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.

_{1}, A_{2},..., A_{n}), r(R) is a block over R. For each _{x}) is a block with R_{x} = ({x}; A_{1}, A_{2},..., A_{n}) such that:

_{x})

Here, for simplicity we use the notation:

and _{1},A_{2},...,A_{n }).

_{1},A_{2 },..., A_{n }), r(R) is a block over R and

^{+}

If

The block r satisfies

where:

Let block scheme R= (id; A_{1}, A_{2},..., A_{n}), we denoted the subsets of functional dependencies over R:

^{3}

_{h} is called the complete set of functional dependencies if:

A more specific way:

F_{hx} is the same with every

^{+}

We denote the set of all subsets of a set

_{1}, A_{2},..., A_{n} ), F_{ }is the set of functional dependencies on R, K

i)

ii)

F

(i)

(ii)

We choose the operations and basic multivalued logical function:

•

•

•

• With each value _{b}:

The functions I_{b},

Let P = {x_{1}, x_{2}, ..., x_{n}} is a finite set of Boolean variables,

Each value in

Each variable in P is a CTBĐT.

Each function I_{b},

If a is a multivalued Boolean formula then (a) is a CTBĐT.

If a and b are CTBĐT then

Only formulas created by rules from (1) –(5) are CTBĐT.

We denote MVL(P) as a set of CTBĐT building on the set of variables P = {x_{1}, x_{2}, ..., x_{n}} and set of values B = {b_{1}, b_{2},..., b_{k}} including k values in [0;1],

_{1}, v_{2}, ..., v_{n}} in space ^{n} = _{1}, v_{2}, ..., v_{n}) is the value of formula f for v value assignments.

In the case where there is no confusion, we understand the symbol

An attribute set in P.

A set of logical variables in P.

A multivalued Boolean formula is the logical union of variables in X.

On the other hand, if X = {B_{1}, B_{2}, ..., B_{n}}

We call formula f: Z

• Multivalued derivative formula if Z and V has the associational form, mean:

• Strong multivalued derivative formula if Z has the ecruitmental form and V has the associational form, mean:

• Weak multivalued derivative formula if Z has the associational form and V has the ecruitmental form, mean:

• Duality multivalued derivative formula if Z and V are in recruitment form, mean:

For each finite set CTBĐT F = {f_{1}, f_{2}, ..., f_{m}} in MVL(P), we consider F as a formatted formula

_{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

_{f,m } is the set of assignments v such that f(v) receive value not less than m:

_{F,m } of finite sets of formulas F on P, is the intersection of the m-truth tables of each member formula in F.

_{1},A_{2},...,A_{n }), r(R) is a block on R , _{i1}, v_{i2}, ..., v_{in }} _{i=1..s} in space B ^{nxs} is a value assignment. Thus, with each CTBĐT f _{i1}, v_{i2}, ..., v_{in }) _{i=1..s } is the value of formula f for v value assignments.

Let

Inferred: f (v) = 0.5.

We have two special assignment:

Unit assignment:

_{f, m } is the set of assignments v such that f(v) receive value not less than m:

_{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 |^{nxs} | = k ^{nxs}, we have the following theorem:

^{ }and values m_{1}, m_{2},…, m_{d } trong B,

Proof:

a) With each

and we have:

_{ } then f is the formula to look for.

Indeed, we have:

.Which according to the properties of

So infer:

b) From CTBĐT f we have:

Which we have:

we infer:

and

_{x} get T_{x} as the m-truth block.

Proof:

Use the result of the theorem 3.1 with special cases: m_{1} = m_{2} = … = m_{d} = m we obtained CTBĐT f satisfies two conditions:

and then CTBĐT f_{x} also satisfies two conditions:

_{f,m} = T và T_{fx,m} = T_{x}.

_{1}, A_{2},...,A_{n }), r(R) is a block on R,

Let

The formulars:

The formulars:

We denoted MVP(U) is the set of all multivalued positive Boolean formulas on U.

_{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

(i)

(ii) Symmetry:

(iii)

Thus, we see the mapping _{i} satisfies the reflective, symmetrical and sufficiency properties. Equality relationships with logic of two values

_{1},A_{2},...,A_{n }), r(R) is a block on R,

_{r}:

If block r contains at least a certain element k then:

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.

