is called as the phase. h Definition 4.5 (Phase). The derived contexts of time element s and time–concept element v are set as Ks and Kv, cs(g) and cv(g) are the object concepts of Ks and Kv respectively, and "g 2 Gs Gv. The mapping is defined as p : BðKÞ ! IðKÞ, of which, K representatives Ks and K v ; IðKÞ is the set of all object concept connotations in the formal context K. Thus, the phase of the phenomenon at a certain point of time g is defined as the form ðpðcs ðgÞÞ; pðcv ðgÞÞÞ . The set of all phases under the phenomenon element P is denoted as S :¼ fðpðcs ðgÞÞ; pðcv ðgÞÞÞjg 2 Gs Gv g, which is called as the phase space. For example, at time point 221, the moon is located at 0 on the morning of the second day of the lunar month, that is, the phase in this time is (morning of the second day of a lunar month, 0). By referring to Proposition 4.2, the component phase is also placed into the time sequence, so that the proposition below exists.
Proof. "g1, g2 2 Gs Gv is set and the mapping as l:Gs Gv ? p(B(Ks)) p(B(Ks)). It is proven by the proof process of Proposition 4.2 that g1 6s g2 ) f(g1) f(g2), i.e., ðGs Gv ; 6sv Þ and (p(B(Ks)), 6) are order preserving, and it is proven by the same methods that ðGs Gv ; 6sv Þ and (p(B(Kv)), 6) are order preserving. Then, by Definition 4.4 it is shown that (p(B(Ks)), 6) and (p(B(Kv)), 6) are order preserving with (p(B(Ks)) p(B(Kv)), 6) respectively, and the posets ðGs Gv ; 6sv Þ and ðpðBðK s ÞÞ pðBðK v ÞÞ; 6Þ are order preserving. It can be seen from Propositions 4.2 and 4.3 that both the phenomenon and component phase have time attributes, and change over time sequence. The knowledge events occurring in the time sequence are uniformly known as flow events. The flow of any object is essentially the transmission of resources along the joint lines in a network system. Thus, such flow can be informally defined as a two-tuple {resources, joint line}. For example, the flow of the central nervous system can be composed of two-tuple {pulse, neurons connection}. The flow of the Internet can be composed of twotuple {packet, cable}. In order to characterize the abstract flow of the knowledge in the knowledge system, the time flow can be defined as two-tuple {flow event, time sequence}. The formal definition is given for the flow in the phenomenon system below. h
Definition 4.6 (Flow). P :¼ ðGs ; Gv ; D; s; vÞ is set as the phenomenon element, of which, the mapping is set as r:Gs Gv ? E, with E representing the set of flow events (phenomenon, phase). The set F :¼ {(r(g), 6s)jg 2 Gs Gv} is called the flow. If the mapping r is set as c, and c:Gs Gv ? P (P is the phenomenon space), then F is called as the phenomenon-time flow. If the mapping r is set as l:Gs Gv ? S (S is the phase space), then F is called as phase–time flow. 5. Time flow chart 5.1. Introduction of an example Data used in this example is a summary made by China Auto Market in relation to the sales status of automobiles of different models and brands on the Chinese mainland. We utilized the sales data from the first quarter of 2008 to the second quarter of 2011, for the Volkswagen Jetta, Toyota Corolla and Audi A6L, as shown in Fig. 2. In this example, we will formalize the data into phenomenon context and phenomenon and visualize the passage of time by applying the concepts and theories proposed in this paper. We first set the time element: s :¼ ðGs ; Ms ; W s ; Is Þ, where Gs refers to the aggregation of all time points within a given interval. Choose ordinal real numbers with specific semantics as the elements of this aggregation, so that 103 refers to the third quarter of 2010. Consequently, all the time points within the interval from the first quarter of 2008 to the second quarter of 2011 form a poset (Gs, 6) with 6 as the partial ordering relation. In Ms :¼ {year, quarter}, Ws refers to the aggregation of the attribute values of years and quarters. The formal context of the time element s is shown in Table 8. Meanwhile, set the concept element: v :¼ ðGv ; M v ; W v ; Iv Þ, where Gv :¼ {1, 2, 3, . . . , 14} refers to the aggregation of all events, and Mv :¼ {VW Jetta, Toyota Corolla, Audi A6L} refers to the attribute aggregation of the sales volumes of the three models, and Wv refers to the aggregation of the corresponding sales values of various models. The formal context of v is shown in Table 9.
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Table 7 Phenomenon context. Gv Gs
Time element 2nd day
(221, 1) (307, 2) (412, 3)
(221,1 )
Concept element 3th day
(307,2 )
4th day
Morning
(412,3 )
Evening of
Morning of
Morning of
the 2nd day
the 3th day
the 4th day
0°
15°
45°
Fig. 1. Representations of concept lattice formed in the phenomenon context on the time sequence.
Consequently, according to Definition 4.3, the phenomenon context KP can be found, as shown in Table 10. The concept lattice can be formed, based on the given phenomenon context, by using the grid scale that consists of two ordinal scales,one of the ordinal scales refers to the ascending relationship of a certain model within the sales interval, e.g., for Toyota Corolla, the sales interval is [24,340, 54,459] and its attribute can be defined between P20,000 and P50,000. The other ordinal scale refers to the ascending relationship of a certain model by quarter. Figs. 3–5 indicates the concept lattices of three models respectively. For the sake of simplicity, the phenomenon mark c(Gs) can be abbreviated as Gs, e.g. c(81, 1) can be abbreviated as 81. In Figs. 3–5, the nodal points with marks are the object concepts indicated in black circles, referring to the phenomenon c(g) at a time point g. For example, the nodal point marked 83 in Fig. 4 refers to the phenomenon that the sales volume of the Toyota Corolla exceeded 30,000 in the third quarter of 2008. Fig. 6 indicates the sales status of Corolla of Toyota with time variation in 4 years, and such legend indicating the flow process of flow events (see Definition 4.5) is called ‘‘time flow chart’’. The graduations on the sales volume dimension refer to P20,000, P30,000, P40,000 and P50,000, respectively, while those on the quarter dimension refer to P1, P2, P3 and P4 respectively. 5.2. Analysis on effectiveness and feasibility of time flow chart As shown in Figs. 3–5, the hierarchical order relation between phenomena is reflected by Hasse diagrams of traditional concept Audi A6L Toyota Corolla VW Jetta
70000 60000 50000 40000 30000 20000 10000 0
2Q.2008 4Q.2008 2Q.2009 4Q.2009 2Q.2010 4Q.2010 2Q.2011
Fig. 2. Sales chart of three models within 4 years.
Noon
Evening
0
15
45
Table 8 Time element of car sales. Time
Year
Quarter
81 82 83 84 91 92 93 94 101 102 103 104 111 112
2008 2008 2008 2008 2009 2009 2009 2009 2010 2010 2010 2010 2011 2011
1 2 3 4 1 2 3 4 1 2 3 4 1 2
lattice with the dimensions of time and sales volume. Such relation mainly indicates the specialization and generalization between ‘‘concepts’’ in the concept lattice, and it is impossible to completely visualize the flow process of phenomena in the time order. Specifically, taking the concept lattice as indicated in Fig. 4 for example, Black Nodes (81, 91) and (101, 111) are connected by a black line segment, and the structure of concept lattice only indicates that phenomenon c(101) or c(111) is the parent concept of phenomenon c(81) or c(91). Moreover, Phenomena c(81), c(91) or c(101), c(111) coincide at the same node due to the selected scale graduations. It can be observed from above two points that the Hasse diagrams based on traditional concept lattice is neither able to show the successive relation between concepts in respect of time, nor to indicate that the phenomena change according to the ordered time dimension. For the purpose of illustrating the flowability and temporal sequentiality of phenomena, it is necessary to unify the time order of phenomena and linearity in form. It can be observed from Proposition 4.1 that the posets ðGs ; 6s Þ and ðGs Gv ; 6sv Þ are orderpreserving, and it should be noted that Gv is defined as the natural number set in this example, i.e., the partial ordering relation ‘‘6v’’ in poset ðGv ; 6v Þ satisfies the condition of less-than relation
Table 9 Concept element of auto sales. VW Jetta
Audi A6L
Toyota Corolla
58,018 51,705 41,955 40,593 55,647 55,969 55,254 59,848 52,994 53,765 53,004 62,759 50,353 46,409
20,424 19,374 20,226 19,049 23,457 22,936 25,895 26,477 28,632 28,717 29,952 26,098 22,228 25,387
41,901 39,710 38,938 36,593 42,690 35,892 43,823 42,525 31,184 31,410 54,459 46,645 38,559 24,340
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J. Zhao et al. / Knowledge-Based Systems 33 (2012) 136–144 Table 10 Phenomenon context.
