MULTIVALUED POSITIVE BOOLEAN DEPENDENCIES BY GROUPS IN THE DATABASE MODEL OF BLOCK FORM

The article proposed the concept of multivalue positive Boolean dependencies by groups in the database model of block form, proved the equivalence of three types derived: m-deduction by logic, m-deduction in groups by block, m-deduction in groups by block not exceeding p elements,, necessary and sufficient conditions for a block to be a m-tight representation by groups of a set of multivalue positive Boolean dependencies by groups on the block ... In addition, some properties multivalue positive Boolean dependencies by groups have been stated and proved here.


I. INTRODUCTION
In recent years, research to expand the relational data model has been interested by many scientists around the world.. Following this research direction, there are some proposed database models such as: Multidimensional data model [1], [2], [3], data block [4], [5], data warehouse [6], [7],… the database model of block form [8].
In a database model of block form, the concepts: blocks, block diagrams, slices, relational algebra over blocks, functional dependencies, closures of index attribute set ... have been studied [8]. However, the study of extended logical dependencies in this data model is limited, many types of dependencies have not been studied. This article wants to propose and study the properties of a new type of logical dependency in a database model of block form: that is multivalued positive Boolean dependency by groups

II.1 The block, slice of the block
Definition II.1 [8] Let R = (id; A 1 , A 2 ,..., A n ) is a finite set of elements, where id is non-empty finite index set, A i (i=1..n) are attributes. Each attribute A i (i=1..n) there is a corresponding value domain dom(A i ). A block r on R, denoted r(R) consists of a finite number of elements that each element is a family of mappings from the index set id to the value domain of the attributes A i (i=1..n). In other words: t r(R)  t = { t i : id  dom(A i )} i=1..n . Then, block is denoted r(R) or r(id; A 1 , A 2 ,..., A n ), if without fear of confusion we simply denoted r.
Definition II.2 [8] Let R = (id; A 1 , A 2 ,..., A n ), r(R) is a block over R. For each x id we denoted r(R x ) is a block with R x = ({x}; A 1 , A 2 ,..., A n ) such that: Then r(R x ) is called a slice of the block r(R) at point x.
Definition II.3 [8] Let R = (id; A 1 ,A 2 ,...,A n ), r(R) is a block over R and , X Y is a notation of functional dependency. A block r satisfies X  Y if  t 1 , t 2  r such that t 1 (X) = t 2 (X) then t 1 (Y) = t 2 (Y).
A more specific way: F hx is the same with every x  id mean: M', N' respectively, formed from M, N by replacing x by y. II.3 Closure of the index attributes sets Definition II.6 [10] Let block scheme =(R,F), R=(id; A 1 , A 2 ,..., A n ), F is the set of functional dependencies on R. With each , we define closure of X for F denoted X + as follows:

III. MULTIVALUED BOOLEAN FORMULARS III.1 The operations and multivalued logical function
Definition III.1 [11] For the set of Boolean values B = {b 1 ,b 2 , ...,b k } including k values in [0;1], k  2 are in ascending order and satisfy the following conditions: We choose the operations and basic multivalued logical function: Definition III.2 [11] Let P = {x 1 , x 2 , ..., x n } is a finite set of Boolean variables, B is the set of Boolean values. Then the multivalued boolean formulas (CTBĐT) also known as multivalued logic formulas are constructed as follows: (iv) If a is a multivalued Boolean formula then (a) is a CTBĐT.
(v) If a and b are CTBĐT then a b, a b and  a are CTBĐT.
(vi) Only formulas created by rules from (i) -(v) are CTBĐT.
Definition III.3 [11] We define a b equivalent to CTBĐT ( a) b and then: a b = max (1-a, b).
Definition III.4 [11] Each vector of elements v = {v 1 , v 2 , ..., v n } in space B n = B x B x ... x B is called a value assignment. Thus, with each CTBĐT f  MVL(P) we have f(v) = f(v 1 , v 2 , ..., v n ) is the value of formula f for v value assignments.
We understand the symbol X  P at the same time performing for the following subjects: -An attribute set in P.
-A set of logical variables in P.
-A multivalued Boolean formula is the logical union of variables in X.
For each finite set CTBĐT F = {f 1 , f 2 , ..., f m } in MVL(P), we consider F as a formatted formula F = f 1  f 2  ...  f m . Then we have:

III.2 Table of values and truth tables
With each formula f on P, table of values for f, denote that V f contains n+1 columns, with the first n columns containing the values of the variables in U, and the last column contains the value of f for each values signment of the corresponding row. Thus, the value table contains k n row, n is the element number of P, k is the element number of B.
Definition III.5 [11] Let m  [0;1], truth table with m threshold of f or the mtruth table of f, denoted T f,m is the set of assignments v such that f(v) receive value not less than m: Then, the m-truth table T F,m of finite sets of formulas F on P, is the intersection of the m-truth tables of each member formula in F. T F,m = .
We have: v T F,m necessary and sufficient are f F: f(v)  m.

III.3 Logical deduction
Definition III.6 [11] Let f, g are two CTBĐT and value m B. We say formula f derives formula g from threshold m and denoted f ╞ m g if T f,m  T g,m . We say f and g are two m-equivalent formulas, denoted f ≡m g if T f,m = T g,m .
With F, G in MVL(P) and value m[0;1], we say F derives G from threshold m, denoted F ╞ m G, if T F,m  T G,m .
Moreover, we say F and G are m-equivalents, denoted F ≡m G if T F,m = T G,m .

III.4 Multivalued positive Boolean formula
Definition III.7 [11] Formula f MVL(P) is called a multivalued positive Boolean formula (CTBDĐT) if f(e) = 1 with e is the unit value assignment: e = (1, 1, ..., 1), we denoted MVP(P) is the set of all multivalued positive Boolean formulas on P.

IV. The multivalued truth block by groups of the data block
Let R = (id; A 1 , A 2 ,..., A n ), r(R) is a block over R, we convention that each value domain d i of attribute A i (is also of index attribute x (i) , xid), 1 i  n, contains at least p (p2) elements. Then, with each value domain d i , we consider the mapping  i :(d i ) p  B, satisfies the following properties: (i) Reflectivity:a  (d i ) p :  i (a)= 1, if in a contains at least two identical components.
Thus, we see the mapping  i is an evaluation on a group containing p (p2) values of d i satisfying reflection and commutative properties. Equality relation is a separate case of this relation.

Example IV.1
Let R = ({1, 2}, A 1 , A 2 ); then the index attribute of R are U = {1 (1) , 1 (2) , 2 (1) , 2 (2) }, with: A 1 : Weight of the ball (C: high, K: quite high, M: average, S: low), A 2 : Color of the ball (Đ: red, V: yellow, X: blue, N: brown). r is a block over R, includes 4 elements: t 1 , t 2 , t 3 , t 4 as follows: With p = 3, corresponding to each group has 3 balls, then: We consider the mapping  i : a  (d 1 ) 3 , we assign  1 (a)=1 if in a we have at least 2 balls of the same weight,  1 (a) = 0.5 if in a we have 3 balls with different weights for each pair and 1 ball with high weight, the remaining cases we have  1 (a) = 0.

Definition IV.2
Let R = (id; A 1 ,A 2 ,...,A n ), r(R) is a block over R, each value domain d i of attribute A i (is also of index attribute x (i) , xid,1 i  n), contains at least p elements,  i is an evaluation on groups containing p (p2) values of x (i) , xid, 1 i  n. For each group of p elements: u 1 , u 2 , ..., u p arbitrary (not necessarily distinguish) on the block, we call (u 1 , u 2 , ..., u p ) is the value assignment: (u 1 , u 2 , ..., u p ) = (t x1 , t x2 , ..., t xn ) with t xi =  i (u 1 .x (i) , u 2 .x (i) , ..., u p .x (i) ), xid, 1 i n. Then, for each block r we denote the multivalued truth block by groups of block r as T p r : T p r = { (u 1 , u 2 , ..., u p ) | u j  r, 1 j  p }.
(2) From (1) and (2)  If t  e, we build the block r including p elements as follows: From the properties of the mapping  i : (d i ) p  B with each index attribute x (i) , xid, 1 i  n we have:  a xi (d i ) p : a xi = (a xi1 , a xi2 , ..., a xip ) such that the  i (a xi ) = t xi .
Then, with each index attribute x (i) in U = , We fill in the column of this index attribute of block r values a xi1 , a xi2 , ..., a xip . According to the way of building block r, we have: T p r = {e, t}  T F,m with e is the unit value assignment. Thus r is a block with p elements and m-satisfying by groups set PTBDĐTTNB F.
Under the assumption if r is m-satisfying by groups F then r will m-satisfy by groups f, this means: T p r = {e, t}  T f,m , infer: t  T f,m .

