Sciresol Sciresol https://indjst.org/author-guidelines Indian Journal of Science and Technology 10.17485/IJST/v13i25.759 Some properties of multivalued positive Boolean dependencies in the database model of block form Thang Trinh Dinh thangdhsp2@hpu2.edu.vn 1 Tuyen Tran Minh 2 Truc Trinh Ngoc 1 Phuong Pham Thi 1 Hanoi Pedagogical University No2 University Union Vietnam 13 25 2020 Abstract

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.

Keywords Multivalued positive Boolean dependencies block Boolean dependencies block schemes None
Introduction 1.1 The block, slice of the block

Definition 1 .1 1

Let R = (id; A1, A2,..., An) is a finite set of elements, where id is non-empty finite index set, Ai (i=1.. n) is the attribute. Each attribute Ai (i=1.. n) there is a corresponding value domain dom (Ai). 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 Ai (i = 1.. n).

We have:

tr(R)t=ti: id domAii=1.n

Then, block is denoted r(R) or r (id; A1, A2,...,An), if without fear of confusion we simply denoted r.

Definition 1.2 1

Let R = (id; A1, A2,..., An), r(R) is a block over R. For each xid we denoted r(Rx) is a block with Rx = ({x}; A1, A2,..., An) such that:﻿

txrRxtx=txi=t| xi i=1..n,  where  tr(R), t=ti: id domAii=1..n

Then r(Rx) is called a slice of the block r(R) at point x.

1.2 Functional dependencies

Here, for simplicity we use the notation:

x(i)=x; Ai; id(i)=x(i)xid

and x(i)(xid,i=1n) is called an index attribute of block scheme R = (id; A1,A2,...,An ).

Definition 1.3 2

Let R = (id; A1,A2 ,..., An ), r(R) is a block over R and X,Yi=1nid(i),XY is a notation of functional dependency. A block r satisfies XY if:

t1,t2R such that t1(X)=t2(X) then t1(Y)=t2(Y)

Definition 1.4 2

Let block scheme α=(R,F),   R=id;A1,A2,,An F is the set of functional dependencies over R. Then, the closure of F denoted F+ is defined as follows:

F+={XYFXY}

If  X=x(m)id(m),Y=y(k)id(k) then we denoted functional dependency XY is simply x(m)y(k) .

The block r satisfies x(m)y(k) if t1,t2r such that t1x(m)=t2x(m) then t1y(k)=t2y(k),

where: t1x(m)=t1x;Am,t2x(m)=t2x;Am,  t1y(k)=t1y;Ak,  t2y(k)=t2y;Ak.

Let block scheme R= (id; A1, A2,..., An), we denoted the subsets of functional dependencies over R:

Fh=XYX=iAx(i),Y=jBx(j)A,B{1,2,,n},xid

Fhx=Fh | i=1nx(i) =XY  Fh | X, Y  i=1nx(i)

Definition 1.5 3

Let block scheme α=R,Fh,  R=id;A1,A2,,An, then Fh is called the complete set of functional dependencies if:

Fhx=Fh|i=1nx(i)         is the same with every   xid

A more specific way:

Fhx is the same with every xid mean: x,yid: MNFhxM'N'Fhy with  M', N' respectively, formed from M, N by replacing x by y.

1.3 Closure of the index attributes sets:

Definition 1.6 3

Let block scheme α=(R,F),R=id;A1,A2,,An , F is the set of functional dependencies on R.

With each Xi=1nid(i), we define closure of X for F denoted X+ as follows:

X+=x(i)Xx(i)F+, xid, i=1..n

We denote the set of all subsets of a set i=1nid(i) as set SubSet ( i=1nid(i)).

1.4 Key of the block scheme <inline-formula id="if-e9246a192b95"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math></inline-formula> = (R, F)

Definition 1.7 4

Let block scheme α = (R, F), R = (id; A1, A2,..., An ), F is the set of functional dependencies on R, K i=1nid(i). K called a key of block schema α if it satisfies two conditions:

i) Kx(i)F+, xid, i=1..n.

ii) K'  K  then   K'  has no properties  i).

If K is a key and KK'' then K’’ called a super key of the block scheme R for F.

Multivalued Boolean formulas 2.1 Multivalued boolean formulas

Definition 2.1 5

For the set of Boolean values B=b1,b2,,bk including k values in [0;1], k2 are in ascending order and satisfy the following conditions:

(i) 0 B .

(ii)  b B  1- b B .

We choose the operations and basic multivalued logical function:  a, b B

ab=min(a,b),

ab=max(a,b),

¬a=1-a

• With each value  b B, we define the function Ib:

xB:  Ib(x)=1 if x=b and Ib(x)=0 if xb

The functions Ib,  b B called generalized negative functions.

Definition 2.2 5

Let P = {x1, x2, ..., xn} 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:

Each value in  B is a CTBĐT.

Each variable in P is a CTBĐT.

Each function Ib,  b B is a CTBĐT.

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

If a and b are CTBĐT then ab, ab and ¬a is a CTBĐT.

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 = {x1, x2, ..., xn} and set of values B = {b1, b2,..., bk} including k values in [0;1], k2 .

Definition 2.3 5

We define ab equivalent to CTBĐT (¬a)b and then: ab = max(1- a, b).