Sales Volume 19049-29952
Gv Gs
Year
Quarter
VW Jetta
Audi A6L
Toyota Corolla
(81, 1) (82, 2) (83, 3) (84, 4) (91, 5) (92, 6) (93, 7) (94, 8) (101, 9) (102, 10) (103, 11) (104, 12) (111, 13) (112, 14)
2008 2008 2008 2008 2009 2009 2009 2009 2010 2010 2010 2010 2011 2011
1 2 3 4 1 2 3 4 1 2 3 4 1 2
58,018 51,705 41,955 40,593 55,647 55,969 55,254 59,848 52,994 53,765 53,004 62,759 50,353 46,409
20,424 19,374 20,226 19,049 23,457 22,936 25,895 26,477 28,632 28,717 29,952 26,098 22,228 25,387
41,901 39,710 38,938 36,593 42,690 35,892 43,823 42,525 31,184 31,410 54,459 46,645 38,559 24,340
Quarter 1-4
≥ 19000
≥1
≥ 22000
≥2 81
≥ 25000
≥3
91 111
≥4
82 82
101
83
112
84 93
102 103
94 104
Fig. 5. Concept lattice for sales status of the Audi A6L.
Sales Volume 41955-62759
Quarter 1-4
≥ 40000
2008
≥2
≥ 55000
2009
≥3 101 111 81 91
≥4
112
γ(81)
2010
γ(83)
2011
82 102 92
Sales Volume
≥1
≥ 50000
83 103
γ(82)
Time flow
84
93
γ(93) γ(94)
γ(91) 94
γ(92)
Quarter
γ(84)
γ(104)
104
γ(101)
Fig. 3. Concept lattice for sales status of the Volkswagen Jetta.
γ(102)
2008
γ(103)
γ(111) Sales Volume 24340-54459
Quarter 1-4
≥ 20000
γ(112)
2010
≥1
≥ 30000
≥2
≥ 40000
2011
≥3
≥ 50000
2009
101 111 82 92 102
81 91
Fig. 6. Time flow chart for the sales status of Toyota Corolla.
83
93 103
≥4
112
84 94 104
Fig. 4. Concept lattice for sales status of the Toyota Corolla.
between natural numbers. In addition, it can be observed from Proposition 4.2 that the posets ðGs Gv ; 6sv Þ and (B(KP), 6) are order-isomorphic in case that the concept lattice B(KP) is generated by ordinal scale, which means the hierarchical order of concept lattice consisting of phenomena are order-preserving with the lessthan relation of natural numbers, and even isomorphic, which provides the theoretical basis for temporal sequentiality. Hence, the poset ({(81, 1), (91, 5), (101, 9), (111, 13)}, 6sv) (subset of the time point set Gv Gs, as shown in Table 10) is orderpreserving with the poset ({1, 5, 9, 13}, 6v) (subset of the time domain Gs), and is isomorphic with ({c(81), c(91), c(101), c(111)}, 6).
Therefore, it can be guaranteed that the following facts are reasonable under the framework specified in this paper: the starting time point for Corolla of Toyota is (81, 1), and its corresponding phenomenon is c(81) (c(81), implying the sales volume is greater than 40,000. When the time point reaches (101, 9), its corresponding phenomenon turns out to be c(101) (c(101), implying the sales volume is greater than 30,000 and less than 40,000, and so forth. Moreover, it should be explained that the definition of time points in this example is based on semantics, e.g., ‘‘81’’ represents the first quarter in the year of 2008. However, the following problems would be usually encountered as dealing with more complicated time semantics: (1) it is impossible to use valid numbers to represent the particular implications; (2) there is no guarantee that time semantics indicates the satisfaction of less-than relation of real numbers or natural numbers; (3) the representation of time semantics may be repeated. With consideration of the aforesaid problems, the time point is represented as ‘‘Gs Gv ’’ as required in this paper.
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The time flow chart (Fig. 6) proposed in this paper shows that the flowing mode of phenomena with time variation is different from that as indicated in the traditional Hasse diagram (Fig. 4). According to Hasse diagram, different marks may coincide at the same node, which implies that different phenomena may have the same connotation. However, the time points for different phenomena are not coincident. For example, Phenomena ‘‘101’’ and ‘‘111’’ coincide at the same time point in Fig. 4. Although both imply ‘‘the sales volume in the first quarter is greater than 30,000’’, they represent 2010 and 2011 respectively. In this way, it is impossible to completely visualize the time order in the same concept lattice, while the bidimensional Hasse diagram is represented with three dimensions (year, quarter and sales volume) in the time flow chart. It is noteworthy that the phenomenon context is cut into slices by different time values in accordance with the dimension of ‘‘year’’, and each slice represents the set of phenomena occurring in different quarters of the year. For the phenomenon element P :¼ ðGs ; Gv ; r; s; vÞ, when the value of Map ‘‘r ’’ is c:Gs Gv ? P, the corresponding phenomenon-time flow is: F :¼ {(r(g), 6s)jg 2 Gs Gv}; when the value of Map ‘‘r ’’ is l:Gs Gv ? S, the corresponding phase–time flow can be also obtained (the computing process is omitted here). In conclusion, the time flow chart based on time order concept is effective and feasible to demonstrate the flow process of various flow events (see Definition 4.5) under the phenomenon context in the time order. 6. Conclusions and next steps The phenomenon system is formalized in this study, based on formal concept analysis and rough set theory. Phenomenon are the basic materials that constitute knowledge, and the concept element is the formal representation of the knowledge level in the phenomenon element. Therefore, with studies on the correlation of the concept element, the correlation can be established in the knowledge intensive levels between different phenomenon elements of the system, laying a foundation for the future study of the formation of knowledge. Meanwhile, the research on the diachronism and formal definitions of phenomenon, state, phase and flow in the phenomenon system also provide a basis for future studies of the motion and evolution of knowledge-type complex systems. In order to further construct the structure of the knowledge granule, the following basic attributes must be satisfied: (1) granule intension: the identification of a certain granule in a given environment that is distinct from other granules is reflected as equivalence and correlation between the internal elements of the granule, as well as the boundary for the higher level of granular knowledge; (2) granule extension: the interaction between granules, or the local requirements that granules in the knowledge system must follow. The interaction among a large number of granules is reflected as the effect of granules on the environment. By studying the phenomenon element and its implied knowledge
correlation, intensive attributes in the granule are satisfied in this study. All correlated concept elements are the basic elements to constitute the concept granule, and its boundaries are determined by the equivalence class of specific knowledge in the concept element. The next step is to construct a higher level of knowledge granule structure, relative to phenomena elements, by studying granule extensive attributes between granules, and the rules of inference and calculation between phenomena. References [1] A. Skowron, J. Stepaniuk, J.F. Peters, Rough sets and infomorphisms: towards approximation of relations in distributed environments, Fundamenta Informaticae 54 (2003) 263–277. [2] A. Skowron, Jan Bazan, Marcin Wojnarski, Interactive rough-granular computing in pattern recognition, Pattern Recognition and Machine Intelligence 5909 (2009) 92–97. [3] B. Ganter, R. Wille, Formal Concept Analysis: Mathematical Foundations, Springer, 1999. [4] B. Ganter, R. Wille, Conceptual scaling, in: F. Roberts (Ed.), Applications of Combinatorics and Graph Theory to the Biological and Social Sciences, Springer Verlag, 1989, pp. 139–167. [5] J. van Benthem, Temporal logic, in: D.M. Gabbay, C.J. Hogger, J.A. Robinson (Eds.), Handbook of Logic in Artificial Intelligence and Logic Programming, Epistemic and Temporal Reasoning, vol. 4, Clarendon Press, Oxford, 1995, pp. 241–350. [6] D.M. Gabbay, I. Hodkinson, M. Reynolds, Temporal Logic C Mathematical Foundations and Computational Aspects, vol. 1, Clarendon Press, Oxford, 1994. [7] D.M. Gabbay, C.J. Hogger, J.A. Robinson, Handbook of Logic in Artificial Intelligence and Logic Programming, Epistemic and Temporal Reasoning, Clarendon Press, Oxford, 1995. [8] G. Liu, Rough set theory based on two universal sets and its applications, Knowledge-Based Systems 23 (2) (2010) 110–115. [9] K.E. Wolff, A conceptual view of knowledge bases in rough set theory, in: Proceedings of the Second International Conference on Rough Sets and Current Trends in Computing, vol. 7, 2000, pp. 188–192. [10] K.E. Wolff, Conceptual interpretation of fuzzy theory, in: H.J. Zimmermann (Ed.), EUFIT’98, 6th European Congress on Intelligent Techniques and Soft Computing, 1998, pp. 555–562. [11] J. Zhao, L. Liu, Construction of concept granule based on rough set and representation of knowledge-based complex system, Knowledge-Based Systems 24 (6) (2011) 809–815. [12] L.A. Zadeh, Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic, Fuzzy Sets and System 19 (1997) 111– 127. [13] L. Wei, J.J. Qi, Relation between concept lattice reduction and rough set reduction, Knowledge-Based Systems 23 (8) (2010) 934–938. [14] Qiang Wu, Zongtian Liu, Real formal concept analysis based on grey-rough set theory, Knowledge-Based Systems 22 (1) (2009) 38–45. [15] R. Yan et al., Research on the model of rough set over dual-universes, Knowledge-Based Systems 23 (8) (2010) 817–822. [16] R.A. Kowalski, M.J. Sergot, A logic-based calculus of events, New Generation Computing 4 (1986) 67–95. [17] Y.Y. Yao, Concept lattices in rough set theory, in: S. Dick, L. Kurgan, W. Pedrycz, M. Reformat (Eds.), Proc. of the 2004 Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS 2004). IEEE, 2004, pp. 796–801. [18] Y.Y. Yao, Granular computing: basic issues and possible solutions, in: The 5th Joint Conference on Information Sciences, New Jersey, USA, vol. 1, 2000, pp. 186–189. [19] Y.Y. Yao, A partition model of granular computing, Transactions on Rough Sets 1 (2004) 232–253. [20] Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences 11 (1982) 341–356. [21] Z. Pawlak, A. Skowron, Rough sets: some extensions, Information Sciences 177 (2007) 28–40.
