Some properties of multivalued positive Boolean dependencies in the database model of block form

Objectives: The article proposed a new type of dependency on blocks and slices. Then found and proved the properties of this new dependency.Method: Logical inference methods were used. Findings: A new type of data relationship has been proposed: Multivalued positive Boolean dependencies on block and slice in the database model of block form. From this new concept, the article stated and demonstrated the equivalence of the three types of deduction, namely: m-deduction by logic, m-deduction by block, m-deduction by block has no more than two elements. Next are the necessary and sufficient criteria of the tight m-expression for the set of multivalued positive Boolean dependencies on block and slice, the sufficient properties for a set of functions {I,∧,∨}. The properties related to this new concept when the block degenerated into relation.Novelty: The proposed new dependency with their properties on the block and on the slice are completely new.


The block, slice of the block
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) is the attribute. 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).
We have: Then, block is denoted r(R) or r (id; A 1 , A 2 ,...,A n ), if without fear of confusion we simply denoted r.
https://www.indjst.org/ Definition 1.2 (1) 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.

Functional dependencies
Here, for simplicity we use the notation: } and x (i) (x ∈ id, i = 1 . . . n) is called an index attribute of block scheme R = (id; A 1 ,A 2 ,...,A n ). Definition 1.3 (2) Let R = (id; A 1 ,A 2 ,..., A n ), r(R) is a block over R and X, Y ⊆ ∪ n i=1 id (i) , 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) Definition 1.4 (2) Let block scheme α = (R, F), R = (id; A 1 , A 2 , . . . , A n ) F is the set of functional dependencies over R. Then, the closure of F denoted F + is defined as follows: The block r satisfies = t 2 (y; A k ). Let block scheme R= (id; A 1 , A 2 ,..., A n ), we denoted the subsets of functional dependencies over R: x (i) } Definition 1.5 (3) Let block scheme α = (R, F h ) , R = (id; A 1 , A 2 , . . . , A n ), then F h is called the complete set of functional dependencies if: is the same with every x ∈ id A more specific way: F hx is the same with every x ∈ id mean: ∀x, y ∈ id : M, N by replacing x by y.

Closure of the index attributes sets:
,we define closure of X for F denoted X + as follows: We denote the set of all subsets of a set ii) ∀K ′ ⊂ K then K ′ has no properties i).
If K is a key andK ⊆ K ′′ then K" called a super key of the block scheme R for F.
We choose the operations and basic multivalued logical function: The functions I b , b ∈ B called generalized negative functions.
Definition 2.2 (5) 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: • A multivalued Boolean formula is the logical union of variables in X.

Table of values and truth tables
With each formula f on P, 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.

Logical deduction
Definition 2.6 (5) Let f, g is two CTBĐT and value m ∈ B. We say formula f derives formula g from threshold m and denoted f |= m g i f T f ,m ⊆ T g,m . We say f and g are two m-equivalent formulas,

The m-truth block of the data block
.
We have two special assignment:

Then, the m-truth block T F,m of a finite set of formulas F on U, is the intersection of the m-truth blocks of each formula of member f in F.
With | B| = k then | B nxs | = k nxs , we have the following theorem: also satisfies the following two properties: Therefore, if we set: ) j=1..s https://www.indjst.org/ then f is the formula to look for. Indeed, we have: .Which according to the properties of and then CTBĐT f x also satisfies two conditions: . Then: • The formulars: ) , are not the CTBDĐT.
We denoted MVP(U) is the set of all multivalued positive Boolean formulas on U.

Definition 3.4
Let R = (id; A 1 , A 2 ,...,A n ), r(R) is a block on R, we denote d i is the value domain of the attribute A i (is also of index attribute Then, for each value domain we consider mapping: α i : d i x d i → B satisfies the following conditions: Thus, we see the mapping α i are the relationships above d i satisfies the reflective, symmetrical and sufficiency properties. Equality relationships with logic of two values B ={0,1} is the separate case of the above relationship.

