Concept Lattices in Green Farmland Databases and Concept Intent Reduction for the Mass Food Production 1

Formal Concept Analysis (FCA) in green farmland databases and Concept Intent Reduction for the mass food production provides a method for extracting concepts from binary contexts. However, FCA-concepts cannot describe negations and disjunctions of attributes. Hence, we take the logic operators into consideration in the process of constructing concepts and obtain new extended concepts, which are more expressive than FCA-concepts. This study mainly discusses the connections between FCA-concepts and concepts with logic values in green farmland databases and concept intent reduction for the mass food production and provides a method for reducing concepts. The reduction does not lose essential information. Results can be used in data mining and construction of architecture ontology.


INTRODUCTION
In Computer Science and architecture, the semantics of data can be advantageously exploited to better retrieve and efficiently discover relevant patterns which are a concise and semantically rich representation of data.Patterns can be clusters, concepts and so on.In this study we present alternate ways to extract concepts from binary green farmland databases.
Recently, a main axis of research on FCA has aimed at extending the classical FCA, either by scaling methods (Ganter and Wille, 1999;Lei et al., 2009) or by extending the definition of the Galois connection (Huchard et al., 2007;Jiang et al., 2010).One of the disadvantages of scaling methods is that it is very sensitive to user's selection of scale attributes.Our main contributions in this study are to propose an approach towards constructing concepts, which can extract more interesting concepts from binary relations, without using any scaling method.For example, for the binary relation of an educational film "Living Beings and Water" described, the classical FCA can only describe that Leech, Bream, Frog, Spike-weed and Reed live in water, but not describe that all of Dog, Bean and Maize do not live in water.Similar to conjunction, negation and disjunction are also important logic operators, which are common in human language (e.g., not young).Therefore, it is necessary to extend the classical FCA-concepts to improve the expressiveness of concepts.Hence, we take the logic operators into consideration in the process of constructing concepts and obtain new extended concepts, which are more expressive than the classical FCA-concepts.
This study mainly provides some properties of the extended concepts and a method for reducing their intents.The reduction not only simplifies concept representation, but does not lose essential information.

CONCEPTS WITH LOGIC VALUES AND CORRESPONDING CONCEPT LATTICES
Given a binary relation R = (G, M, I), we can built complex statements from attributes in M inductively with negation (), disjunction (), conjunction () and implication ().Because for any statements  and , "      ", we can define -statements with negation and disjunction as follows:  = a|a|12, where aM.Let  be the set of -statements.For any rG and aM, r always possesses or does not have a, that is, r always satisfies the property a(a).Hence, we delete all of -statements with the form a(a) from .
Example 1: Given the following binary relation R = (G, M, I), we obtain some -statements such as a, a, a b, b (c) and a (b) c (Table 1).Then, we define the satisfaction relation between objects and -statements as follows: for any object r G and -statement , we say that r satisfies , denoted by r| = R , if: Which correspondingly induces the following two maps  and  between the power P(G) of G and the power set P() of .For any XG, (X) is defined as follows: (X) = {: rG, r| = R}, which is the set of -statements satisfied by the objects in X.For any Y, (Y) is defined as follows: (Y) ={rG: , r|=R }, which is the set of objects which satisfy all statements in Y.
We call X and Y the extent and the intent of (X, Y), respectively.LL(R) denotes the set of all R L -concepts.
Example 2: The binary relation R in Example 1 has 16 R L -concepts.For example, ({r 1 , r 3 }, {a,c}), which is the reduced form.
Similar to the classical FCA-concepts, the intersection of any number extents (intents) of R Lconcepts is always an extent (intent).Generally, the union of extents of the classical FCA-concepts does not result in an extent, while the union of extents of R Lconcepts is exactly the extent of some R L -concept.The reasons are as follows: from the logic point of view, each intent is actually equivalent to the conjunction of some -statements, that is, each intent is represented as a Conjunctive Normal Form (CNF).Thus, given two RL-concepts, we immediately obtain two CNFs from their intents and the disjunction of them can be transformed into a CNF.The set of objects satisfying the CNF is exactly the union of the two extents.Based on the above description, there is therefore the following proposition: Proof: On the one hand, it is easily inferred that X1(X2X3)  (Y1(Y2Y3)) holds.On the other hand, (Y1(Y2Y3))  X1  (X2X3).Or else, there is an object r(Y1(Y2Y3)) but rX1(X2X3) and hence rX1 and rX2X3.By rX1, there is 1Y1 satisfying r0|R1.By rX2X3, there is rX2 or rX3.Hence, there are the three cases as follows: Case 1: rX2 and rX3.There is 2Y2 satisfying r0|R2.Hence, there is r0|R12.Because 12 Y1(Y2Y3), there is r0|=R12, which leads to a contradiction.