Contents lists available at SciVerse ScienceDirect
Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys
Extended representation of the conceptual element in temporal context and the diachronism of the knowledge system Jian Zhao a,b, Lei Liu a, Liang Hu a,⇑ a b
College of Computer Science and Technology, Jilin University, Changchun 130012, PR China College of Software, Changchun Institute of Technology, Changchun 130012, PR China
a r t i c l e
i n f o
Article history: Received 27 July 2011 Received in revised form 8 March 2012 Accepted 8 March 2012 Available online 15 March 2012 Keywords: Granular computing Formal concept analysis Rough set Diachronism Knowledge system
a b s t r a c t Based on granular computing, the conceptual element was established as the conceptual category of knowledge, with formal concept analysis as a basic tool, and the indiscernibility relation was established for rough set theory. The conceptual element in temporal context was then extended to the phenomenon element yielding temporal granular representation. Through the introduction of partially ordered timeseries, the diachronism of the knowledge system was studied. An isomorphic mapping was conducted between the sequence structure and concept lattice structure, consisting of a phenomenon element and other time-series materials, providing visualization of the movement of the phenomenon element over time. Dynamic modeling was applied to knowledge granules in the open problem domain of context and time, providing a theoretical basis for reference. 2012 Published by Elsevier B.V.
1. Introduction In general, knowledge is considered to be formed by information gathering, processing and interpretation, from the vast and complex information environment. The source of knowledge is usually taken as conceptual descriptions of the visual experience of objective targets, made by observers in the real world, such as descriptions of observed objects like celestial bodies or atoms, their structure and laws of motion. Such sources and characteristics of human knowledge are described such that time and space are the transcendental forms of human consciousness, with their effects on the corresponding objects of sensation forming the phenomenon while the phenomenon is comprehensively judged to form the knowledge. In other words, although the materials for the formation of a phenomenon are acquired, the forms still exist firmly in the consciousness, due to the a priori effect of time and space on the phenomenon. For example, the knowledge formed by general observers making judgments of objects is fixed, and essentially consistent. However, the sources of knowledge also contain descriptions of symbols or conceptual structures which cannot be obtained by visual experience of the language system or ideology. Fixed and consistent knowledge of such phenomena cannot be obtained, and ambiguity and distortion occur due to differences in human ⇑ Corresponding author. E-mail addresses: [email protected] (J. Zhao), [email protected] (L. Liu), [email protected] (L. Hu). 0950-7051/$ - see front matter 2012 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.knosys.2012.03.006
subjective factors or changes in perspective. In order to obtain understandable knowledge from such phenomena in a given field, various phenomena must be correlated with differences and generalities removed to form a knowledge community. Thus, the entire field context can be understood as a system, defined by Ervin Laszlo as: ‘‘the system is an organic whole with structures and specific functions composed of a number of interrelated and interactive elements’’. This system consists of interrelated intellectual phenomena, and the interaction can be understood as correlated phenomena brought into the knowledge community with boundaries, out of which non-correlated phenomena are abandoned. Based on the above precepts, the basic element constituting the knowledge system was named a phenomenon element in this study, and the system was called a phenomenon system, where the phenomenon is taken as material for the formation of knowledge. The states of the phenomena in this system will subsequently be changed and re-integrated to form a new knowledge community, when changes occur in the boundary or the nature of the field context. This activity reflects that the system is in constant motion, showing the historical state at a certain point in time. The phenomena are flowing through time, and in this study, this nature of the phenomenon system is called the diachronism of the phenomenon system. The formal representations were carried out on the concept and time categories of the phenomenon system, as well as on their diachronism. First, accuracy and continuity could not be achieved due to limitations of the knowledge acquisition in the phenomenon system, such that all materials for the formation of knowledge
J. Zhao et al. / Knowledge-Based Systems 33 (2012) 136–144
were represented as granular [18], so the knowledge formed was called a knowledge granule. In this study the focus was on the intensive and extensive attributes of the knowledge granule. When the granule is taken as a whole, its intensive attribute is considered to be determined by the attributes of all sub-granules. When the granule is taken as a part, its extensive attribute is considered to determine how the granule will be sensed and how it will be correlated with other granules. A detailed discussion on the association can be found by referring to this author’s articles [11]. Formal concept analysis (FCA) was used as a basic tool to describe the knowledge granule in this study. The basic unit is called a concept to describe the knowledge object in the FCA, and the structure of concept lattice was adopted to describe the sequence relations between concepts. However, the knowledge granule has its own structure and compositions, making it the sub-system under the phenomenon system, which precisely reflects the relationship of generalization and specialization between the internal components of the knowledge granule. In addition, the attributes of the knowledge granule may have multiple characteristics. The color brown for example, is not only different from red, blue and yellow but also from such characteristics as triangle, wooden and large. Thus the formation of knowledge is largely ambiguous, with fuzzy boundaries, necessitating that the overall system of knowledge granules has the capacity to roughly identify, classify, and organize the various knowledge granules. Therefore, it is necessary to classify the represented characteristics of knowledge granules by equivalence class. Concepts including indiscernibility relation in rough set theory are used to facilitate such classifications. Rough set theory and scale theory of FCA are combined to construct concept elements as the basis for formation of the phenomenon, and thus the knowledge system is more accurately represented. Meanwhile, in order to study the motion in the knowledge system, we must describe its state changes during the passage of time. In this study, the phenomenon change is understood to occur in a specific, discrete time slice, and the formal representation of the time slice and time sequence are given based on FCA. The phenomenon element is constructed as a basic element of the knowledge system,using the time elements embedded in the concept element. The remainder of this study was organized within Sections 2–5, as described below. Section 2 contains comparisons of the strengths and limitations between this study’s theory and those of related works, within this field. The basic structure of the knowledge granule was given in Section 3, as well as the definition of the conceptual element with granular structure. The basic form of granular time was presented in Section 4, where the conceptual element was extended to the phenomenon element in temporal context, and the formal definition of the phenomenon system was given. Additionally, related definitions of the state, phase and flow of phenomenon knowledge in the evolutionary process were also provided. Based on the partial order of concept lattice, time series was constructed, so as to study the diachronism of the phenomenon system. Visualization of the movement of the phenomenon element was provided by descriptive an example in Section 5. 2. Relevant work Several influential and effective methods and theories have emerged in areas like knowledge representation, processing and description and the solution of structured problems in the objective world. 2.1. Ideas regarding granular computing and rough set Granular computing is an emerging discipline in problem solution methods and information processing areas, according to which
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[2,12,18,19], human intelligence is provided with global analysis capacity, or the ability to process problems in different granular worlds and observe and analyze objective phenomena with different granularities. The work of Zadeh [12] and Yao [17–19] have provided the basic representation of granular structure, utilizing tools such as fuzzy set, neighboring system, or rough set, depending on which relevant approximation theory and knowledge representation method was developed. Rough set theory [1,8,15,20,21], put forward by Polish Scholar Z. Pawlak, views knowledge as the division of given universe U. In other words, it granulizes irrelevant knowledge through equivalence class and utilizes the granular knowledge as the main processing object. Definition 2.1. Set a universe U and a cluster of equivalence relation A on U, a 2-tuple (U, A) is called a knowledge base with respect to the universe U. If there exists an equivalence relation B # A(B – U), then a \B is called an indiscernibility relation of B, denoted by IND(B) T and "u 2 U, [u]IND(B) = "R2B[u]IND(R). U/IND(B) = {[u]IND(B)j"u 2 U} indicates the knowledge related to IND(B) and is called a B-baseset (or B-knowledge) of the knowledge base (U, A) with respect to the universe U. This theory is an expansion of the classic set theory. It inserts knowledge used for classification into the set: the degree of a given object belonging to set X can be judged according to upper approximation and lower approximation. Definition 2.2. Upper and lower approximationsSet a knowledge base (U, A). So "G # U and an equivalence relation B A; the lower and upper approximations of subset G with respect to the equivalence relation B are respectively defined [21]:
BðGÞ :¼ fg 2 Uj½gB # Gg ¼ [fF # Gj8F 2 U=Bg BðGÞ :¼ fg 2 Uj½gB \ G – Ug ¼ [fF 2 UjF \ G – Ug Theories like granular computing have certain advantages in modeling approximate representations of knowledge semantics and structured problems in the objective world. However, the dynamic contact among knowledge granules, as well as the research on the overall structure of the knowledge space constituted by all knowledge granules and its evolution in time has been ignored, and therefore, it has certain limitations in processing open problem domain. Based granular computing, a study was carried out on the overall structure of a knowledge system consisting of knowledge granules, as well as its evolution over time, so as to provide a better understanding of open knowledge granulation and evolution in context and time. 2.2. Formal concept analysis The Classical FCA theory usually begins with the basic concepts of some particular formal context, such as a single-value context, which focus on the object sets, attribute sets, and their relationships. Definition 2.3 [3]. A single-value context is a 3-tuple K :¼ (G, M, I), where G and M are sets, and the binary relation I # G M and elements of G and M are called the objects and attributes accordingly. (g, m) 2 I or gIm indicates the object g has the attribute m. Definition 2.4 [3]. Suppose A is a subset of object set G, and define f(A) :¼ {m 2 Mj"g 2 A, gIm} (the set of common attributes of objects in A). Accordingly, suppose B is a subset of attribute set M, and define. g(B) :¼ {g 2 Gj"m 2 B,gIm} (the set of the objects of all attributes in B).
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Definition 2.5 [3]. A formal concept of the formal context (G, M, I) is a 2-tuple (A, B), where A # G, B # M and f(A) = B, g(B) = A. A is called the extent of the concept (A, B), while B is the intent thereof. B(G, M, I) is used to indicate the set of all concepts on the context (G, M, I). Let c(g) = (g(f(g)), f(g)) is an object concept; l(m) = (g(m), f(g(m))) is an attribute concept. Concepts form a ‘‘concept lattice’’ structure via ‘‘hierarchical sequence’’, which represents the relation between generalization and specialization among the concepts. FCA highlights the formalized and structured representation of knowledge. However, it has certain limitations in processing fuzzy and uncertain knowledge, in terms of semantic representation of knowledge. In order to deal with fuzzy or uncertain knowledge, it is necessary to use multivalued context in formal concept analysis for combining two theories in the form [9,10,13,14,17]. The definition of the multivalued context is first given as follows.
defined sufficiently based on RS. Thus, redefinition and a better understanding were provided for time, process, flow, state and other concepts in the event calculus and other theories.
Definition 2.6 [4]. Let (G, M, W, I) be a multi-valued context, with a ternary relation (I # G M W), while Sm(m 2 M) is called the conceptual scales. If m(G) # Gm, then the objects in the scale are the values of the attribute m, while the attributes in the scale are those with single-value context. The single-value context (G, N, J) is derived, where N ¼ [m2M M m and gJ(m, n) , m(g) = w and w Imn. The formal context of the multi-valued attributes can depict the formal context of the exact relationship between the objects and the attributes, and more accurately reflect the connectivity and transmissibility between the concepts, with the condition that the knowledge structure is basically constant. Two theories were combined in this study, where a rough formal concept was introduced for a semantic representation of phenomenon elements. Specific propositions will be given in Section 3.
In general, granular structure can be established in a special domain based on specific context. A certain constraint is defined in this domain, where granular connotation is expressed by constraint rules, reflecting the general characteristics of all elements in the domain, while granular extension is the element set in this domain satisfying the constraint, or the set of elements covered in granular structure. However, context relation represents the external attribute of interaction among granular structures. The formalized representation of the conceptual category of knowledge is carried out through granular structure as described below. First, considering that the conceptual categories of most phenomena are ambiguous, containing indistinct boundaries, a knowledge domain is required with the capacity for rough identification, classification and organization of scattered grains. To that end, the indiscernibility relation of rough set theory was used to process incomplete and inadequate information, and then the knowledge domain was divided into different equivalence classes, so the phenomena are classified according to different attributes or characteristics. Meanwhile, the granular structure forms of the conceptual category in phenomena are given by combining the scale theory of FCA. The isomorphism between both theories is first given by the following propositions, and a detailed proof of the proposition can be found by referring to the author’s article [11].
2.3. Ontology engineering Meanwhile, a number of theories and methods have been proposed on the research of timing materials in the knowledge representation field, such as Temporal Logic [5–7] and Event Calculus [16]. According to situation calculus, it is stated that low is a function and predicated to change from one scene to another. The event calculus is designed to allow reasoning in the time interval, including talking time and time interval capability. For example, Initiates(e, f, t) indicates that the event e occurs in the time t, leading to the flow f is true. Terminates(w, f, t) indicates that the flow f is no longer true. Happens(e, t) indicates that the event e occurs at the time t. Clipped(f, t, t2) indicates that f is terminated by a certain event in a moment from t to t2. The following facts were provided by an event calculus axiom: a flow will be true in a moment, if this flow is induced by an event in a certain past moment and not terminated by a certain event. The axiom formalization is shown as follows:
EVENT CALCULUS AXIOM Tðf ; t 2 Þ () 9e; t
3. Knowledge granule and concept element In granular structure, there are three basic attributes that must be satisfied, including: (1) context attributes indicating the existence of a granule in special environment; (2) internal attributes reflecting elemental interaction in a granule, (3) external attributes reflecting the interaction between a granule and others. Based on the above attributes, the basic form of granular structure is given below. Definition 3.1. Granular structureA granular structure is described with a triple (EG, IG, R), where EG and IG are the sets called as granular extension and granular connotation respectively. Binary relation R # EG IG is called as context relation.
Proposition 3.1. Any knowledge base (U, A) is given, where U is the universe and A :¼ {Bmjm 2 M}. S(U, A) :¼ ((U, A, W, I), (SBjB 2 A)) is known, where SB is the nominal scale with its derived context (U, N, J). c is set to represent the mapping of object concept in the context (U, N, J), and thus, if (u, v) 2 IND(P)((u, v) 2 U2, P # A), then c(u) = c(v). It is known by Proposition 3.1 that if a knowledge base (U, A) is given, then a many-valued context can be generated under the effect of scale operator, and the derived context of a many-valued context is
Happensðe; tÞ ^ Initiatesðe; f ; tÞ
^ ðt < t2 Þ ^ :Clippedðf ; t; t2 Þ Clippedðf ; t; t 2 Þ () 9e; t 1
Happensðe; t 1 Þ ^ Terminatesðe; f ; t 1 Þ
^ ðt < t 1 Þ ^ ðt 1 < t2 Þ Thus, equation Happens(Rise(Moon), 11:00) can be used to illustrate that the moon rises at 11:00. First-order logic is commonly used as a research tool in the majority of these theories. On this basis, direct descriptions of an object and its category affiliation, scenarios, processes, etc. can be obtained with easy maneuverability. However, limitations exist in the semantic representation of objects, the studies of object evolution caused by changes in the semantic environment, as well as the representation of time order. The time-series was defined on the basis of FCA lattice structure. Meanwhile, knowledge was
U; fðB; ½uB ÞjB 2 A; ½uB 2 U=Bg; JÞ; i.e., the knowledge base (U, A) can be expressed with the derived context of a many-valued context of a formal concept. Then, the consistency between basic concepts of both theories is a basis for establishing granular structure. Based on this proposition, the formalized description of a conceptual category is given and called a concept element. Definition 3.2 (Concept element). A knowledge base (U, A) is given as well as a many-valued context (G, M, W, I), of which, M # A, G # U/M,W :¼ {[u]Bju 2 G, B 2 M} and I # G M W. The concept element is the derived context of a nominal scale as (G,{(B, [u]B)jB 2 M, [u]B 2 U/B}, J). uJ(B, [v]B) , [u]B = [v]B is satisfied, where "u, v 2 U and u – v. It is abbreviated as v :¼ (G, N, J), where N :¼ {(B, [u]B) jB 2 M, [u]B 2 U/B}.