If r(f,m) then we say PTBDĐT f is m-right in the block r.

_{1},A_{2},...,A_{n } ), r(R) is a block on R,

Proof

i) Under the assumption we have

So we have

ii) Under the assumption

Therefore:

_{1},A_{2},...,A_{n }), r(R) is a block on R,

_{ }.

_{ }then r m-satisfying set of multivalued positive Boolean dependency F: r(F,m).

Proof

i) Under the assumption we have:

So we have:

ii) Under the assumption

So we have:

From the proposition 3.1 and 3.2 we have the following necessary and sufficient conditions:

_{1},A_{2},...,A_{n }), r(R) is a block on R,

_{ }.

For the set PTBDĐT F and PTBDĐT f:,

We say F m-deduced f by the block and denoted

We say F m-deduced f by the block contains no more than 2 elements and denoted

We have the following equivalent theorem:

_{1},A_{2},...,A_{n }), r(R) is

_{m} f (m-deduction by logic),

(ii) F |-_{m} f (m-deduction by block),

(iii) F |-_{2,m } f (m-deduction by block has no more than 2 elements).

Proof

(i) => (ii): Under the assumption we have _{ } From (1) and (2) we infer:

(ii) => (iii): Obviously, because inference by the block has no more than 2 elements is the special case of inference by block.

(iii) => (i): Suppose

Indeed, if t = e then we have

Thence inferred r (F, m). Under the assumption we have r (F, m) => r (f,m), so that ^{(1).}

From (1) we infer

_{1},A_{2},...,A_{n }), r(R) is a block on R, m _{x} the following three propositions are equivalent:

(i) _{x} |=_{m} f_{x} (m-deduction by logic),

(ii) _{x} |-_{m } f_{x} (m-deduction by slice r_{x}),

(iii) _{x} |-_{2,m } f_{x} (m-deduction by the slice r_{2x} has no more than

In the case id = {x}, then the block degenerated into a relation and the above m-equivalence theorem becomes the m-equivalent theorem in the relational data model. Specifically, we have the following consequences:

_{1},A_{2},...,A_{n }), r(R) is a block on R, m

_{m} f (m-deduction by logic),

_{m } f (m-deduction by relation),

_{2,m } f (m-deduction by relation has no more than 2 elements).

_{1},A_{2},...,A_{n }), r(R) is a block on R, ^{+} is the set of all PTBDĐT are m-deduction from

_{1},A_{2},...,A_{n }), r(R) is a block on R,

So, we have:

_{1},A_{2},...,A_{n }), r(R) is a block on R,

By definition, we have:

Apply the result of the theorem of three equivalent propositions for PTBDĐT, we have:

From (1) and (2) we infer: (MBDĐT(r,m),m)^{+} = MBDĐT(r,m).

So two sets (MBDĐT(r,m),m)^{+} and MBDĐT(r,m) are two sets PTBDĐT m-equivalents on blocks.

_{1},A_{2},...,A_{n }), r(R) is a block on R, _{x} we have:

_{1},A_{2},...,A_{n }), r(R) is a block on R,

_{1},A_{2},...,A_{n }), r(R) is a block on R,

If r is m-tight representation set PTBDĐT

_{1},A_{2},...,A_{n }), m

Proof

Suppose _{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:

We denoted: F = MBDĐT(r,m), from the above proof we have:

From (3) and (4) we infer:

Thus:

_{1},A_{2},...,A_{n }), r(R) is a block on R, _{x } we have:

_{1},A_{2},...,A_{n }), r(R) is a block on R, _{r } = T_{MBDĐT(r,m),m.
}

_{1},A_{2},...,A_{n }),

Proof

By definition, we have: r is m-tight representation set

Other way:

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

Let R = (id; A_{1},A_{2},...,A_{n }),

This consequence is exactly what we already know in the relational data model.

We denoted

Let R = (id; A,A,...,A), _{x} is m-tight representation set

Proof

_{x} is m-tight representation set

Indeed, under the assumption we have: r is m-tight representation set PTBDĐT

Thence inferred:

Which we have:

So r_{x} is m-tight representation set

_{x} is m-tight representation set

Indeed, under the assumption r_{x} is m-tight representation set

Inferred:

Which we have:

So r is m-tight representation set PTBDĐT

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

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.

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).