REDUCTION OF R L -CONCEPTS
Now, we discuss the intents of R L -concepts and provide a method for reducing the intents of R Lconcepts.Our idea is as follows: We regard statements as formulas in predicate logic and then apply the inference rules to reducing the intents of R Lconcepts.
We assume that the -statements in intents are drawn from a linearly ordered set.That is, we write the -statement according to the numbers of positive attributes and negative ones contained in these statements.For example, an intent Y can be represented as a set {a, b, ab, bc, bc(d)} or {b, a, bc, ab, bc(d)} but not {ab, a, b, bc, bc(d)}.Now, we introduce formal inference rules as follows: For any three -statements , 1 and 2, the inference rules are sound, that is, if |1, then for any object rG, r|=R implies r|=R1.Given a R L -concept (X, Y), for any Y, we define Thus, we provide the following algorithm for reducing intents.In fact, the union/intersection of extents of RLconcepts always results in an extent.The relation R in EXAMPLE 1 has 7 concepts.We use the reduced labelling in order to improve the of the lattice.The line diagram in Fig. 1 represents the concept lattice with logic operators.Figure 2 is the concept lattice.

Algorithm for reducing intents:
By proposition 1, it is easily inferred that LL(R) is a distributive lattice.Furthermore, for R L -concepts, the set of any number objects is always an extent of some R L -concept.Hence, given a R L -concept (X, Y), (G-X, (G-X)) is also a R L -concept, called complement R Lconcept of (X, Y) and denoted by ~(X, Y).Thus, each R L -concept has corresponding complement R L -concept and hence LL(R) is a complement distributive lattice.
By Stone representation theorem, LL(R) is isomorphic to the power-set lattice P(Atom(R)).

THE CONNECTION BETWEEN CONCEPT LATTICES
Now, we discuss the connection between L(R) and LL(R).From example 3, we find that the extent of each classical FCA-concept in R is also the extent of some RL-concept.Thus, for any classical FCA-concept (X, Y), we can construct a R L -concept (X, (X)).Hence, there is a natural mapping from L(R) to LL(R), which is an infimum-preserving order-embedding, as shown in proposition.
Proof: Firstly, it is easily inferred that  is an orderembedding and -preserving.Conversely, for any RLconcept (X, Y), YM is the maximal set of attributes common to the objects in X and then YM is the intent of some concept of R. Thus, we can obtain a supremum-preserving order-preserving map from L(R) to LL(R), which can be proved in a similar fashion.
Proposition 4: There is a map from LL(R) to L(R) such that for any (X, Y) LL(R): ((X,Y)) = (h(YM), YM) which is surjective supremum-preserving orderpreserving.

CONCLUSION
This study provides a kind of extended FCAconcepts.Compared with the classical FCA-concepts, the extended concepts are more expressive.We provide some interesting connections between L(R) and LL(R).
Our results can be used to extract discovery from data and construct architecture ontology.
Two of the interesting problems are how to define concepts in relations with imprecise information.Future works focus on these questions.

Fig. 1 :
Fig. 1: Concept lattice with operators Y) Output: The reduced form Core(Y) of the intent Y Process 1. Core(Y)  { } 2. Begin 3.While Y  Do 4. Begin 5.For the first -element Y, computing  Y () 6. Y  Y   Y () 7. Core(Y)  {} 8. End 9. End Example 3: Let Y = {a, b, ab, bc, bc(d)} be an intent.By the algorithm, we can obtain Core(Y) = {a, b} by the following two loops: Loop 1: The attribute a is the first element in Y and  Y (a) ={ a, ab}.Hence, new Y = {b, bc, bc(d)} and Core(Y) = {a}.Loop 2: The attribute b is the first element in Y and  Y (b) = {b, bc, bc(d)}.Hence, new Y =  and Core(Y) = {a, b}.

Table 1 :
Relation between objects and -statements Objects