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It is known from granular structure of concept element that the constraints among all elements with similarity in the domain U can be reflected in granular connotation {(B, [u]B)jB 2 M, [u]B 2 U/B} through the indiscernibility relationship. While granular extension G is a set of elements satisfying this indiscernibility relationship, and specific characterization of constraints are given by J. Thus, conceptual category is a granular structure satisfying granular attributes.
4. Expansion of the concept element and diachronism of a phenomenon system 4.1. Formal representations of basic elements in a phenomenon system In order to ensure the individual continuity and uniqueness of a certain phenomenon unit in a phenomenon system, which will avoid the occurrence of multiple replications, it was necessary to clarify the definition of its state and the changes occurring in the time dimension. For example, in order to determine if a fresh or rotten apple is the same object, it is required to give reconfirmation according to the time and state. Thus, the phenomenon unit was a necessary part of the inner, embedding time sequence. First, we considered how to define the state of a phenomenon unit. The knowledge object studied is essentially a conceptual description of the visual experience of objective targets, by observers in the real world, which exists as a phenomenon within the structure of a granule surrounding the knowledge core. However, the knowledge core is essentially accumulated and formed by the knowledge in the core, from correlated concept elements. When the current content of specific knowledge is investigated, the new knowledge restructuring will be carried out by changing perspectives and intercepting different equivalence class slices. Thus, the state of knowledge in flux can be considered as the set of equivalence class slices with different perspectives. However, if such change is placed in the field of a complex system for observation, it won’t involve human factors, but will proceed with a self-organization approach, so the state of knowledge is required to be described as the set (knowledge core) of knowledge slices shared in different concept elements of the granule at different points in time. Then, a further study of the composition of knowledge granule is required to determine how to embed the time element in the concept element. Time is continuous in the real world, while discrete in this context set. For example, the maturation process of an apple occurs in continuous time, while knowledge such as: The nature of the Soviet Union will change from one state to another due to changes in politics and world trends. Therefore, the motion of knowledge exists in a time slice rather than the uniform time of the physical world. Thus, the occurrence of a general event in the knowledge system is considered to consist of state slices and time slices of knowledge. In order to discuss discrete time, the time must be represented by granular structure. The granular time is defined as a time element, and it is represented formally as a multivalue context, for example: s :¼ ((GT, MT, WT, IT), (Stjt 2 MT)). The formal context of time elements is shown in Table 1. Time is the element of time domain Gs, representing the time slice, which is known as the time object. Gs indicates the set of all time points in an interval. Its elements can be expressed by ordered numbers (1, 2, 3, . . .), or by strings such as ‘‘851914 (representing 14 o’clock on May 19, 2008)’’ which can also be used to indicate time points with specific meanings. Other time attributes are the specific descriptions of time objects. In this multi-valued context, various scales can be used according to different granularities. The nominal scales can be applied to accurately indicate the granulation of time, and other scales (such as ordinal scales) may be used to indicate time in a less precise, more fuzzy manner.
Space–time in the physical world generally consists of continuous four-dimensional space–time, where an event’s occurrence is constituted by all aspects of a specific space–time slice. For example, the Cold War is an event occurring at a number of space–time points, of which, sub-events such as the arms race, the space race, or the disintegration of the Soviet Union occur in a certain phase within this space–time slice. Each phase can also be continuously decomposed into increasingly more specific phases or sub-events. Each event has attributes in both time and space, while the space–time attributes also exist with boundaries. For example, the Cold War Period has a limited time boundary of 1947–1991, and the largest space boundary. At a certain spatial location, the Soviet Union has the largest time boundary, while the space boundary changes over time. The descriptions and hierarchical relations of the event can be analogous to the concept lattice structures of FCA, while the overall event and its space–time specifications can also be indicated by formal context. The major environment of discussion is the phenomenon system. For instance, in describing the Cold War event, the main focus is on the knowledge level world, like politics, economy, culture, ideology and so forth. Therefore, its space–time slice is different from the continuous space–time environment in the physical world, and its time is granular and discrete, while the space dimension is actually the state of a concept element at a certain point of time. In order to found a basis for descriptions on the motion characteristics of the phenomenon system, an analogy is carried out of the physical system, using related concepts of the physical system, such as state, process and phase. Above all, the concept of state is discussed. In the real-world system, it is very difficult to define the state. For example, the state of a moving person can be described in many aspects, such as physical, biological and psychological, etc. Generally, the ‘‘idealized’’ approach is adopted in order to simplify the study. In the mechanical motion system of physics, for example, a motional element can be idealized to a particle, while the state of a particle can be simply described with location and momentum. In the phenomenon system, the extended forms of concept element are first given with the time dimension included, so as to determine the state of concept element under the time dimension. Definition 4.1 (Time–concept element and its state). A knowledge base (U, A) and a time domain Gs are given as well as the concept element v :¼ ðGv ; Mv ; W v ; Iv Þ. Then, time–concept element is defined as v :¼ ðG; M v ; W v ; Iv Þ, where G = Gv Gs. The derived context is as K v :¼ ðG; fðB; ½uB Þ j B 2 M v ; ½uB 2 U=Bg; JÞ, the state of concept element v is defined as the connotation of object concept cv(g) of formal context Kv. In Definition 4.1, time is understood as being granular: a time granule is a time slice (time domain Gs), in which a number of time points are contained. The time is included in the inner of layer of the general concept element, i.e. G = Gv Gs is used to indicate the object domain of time–concept element. This form can represent the element changing with time points in the domain Gv, such as the description on objects in the time context like the dawn moon. The internal instant state of a concept element occurs in the current time, a specific time point in the time domain Gs. While discussing the situation of two states of the concept element are the same, it is understood that the concept elements at two different time points have equal values. Under the framework of formal Table 1 Formal context representation of time elements. Time
Year
Month
Day
Daytime
1 2 3
2008 2010 2010
May January April
19 12 14
14 16 7