Consequence IV.1
Let R = (id; A 1 ,A 2 ,...,A n ), r(R) is a block over R, each value domain d i of attribute A i (is also of index attribute x (i) , , contains at least p (p2) elements,  i are evaluations on groups containing p value of the index attribute x (i) , xid, 1 i n, set PTBDĐTTNB F and PTBDĐTTNB f. Then on r x the following three propositions are equivalent: (i) F x |= m f x (m-deduction by logic), (ii) F x ├ p m f x (m-deduction in groups by slice r x ), (iii) F x ├ p p,m f x (m-deduction in groups by slice r px have no more than p elements).
In the case of index set id = {x}, then the block r degenerates into a relation and the above equivalence theorem becomes the equivalent theorem in the relational data model. Specifically, we have the following consequences: Consequence IV. 2 Let R = (id; A 1 ,A 2 ,...,A n ), r(R) is a block over R, each value domain d i of attribute A i (is also of index attribute x (i) , xid,1 i  n), contains at least p (p2) elements, We infer: (NMBD(r,m),m) +  NMBD(r,m) (3) On the other hand, suppose we have: g  (NMBD(r,m),m) + , We need proof g  NMBD(r,m). Indeed, the hypothesis: g  (NMBD(r,m),m) + = { f | f MVP(U), T (NMBD(r,m),m)  T f,m }  g MVP(U), T (NMBD(r,m),m)  T g,m . Which by definition of NMBD (r,m) we have: T p r  T (NMBD(r,m),m)  T p r  T g,m  block r is m-satisfying by groups PTBDĐTTNB g. From there we have: g  NMBD(r,m).
 (NMBD(r,m),m) +  NMBD(r,m) (4) From (3) and (4)   Proof ) Suppose r is m-tight representation by groups set PTBDĐTTNB  we need proof r x is m-tight representation by groups set  x , xid.
Indeed, under the assumption we have: r is m-tight representation by groups set PTBDĐTTNB , using the results of theorem IV.5 we have: T p r = T ,m . Thence inferred: (T p r ) x = (T ,m ) x , xid. Which we have: T p rx = (T p r ) x = (T ,m ) x = T x,m , xid  T p rx = T x,m  r p x ( x ,m), xid. So r x is m-tight representation by groups set  x ,xid. ) Suppose r x is m-tight representation by groups set  x ,x  id we need proof r is m-tight representation by groups set PTBDĐTTNB .
Indeed, under the assumption r x is m-tight representation by groups set  x ,xid T p rx = T x,m , xid. Inferred: (T p r ) x = T p rx = T x,m = (T ,m ) x , xid. Which we have: T p r = , T ,m = .  T p r = T ,m . So r is m-tight representation by groups set PTBDĐTTNB .

V. CONCLUSIONS
From a proposed concept are functions that evaluate values on a group with p elements, The article gave the definition of the multivalued truth block by groups of data blocks. From there build a new type of dependency: it is a multivalued positive Boolean dependency by groups in the database model of block form. From the new concept of dependency is proposed, the authors have stated and proved the equivalent theorem for multivalued positive Boolean dependencies by groups on the block, the necessary and sufficient condition for a block r is m-tight representation set PTBDĐTTNB … From these results we can further study the relationship between other types of extended logical dependencies on the data block.

VI. ACKNOWLEDGEMENTS
The authors thank the teachers, leaders of the Institute of Information Technology and the Management Board of the Hanoi Pedagogical University 2 for creating favorable conditions for us to work and study. This research is funded by Hanoi Pedagogical University 2 (HPU2).