.,A n ), r(R) is a block on R, u, v ∈ r, mappings α i define on each value domain
u.
x (1) , v.x (1) ) , α 2 ( u.x (2) , v.x (2) ) , . . . , α n Then, for each block r, we denote the truth block of block r as T r : If block r contains at least a certain element k then: α(u, u) = 1 ⇒ e ∈ T r . In the case id = {x}, then the block degenerates into a relation and the concept of the truth block of the block becomes the concept of truth table of relation in the relational data model. In other words, the truth block of a block is to expand the concept of the truth table of relation in the relational data model.

we call each multivalued positive Boolean formula in MVP(U) is a multivalued positive Boolean dependency (PTBDĐT) on block. We say block r is m-satisfying the multivalued positive Boolean dependency f and denoted r(f,m) if T r ⊆ T f ,m . The block r is m-satisfying set of multivalued positive Boolean dependency F and denoted r(F,m) if r satisfies all PTBDĐT f in F:
If r(f,m) then we say PTBDĐT f is m-right in the block r.   • We say F m-deduced f by the block contains no more than 2 elements and denoted F | −2,m f if: ∀ r 2 : r 2 (F, m) ⇒ r 2 (f, m).
We have the following equivalent theorem: (ii) => (iii): Obviously, because inference by the block has no more than 2 elements is the special case of inference by block.
x , . . . , t x ∈ id, t ∈ T F,m we need proof : t ∈ T f ,m . Indeed, if t = e then we have t ∈ T f ,m because as we know f is a positive Boolean formula. If t ̸ = e , we built the block r including 2 elements u and v as follows: x , . . . , u x , 1 ≤ i ≤ The existence of the u and v elements as above is due to the properties of the mappings α i mentioned above. Thus r is a block with 2 elements and T r = {e, t} ⊆ T F,m , with e is a element of block whose all component values are equal to 1.
Thence inferred r (F, m). Under the assumption we have r (F, m) => r (f,m), so that T r ⊆ T f,m (1).
From (1) we infer t ∈ T f,m .

Consequence 3.2
For the set PTBDĐT F and PTBDĐT f , R = (id; A 1 ,A 2 ,...,A n ), r(R) is a block on R, m ∈B. Then on r x the following three propositions are equivalent: m f x (m-deduction by the slice r 2x has no more than 2 elements).
In the case id = {x}, then the block degenerated into a relation and the above m-equivalence theorem becomes the mequivalent theorem in the relational data model. Specifically, we have the following consequences: Then on r x we have: If r is m-tight representation set PTBDĐT Σ then we say r is the block m-Armstrong of set PTBDĐT Σ .

Theorem 3.5
Let R = (id; A 1 ,A 2 ,...,A n ), m ∈ B. Then, with every block r(R) different from the empty set on R we have: From T r and value m, According to theorem 3.1 we find a multivalued boolean formula f satisfying conditions: f(e) = 1 và T f ,m = T r . So: e ∈ T r = T f,m infer f is one CTBDĐT and and more due T r = T f,m ⇒ r is m-satisfying f, mean: f ∈ MBDĐT (r, m).
We denoted: F = MBDĐT(r,m), from the above proof we have: ∃f ∈ MBDĐT (r, m) : From (3) and (4) we infer: Apply the results of theorem 3.4 we obtain: So from (5) and (6) we infer: T r = T S,m . Therefore: r is m-tight representation set Σ ⇔ T x = T Σ,m . Proof ⇒) Suppose r is m-tight representation set PTBDĐT Σ we need proof r x is m-tight representation set Σ x , ∀x ∈ id.
Indeed, under the assumption we have: r is m-tight representation set PTBDĐT Σ, using the results of theorem 3.6 we have: T r = T Σ,m .
Thence inferred: (T r ) x = (T Σ,m ) x , ∀x ∈ id. Which we have: So r x is m-tight representation setΣ x , ∀x ∈ id. ⇐) Suppose r x is m-tight representation set Σ x , ∀x ∈ id we need proof r is m-tight representation set Σ.
Indeed, under the assumption r x is m-tight representation set Σ x , ∀x ∈ id ⇒ T rx = T Σx,m , ∀x ∈ id. Inferred: (T r ) x = T rx = T Σx,m = (T Σ,m ) x , ∀x ∈ id Which we have: T r = ∪ x ∈id T rx , T Σ,m = ∪ x∈id T Σx,m ⇒ T r = T Σ,m . So r is m-tight representation set PTBDĐT Σ.

Conclusions
From the proposed new concept: multivalued positive Boolean dependence on block and slice, the article defined the truth block of the data block, prove the completeness of the set of functions {I, ∧, ∨}. In addition, the article also proves the equivalent theorem for multivalued positive Boolean dependencies on block and slice. The necessary and sufficient condition for a block https://www.indjst.org/ is m-tight representation ∑ … If id = {x} then the block degenerated into a relation and the results found on the block are still true on the relation.
We can further study the relationship between other types of logical dependencies on block and slice, extend the set of function dependencies on the block,... contribute to further complete design theory of the database model of block form.