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concept analysis, the value of a concept element is expressed by its scale Sm :¼ (Wm, Mm, Im)(m 2 Mv), i.e., if "m, n 2 Wm are equal, if and only if m, n have the same object concepts in Sm. Thus, the two states of concept element v are the same at the two different time points of g1, g2, or the corresponding object concepts of cv(g1) and cv(g2) have the same intents. Therefore, the state of the concept element here is expressed by the intent of the object concept cv(g), i.e., representing the set of different attributes contained in the object at a specific point in time. Example. The lunar phase is taken as a simple example. The knowledge can be described as the state (lunisolar longitude difference) changes of the moon during the period of a specific time slice such as (from the night of the second day of a lunar month to the daylight of the seventh day of a lunar month. By isolating a part of the knowledge, the time and event contexts are represented in Tables 2 and 3, where the element of time domain Gs, like ‘‘221’’, represents the time of 21 o’clock on the second day of a lunar month. By using the form of the time–concept element, it is indicated as Table 4. The object concept of element (221, 1) in the domain is ({(221, 1)}, {0, East 25}), so its state is represented as {0, East 25}. The time–concept element represents the process and attributes of an event, through which, the event is often manifested itself as a phenomenon. In order to describe the phenomenon in the time dimension under a more complex context, the time slice must be described under multiple dimensions (such as year, month, and date), as extending the time domain Gs to be the time element s, and the phenomenon is described as the form of time element and time–concept element. In the meantime, the state of the time–concept element will be extended to be a phase for describing the state of the phenomenon unit in the system at a specific point of time. A formal definition of a phenomenon unit in the phenomenon system is given in Definition 4.2. Definition 4.2 (Phenomenon element). A knowledge base (U, A) is given, and the time element is set as s :¼ ðGs ; M s ; W s ; Is Þ, and concept element as v :¼ ðGv ; Mv ; W v ; Iv Þ. The two-tuple P :¼ ððGs ; M s ; W s ; Is Þ; ðGv ; M v ; W v ; Iv ÞÞ is called a phenomenon element occurring in the knowledge base (U, A) in the time domain Gs, abbreviated to P :¼ (s, v). It can be seen from the above-mentioned definition of phenomenon element that its form is essentially the combination of formal contexts in two different domains. In order to study it in the
Table 5 Formal contexts of phenomenon element. Gs
221 307 412
Time element
Concept element
Day
Hour
Gv
Angle ()
Location
Second day of a lunar month Third day of a lunar month Fourth day of a lunar month
21 o’clock 7 o’clock 12 o’clock
1 2 3
0 15 45
East 25 East 28 West 13
Table 6 Formal contexts of extended phenomenon element. Gv Gs
(221, 1) (307, 2) (412, 3)
Time element
Concept element
Day
Hour
Angle ()
Location
Second day of a lunar month Third day of a lunar month Foruth day of a lunar month
21 o’clock 7 o’clock 12 o’clock
0 15 45
East 25 East 28 West 13
same domain, the domain of the time–concept element is used to merge Gv and Gs to be Gv Gs. Thus, the following definition is obtained: Definition 4.3 (Phenomenon and phenomenon space). The phenomenon element P :¼ (s, v) is extended to P :¼ ðGs ; Gv ; D; s; vÞ. Among them, D = Gs Gv. Thus, the derived context KP of the formal context of phenomenon has actually become the apposition context of Ks and Kv [3]. KP is called as the phenomenon context, and the object concept of the phenomenon context c(g)(g 2 Gs Gv) as the phenomenon at the time point g. The set of all phenomena under the phenomenon element P :¼ {c(g)jg 2 Gs Gv} is called as the phenomenon space. The above example of the lunar phase is expressed as the Tables 5 and 6 with the form of a phenomenon, of which, the time element represents the time part of phenomenon, while the concept element represents the event part of phenomenon: In the forms of a phenomenon element, a static description of the time is given, as the time element part. However, in order to describe the state changes of a concept element, it is necessary to give a description of the time fluidity or flow, the state is changed successively according to the context of time points. Therefore, it is necessary to include the formal representation of the time sequence to study the diachronism of a phenomenon system. 4.2. Diachronism of a phenomenon system
Table 2 Formal contexts of time. Gv
Angle ()
Location (apparent declination)
1 2 3
0 15 45
East 25 East 28 West 13
Table 3 Formal contexts of event. Gs
Day
Hour
221 307 412
Second day of a lunar month Third day of a lunar month Fourth day of a lunar month
21 o’clock 7 o’clock 12 o’clock
Table 4 Formal context of the time–concept element. G = Gv Gs
Angle ()
Location (apparent declination)
(221, 1) (307, 2) (412, 3)
0 15 45
East 25 East 28 West 13
The time domain here is considered as a poset ðGs ; 6s Þ, in which, 6s is the partial order defined by the time successive relation in Gs, which is called the time sequence here. Meanwhile, "g1, g2, g3 2 Gs is set to satisfy the following attributes: (1) g1 6s g1; (2) g1 6s g2, and g1 – g2, then g2 6s g1 is untenable; (3) g1 6s g2, and g2 6s g3, then g1 6s g3. Gs # R is taken here (R is the real number set), and Gs is set to indicate a closed interval [g, h] of R("g, h 2 Gs), so the time sequence as a partial order is the natural order of real numbers. Similarly, in the domain Gv of the time–concept element, a natural sequence is used as the index set, so the partial order relation 6v of the poset ðGv ; 6v Þ is also in accordance with the natural order of real numbers, where it is equated with the time sequence 6s. In order to discuss the diachronism of the phenomenon, it is necessary to extend the domain Gsto Gv Gs, so that the definition below is obtained. Definition 4.4. Two posets ðGs ; 6s Þ and ðGv ; 6v Þ exist, in which, the partial order is the time sequence. Then, the direct product ðGs Gv ; 6sv Þ of the two sets is also the poset. Among them, 6sv indicates (g1, g2) 6sv (h1, h2) , g1 6sv h1, and g2 6sv h2.
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Proposition 4.1. Mapping of #:Gs ? Gs Gv exists between two posets ðGs ; 6s Þ and ðGs Gv ; 6sv Þ, so the two posets are order preserving. Proof. It is shown by Definition 4.4.
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Proposition 4.3. The posets ðGs Gv ; 6sv Þ; ðpðBðK s ÞÞpðBðK v ÞÞ; 6Þ exist, (of which, the partial order relation 6 is the set-inclusion relation ). If B(KP) is generated by ordinal scale, then these two posets are order preserving.
h
Proposition 4.2. The posets ðGs Gv ; 6sv Þ and (B(KP), 6) exist, of which, B(KP) is the concept lattice of the phenomenon context KP, 6 is the level sequence in B(KP). If B(KP) is generated by the ordinal scale, then the two posets are order isomorphic. Proof. "g1, g2 2 Gs is taken, and then g1 6s g2. The mapping is set as c:Gs ? B(Gs), i.e., c is the object concept mapping of g 2 Gs. The ordinal scale is taken as O :¼ (n, n, 6) to generate derived context, and epitaxial chains are formed in attribute values, so it is proven that: f(g1) f(g2), i.e. g(f(g1)) # g(f(g2)). Therefore, the object concept is shown as c(g1) :¼ (g(f(g1)), f(g1)) 6 c(g2) :¼ (g(f(g2)), f(g2)). That is, g1 6s g2 ) c(g1) 6 c(g2), and vice versa. Therefore, if g1 6s g2 , c(g1) 6 c(g2) exists, and c is known as bijection, then the two partially order sets ðGs ; 6s Þ and (B(KP),6) are order isomorphic under the mapping c. It is also known by Proposition 4.1 that ðGs ; 6s Þ and ðGs Gv ; 6sv Þ are order preserving, thus it is proven that ðGs Gv ; 6sv Þ and (B(KP), 6) are also order isomorphic. It is known from Proposition 4.2 that a structure like concept lattice can be used to represent the time sequence vividly, so that each object concept (i.e. phenomenon) in concept lattice can be considered in the passage of time. Meanwhile, due to the concept lattice generated by using ordinal scales, the partial order relationship can be considered as total ordering. All the following phenomenon context KP discussed will be derived by ordinal scale. The following will use the ordinal scale to obtain the derived context KP of the phenomenon P. The obtained phenomenon context KP is shown in Table 7. The lattice structure is generated by the phenomenon context KP as follows (see Fig. 1). The description of the concept elements discussed above specifies that in a certain time domain, phenomena are formed in concept elements, and divided into different stages by discrete time points. When the knowledge granule at each stage shows two states simultaneously, the state of the time dimension and knowledge are described. As previously discussed, at the time point g 2 Gs Gv, the knowledge state of concept element v is represented by the corresponding object concept cv(g) of Kv. However, the state of the time dimension is described by using the corresponding object concept cs(g) of Ks. Therefore, a combination of these two states can be used to represent the time sequenceknowledge state of concept element at a certain time ("g 2 Gs Gv), which is called as the phase. h Definition 4.5 (Phase). The derived contexts of time element s and time–concept element v are set as Ks and Kv, cs(g) and cv(g) are the object concepts of Ks and Kv respectively, and "g 2 Gs Gv. The mapping is defined as p : BðKÞ ! IðKÞ, of which, K representatives Ks and K v ; IðKÞ is the set of all object concept connotations in the formal context K. Thus, the phase of the phenomenon at a certain point of time g is defined as the form ðpðcs ðgÞÞ; pðcv ðgÞÞÞ . The set of all phases under the phenomenon element P is denoted as S :¼ fðpðcs ðgÞÞ; pðcv ðgÞÞÞjg 2 Gs Gv g, which is called as the phase space. For example, at time point 221, the moon is located at 0 on the morning of the second day of the lunar month, that is, the phase in this time is (morning of the second day of a lunar month, 0). By referring to Proposition 4.2, the component phase is also placed into the time sequence, so that the proposition below exists.
Proof. "g1, g2 2 Gs Gv is set and the mapping as l:Gs Gv ? p(B(Ks)) p(B(Ks)). It is proven by the proof process of Proposition 4.2 that g1 6s g2 ) f(g1) f(g2), i.e., ðGs Gv ; 6sv Þ and (p(B(Ks)), 6) are order preserving, and it is proven by the same methods that ðGs Gv ; 6sv Þ and (p(B(Kv)), 6) are order preserving. Then, by Definition 4.4 it is shown that (p(B(Ks)), 6) and (p(B(Kv)), 6) are order preserving with (p(B(Ks)) p(B(Kv)), 6) respectively, and the posets ðGs Gv ; 6sv Þ and ðpðBðK s ÞÞ pðBðK v ÞÞ; 6Þ are order preserving. It can be seen from Propositions 4.2 and 4.3 that both the phenomenon and component phase have time attributes, and change over time sequence. The knowledge events occurring in the time sequence are uniformly known as flow events. The flow of any object is essentially the transmission of resources along the joint lines in a network system. Thus, such flow can be informally defined as a two-tuple {resources, joint line}. For example, the flow of the central nervous system can be composed of two-tuple {pulse, neurons connection}. The flow of the Internet can be composed of twotuple {packet, cable}. In order to characterize the abstract flow of the knowledge in the knowledge system, the time flow can be defined as two-tuple {flow event, time sequence}. The formal definition is given for the flow in the phenomenon system below. h
Definition 4.6 (Flow). P :¼ ðGs ; Gv ; D; s; vÞ is set as the phenomenon element, of which, the mapping is set as r:Gs Gv ? E, with E representing the set of flow events (phenomenon, phase). The set F :¼ {(r(g), 6s)jg 2 Gs Gv} is called the flow. If the mapping r is set as c, and c:Gs Gv ? P (P is the phenomenon space), then F is called as the phenomenon-time flow. If the mapping r is set as l:Gs Gv ? S (S is the phase space), then F is called as phase–time flow. 5. Time flow chart 5.1. Introduction of an example Data used in this example is a summary made by China Auto Market in relation to the sales status of automobiles of different models and brands on the Chinese mainland. We utilized the sales data from the first quarter of 2008 to the second quarter of 2011, for the Volkswagen Jetta, Toyota Corolla and Audi A6L, as shown in Fig. 2. In this example, we will formalize the data into phenomenon context and phenomenon and visualize the passage of time by applying the concepts and theories proposed in this paper. We first set the time element: s :¼ ðGs ; Ms ; W s ; Is Þ, where Gs refers to the aggregation of all time points within a given interval. Choose ordinal real numbers with specific semantics as the elements of this aggregation, so that 103 refers to the third quarter of 2010. Consequently, all the time points within the interval from the first quarter of 2008 to the second quarter of 2011 form a poset (Gs, 6) with 6 as the partial ordering relation. In Ms :¼ {year, quarter}, Ws refers to the aggregation of the attribute values of years and quarters. The formal context of the time element s is shown in Table 8. Meanwhile, set the concept element: v :¼ ðGv ; M v ; W v ; Iv Þ, where Gv :¼ {1, 2, 3, . . . , 14} refers to the aggregation of all events, and Mv :¼ {VW Jetta, Toyota Corolla, Audi A6L} refers to the attribute aggregation of the sales volumes of the three models, and Wv refers to the aggregation of the corresponding sales values of various models. The formal context of v is shown in Table 9.
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Table 7 Phenomenon context. Gv Gs
Time element 2nd day
(221, 1) (307, 2) (412, 3)
(221,1 )
Concept element 3th day
(307,2 )
4th day
Morning
(412,3 )
Evening of
Morning of
Morning of
the 2nd day
the 3th day
the 4th day
0°
15°
45°
Fig. 1. Representations of concept lattice formed in the phenomenon context on the time sequence.
Consequently, according to Definition 4.3, the phenomenon context KP can be found, as shown in Table 10. The concept lattice can be formed, based on the given phenomenon context, by using the grid scale that consists of two ordinal scales,one of the ordinal scales refers to the ascending relationship of a certain model within the sales interval, e.g., for Toyota Corolla, the sales interval is [24,340, 54,459] and its attribute can be defined between P20,000 and P50,000. The other ordinal scale refers to the ascending relationship of a certain model by quarter. Figs. 3–5 indicates the concept lattices of three models respectively. For the sake of simplicity, the phenomenon mark c(Gs) can be abbreviated as Gs, e.g. c(81, 1) can be abbreviated as 81. In Figs. 3–5, the nodal points with marks are the object concepts indicated in black circles, referring to the phenomenon c(g) at a time point g. For example, the nodal point marked 83 in Fig. 4 refers to the phenomenon that the sales volume of the Toyota Corolla exceeded 30,000 in the third quarter of 2008. Fig. 6 indicates the sales status of Corolla of Toyota with time variation in 4 years, and such legend indicating the flow process of flow events (see Definition 4.5) is called ‘‘time flow chart’’. The graduations on the sales volume dimension refer to P20,000, P30,000, P40,000 and P50,000, respectively, while those on the quarter dimension refer to P1, P2, P3 and P4 respectively. 5.2. Analysis on effectiveness and feasibility of time flow chart As shown in Figs. 3–5, the hierarchical order relation between phenomena is reflected by Hasse diagrams of traditional concept Audi A6L Toyota Corolla VW Jetta
70000 60000 50000 40000 30000 20000 10000 0
2Q.2008 4Q.2008 2Q.2009 4Q.2009 2Q.2010 4Q.2010 2Q.2011
Fig. 2. Sales chart of three models within 4 years.
Noon
Evening
0
15
45
Table 8 Time element of car sales. Time
Year
Quarter
81 82 83 84 91 92 93 94 101 102 103 104 111 112
2008 2008 2008 2008 2009 2009 2009 2009 2010 2010 2010 2010 2011 2011
1 2 3 4 1 2 3 4 1 2 3 4 1 2
lattice with the dimensions of time and sales volume. Such relation mainly indicates the specialization and generalization between ‘‘concepts’’ in the concept lattice, and it is impossible to completely visualize the flow process of phenomena in the time order. Specifically, taking the concept lattice as indicated in Fig. 4 for example, Black Nodes (81, 91) and (101, 111) are connected by a black line segment, and the structure of concept lattice only indicates that phenomenon c(101) or c(111) is the parent concept of phenomenon c(81) or c(91). Moreover, Phenomena c(81), c(91) or c(101), c(111) coincide at the same node due to the selected scale graduations. It can be observed from above two points that the Hasse diagrams based on traditional concept lattice is neither able to show the successive relation between concepts in respect of time, nor to indicate that the phenomena change according to the ordered time dimension. For the purpose of illustrating the flowability and temporal sequentiality of phenomena, it is necessary to unify the time order of phenomena and linearity in form. It can be observed from Proposition 4.1 that the posets ðGs ; 6s Þ and ðGs Gv ; 6sv Þ are orderpreserving, and it should be noted that Gv is defined as the natural number set in this example, i.e., the partial ordering relation ‘‘6v’’ in poset ðGv ; 6v Þ satisfies the condition of less-than relation
Table 9 Concept element of auto sales. VW Jetta
Audi A6L
Toyota Corolla
58,018 51,705 41,955 40,593 55,647 55,969 55,254 59,848 52,994 53,765 53,004 62,759 50,353 46,409
20,424 19,374 20,226 19,049 23,457 22,936 25,895 26,477 28,632 28,717 29,952 26,098 22,228 25,387
41,901 39,710 38,938 36,593 42,690 35,892 43,823 42,525 31,184 31,410 54,459 46,645 38,559 24,340
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J. Zhao et al. / Knowledge-Based Systems 33 (2012) 136–144 Table 10 Phenomenon context.
Sales Volume 19049-29952
Gv Gs
Year
Quarter
VW Jetta
Audi A6L
Toyota Corolla
(81, 1) (82, 2) (83, 3) (84, 4) (91, 5) (92, 6) (93, 7) (94, 8) (101, 9) (102, 10) (103, 11) (104, 12) (111, 13) (112, 14)
2008 2008 2008 2008 2009 2009 2009 2009 2010 2010 2010 2010 2011 2011
1 2 3 4 1 2 3 4 1 2 3 4 1 2
58,018 51,705 41,955 40,593 55,647 55,969 55,254 59,848 52,994 53,765 53,004 62,759 50,353 46,409
20,424 19,374 20,226 19,049 23,457 22,936 25,895 26,477 28,632 28,717 29,952 26,098 22,228 25,387
41,901 39,710 38,938 36,593 42,690 35,892 43,823 42,525 31,184 31,410 54,459 46,645 38,559 24,340
Quarter 1-4
≥ 19000
≥1
≥ 22000
≥2 81
≥ 25000
≥3
91 111
≥4
82 82
101
83
112
84 93
102 103
94 104
Fig. 5. Concept lattice for sales status of the Audi A6L.
Sales Volume 41955-62759
Quarter 1-4
≥ 40000
2008
≥2
≥ 55000
2009
≥3 101 111 81 91
≥4
112
γ(81)
2010
γ(83)
2011
82 102 92
Sales Volume
≥1
≥ 50000
83 103
γ(82)
Time flow
84
93
γ(93) γ(94)
γ(91) 94
γ(92)
Quarter
γ(84)
γ(104)
104
γ(101)
Fig. 3. Concept lattice for sales status of the Volkswagen Jetta.
γ(102)
2008
γ(103)
γ(111) Sales Volume 24340-54459
Quarter 1-4
≥ 20000
γ(112)
2010
≥1
≥ 30000
≥2
≥ 40000
2011
≥3
≥ 50000
2009
101 111 82 92 102
81 91
Fig. 6. Time flow chart for the sales status of Toyota Corolla.
83
93 103
≥4
112
84 94 104
Fig. 4. Concept lattice for sales status of the Toyota Corolla.
between natural numbers. In addition, it can be observed from Proposition 4.2 that the posets ðGs Gv ; 6sv Þ and (B(KP), 6) are order-isomorphic in case that the concept lattice B(KP) is generated by ordinal scale, which means the hierarchical order of concept lattice consisting of phenomena are order-preserving with the lessthan relation of natural numbers, and even isomorphic, which provides the theoretical basis for temporal sequentiality. Hence, the poset ({(81, 1), (91, 5), (101, 9), (111, 13)}, 6sv) (subset of the time point set Gv Gs, as shown in Table 10) is orderpreserving with the poset ({1, 5, 9, 13}, 6v) (subset of the time domain Gs), and is isomorphic with ({c(81), c(91), c(101), c(111)}, 6).
Therefore, it can be guaranteed that the following facts are reasonable under the framework specified in this paper: the starting time point for Corolla of Toyota is (81, 1), and its corresponding phenomenon is c(81) (c(81), implying the sales volume is greater than 40,000. When the time point reaches (101, 9), its corresponding phenomenon turns out to be c(101) (c(101), implying the sales volume is greater than 30,000 and less than 40,000, and so forth. Moreover, it should be explained that the definition of time points in this example is based on semantics, e.g., ‘‘81’’ represents the first quarter in the year of 2008. However, the following problems would be usually encountered as dealing with more complicated time semantics: (1) it is impossible to use valid numbers to represent the particular implications; (2) there is no guarantee that time semantics indicates the satisfaction of less-than relation of real numbers or natural numbers; (3) the representation of time semantics may be repeated. With consideration of the aforesaid problems, the time point is represented as ‘‘Gs Gv ’’ as required in this paper.
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The time flow chart (Fig. 6) proposed in this paper shows that the flowing mode of phenomena with time variation is different from that as indicated in the traditional Hasse diagram (Fig. 4). According to Hasse diagram, different marks may coincide at the same node, which implies that different phenomena may have the same connotation. However, the time points for different phenomena are not coincident. For example, Phenomena ‘‘101’’ and ‘‘111’’ coincide at the same time point in Fig. 4. Although both imply ‘‘the sales volume in the first quarter is greater than 30,000’’, they represent 2010 and 2011 respectively. In this way, it is impossible to completely visualize the time order in the same concept lattice, while the bidimensional Hasse diagram is represented with three dimensions (year, quarter and sales volume) in the time flow chart. It is noteworthy that the phenomenon context is cut into slices by different time values in accordance with the dimension of ‘‘year’’, and each slice represents the set of phenomena occurring in different quarters of the year. For the phenomenon element P :¼ ðGs ; Gv ; r; s; vÞ, when the value of Map ‘‘r ’’ is c:Gs Gv ? P, the corresponding phenomenon-time flow is: F :¼ {(r(g), 6s)jg 2 Gs Gv}; when the value of Map ‘‘r ’’ is l:Gs Gv ? S, the corresponding phase–time flow can be also obtained (the computing process is omitted here). In conclusion, the time flow chart based on time order concept is effective and feasible to demonstrate the flow process of various flow events (see Definition 4.5) under the phenomenon context in the time order. 6. Conclusions and next steps The phenomenon system is formalized in this study, based on formal concept analysis and rough set theory. Phenomenon are the basic materials that constitute knowledge, and the concept element is the formal representation of the knowledge level in the phenomenon element. Therefore, with studies on the correlation of the concept element, the correlation can be established in the knowledge intensive levels between different phenomenon elements of the system, laying a foundation for the future study of the formation of knowledge. Meanwhile, the research on the diachronism and formal definitions of phenomenon, state, phase and flow in the phenomenon system also provide a basis for future studies of the motion and evolution of knowledge-type complex systems. In order to further construct the structure of the knowledge granule, the following basic attributes must be satisfied: (1) granule intension: the identification of a certain granule in a given environment that is distinct from other granules is reflected as equivalence and correlation between the internal elements of the granule, as well as the boundary for the higher level of granular knowledge; (2) granule extension: the interaction between granules, or the local requirements that granules in the knowledge system must follow. The interaction among a large number of granules is reflected as the effect of granules on the environment. By studying the phenomenon element and its implied knowledge
correlation, intensive attributes in the granule are satisfied in this study. All correlated concept elements are the basic elements to constitute the concept granule, and its boundaries are determined by the equivalence class of specific knowledge in the concept element. The next step is to construct a higher level of knowledge granule structure, relative to phenomena elements, by studying granule extensive attributes between granules, and the rules of inference and calculation between phenomena. References [1] A. Skowron, J. Stepaniuk, J.F. Peters, Rough sets and infomorphisms: towards approximation of relations in distributed environments, Fundamenta Informaticae 54 (2003) 263–277. [2] A. Skowron, Jan Bazan, Marcin Wojnarski, Interactive rough-granular computing in pattern recognition, Pattern Recognition and Machine Intelligence 5909 (2009) 92–97. [3] B. Ganter, R. Wille, Formal Concept Analysis: Mathematical Foundations, Springer, 1999. [4] B. Ganter, R. Wille, Conceptual scaling, in: F. Roberts (Ed.), Applications of Combinatorics and Graph Theory to the Biological and Social Sciences, Springer Verlag, 1989, pp. 139–167. [5] J. van Benthem, Temporal logic, in: D.M. Gabbay, C.J. Hogger, J.A. Robinson (Eds.), Handbook of Logic in Artificial Intelligence and Logic Programming, Epistemic and Temporal Reasoning, vol. 4, Clarendon Press, Oxford, 1995, pp. 241–350. [6] D.M. Gabbay, I. Hodkinson, M. Reynolds, Temporal Logic C Mathematical Foundations and Computational Aspects, vol. 1, Clarendon Press, Oxford, 1994. [7] D.M. Gabbay, C.J. Hogger, J.A. Robinson, Handbook of Logic in Artificial Intelligence and Logic Programming, Epistemic and Temporal Reasoning, Clarendon Press, Oxford, 1995. [8] G. Liu, Rough set theory based on two universal sets and its applications, Knowledge-Based Systems 23 (2) (2010) 110–115. [9] K.E. Wolff, A conceptual view of knowledge bases in rough set theory, in: Proceedings of the Second International Conference on Rough Sets and Current Trends in Computing, vol. 7, 2000, pp. 188–192. [10] K.E. Wolff, Conceptual interpretation of fuzzy theory, in: H.J. Zimmermann (Ed.), EUFIT’98, 6th European Congress on Intelligent Techniques and Soft Computing, 1998, pp. 555–562. [11] J. Zhao, L. Liu, Construction of concept granule based on rough set and representation of knowledge-based complex system, Knowledge-Based Systems 24 (6) (2011) 809–815. [12] L.A. Zadeh, Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic, Fuzzy Sets and System 19 (1997) 111– 127. [13] L. Wei, J.J. Qi, Relation between concept lattice reduction and rough set reduction, Knowledge-Based Systems 23 (8) (2010) 934–938. [14] Qiang Wu, Zongtian Liu, Real formal concept analysis based on grey-rough set theory, Knowledge-Based Systems 22 (1) (2009) 38–45. [15] R. Yan et al., Research on the model of rough set over dual-universes, Knowledge-Based Systems 23 (8) (2010) 817–822. [16] R.A. Kowalski, M.J. Sergot, A logic-based calculus of events, New Generation Computing 4 (1986) 67–95. [17] Y.Y. Yao, Concept lattices in rough set theory, in: S. Dick, L. Kurgan, W. Pedrycz, M. Reformat (Eds.), Proc. of the 2004 Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS 2004). IEEE, 2004, pp. 796–801. [18] Y.Y. Yao, Granular computing: basic issues and possible solutions, in: The 5th Joint Conference on Information Sciences, New Jersey, USA, vol. 1, 2000, pp. 186–189. [19] Y.Y. Yao, A partition model of granular computing, Transactions on Rough Sets 1 (2004) 232–253. [20] Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences 11 (1982) 341–356. [21] Z. Pawlak, A. Skowron, Rough sets: some extensions, Information Sciences 177 (2007) 28–40.