Definition 2.4 5

Each vector of elements v = {v1, v2, ..., vn} 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(v1, v2, ..., vn) is the value of formula f for v value assignments.

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

On the other hand, if X = {B1, B2, ..., Bn} P , we denoted:

X=B1B2Bn called the associational form.

X=B1B2Bn called the recruitmental form.

We call formula f: Z V is :

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

f:ZV

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

f:ZV

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

f:ZV

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

f:ZV

For each finite set CTBĐT F = {f1, f2, ..., fm} in MVL(P), we consider F as a formatted formula F=f1f2fm. Then we have:

F(v)=f1(v)f2(v)fm(v)

2.2 Table of values and truth tables

With each formula f on P, table of values for f, denote that Vf 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 kn row, n is the element number of P, k is the element number of  B.

Definition 2.5 5

Let m [0;1], truth table with m threshold of f or the m-truth table of f , denoted Tf,m is the set of assignments v such that f(v) receive value not less than m:

Tf,m=v Bnf(v)m

Then, the m-truth table TF,m of finite sets of formulas F on P, is the intersection of the m-truth tables of each member formula in F.

TF,m=fFTf,m

We have: vTF,m necessary and sufficient are fF : f(v)m .

2.3 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 if Tf,mTg,m. We say f and g are two m-equivalent formulas, denoted fx g if Tf,m=Tg,m.

With F, G in MVL(P) and value m [0;1], we have F derives G from threshold m, denoted F=m G if TF,m TG,m. Moreover, we say F and G are m-equivalents, denoted Fn G if TF,m=TG,m .

2.4 Multivalued positive Boolean formula

Definition 2.7 5

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.

Research results 3.1 The m-truth block of the data block

Definition 3.1

Let R = (id; A1,A2,...,An ), r(R) is a block on R , U=i=1nid(i), |id|=s, We call each vector of elements v = {vi1, vi2, ..., vin } i=1..s in space B nxs is a value assignment. Thus, with each CTBĐT f MVL(U) we have f(v) = f (vi1, vi2, ..., vin ) i=1..s is the value of formula f for v value assignments.

Example 3.1: Let R={1,2},A1,A2,A3 then U=1(1),1(2),1(3),2(1),2(2),2(3) ,  B={0, 0.5, 1}.

Let v=0.510.510.50.5,   f=1(1)1(2)2(1)2(2)1(3)2(3) , then we have f(v) = max(1 - min(0.5, 1, 1, 0.5), min(0.5, 0.5)).

Inferred: f (v) = 0.5.

We have two special assignment:

Unit assignment: e=111...111 and the assignment value 0: Z=000···000

Definition 3.2

Let m [0;1], truth block threshold m of f or m-truth block of f, denoted Tf, m is the set of assignments v such that f(v) receive value not less than m:

Tf,m=v Bnxsf(v)m

Then, the m-truth block TF,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.

TF,m=fFTf,m

We have: vTF,m if and only if fF:f(v)m.

With |  B| = k then |  Bnxs | = k nxs, we have the following theorem:

Theorem 3.1

Let block T=t1,t2,,td Bnxs and values m1, m2,…, md trong B, Idkn×s. Then:

a) There exists one CTBĐT f satisfies the following two properties:

(i) tiT:fti=mi,

(ii) tBnxsT:f(t)=0

b) With every slice : Tx , xid, |id|=s,  CTBĐT fx= f|Tx also satisfies the following two properties:

(iii)txi Tx : fx txi=mi,Tx

(iv)tx BnTx:fxtx=0

Proof:

a) With each tiT: ti=tij1,tij2,,tijnj=1..s,1id , we built formula:

hix(j1),x(j2),,x(in)j=1..s=Itij1x(j1), Itj2x(j2),,Itjinx(in), mij= 1 ..S

and we have: if x(j1),x(j2),,x(jn)j=1..s=ti=tij1,tij2,,tijnj=1..s then:

hiti=mi, hi(t)=0 with tti, 1id

fx(j1),x(j2),,x(jn)j=1..s=h1h2hdx(j1),x(j2),,x(jn)j=1..s

then f is the formula to look for.

Indeed, we have:

fti=h1h2hdti=h1tih2tihitihdti

.Which according to the properties of

hi:  hiti=hitij1,tij2,,tijnj=1..s=Itij1tij1,Itij2tij2,,Itjntijn,mij=1..s=mihi (t)=0 vói  tti, 1id

So infer: fti=mi,1id and tBnxs \T:f(t)=0f is CTBĐT to look for.

b) From CTBĐT f we have: fx=hx1hx2vhxd Then:

txiTx:fxtxi=hx1hx2vhxdtxi=hx1txihx2txivhxi(txi)vhxd

Which we have: hxitxi=hxitxi1,txi22,xtxin=Ixxi1ttxi1,Itxi2txi2,,Itxintxin,mi=mi

hxitx=0 với txtxi,1id .

we infer: fxtxi=mi,1id

and tx Bn\T:fxtx=0fx satisfies 2 conditions required.

Consequence 3.1: With each block T B nxs,T and each value m>0 in B, exists one CTBĐT f take T as the m-truth block, and fx get Tx as the m-truth block.

Proof:

Use the result of the theorem 3.1 with special cases: m1 = m2 = … = md = m we obtained CTBĐT f satisfies two conditions:

(i) tiT:fti=m

(ii) tBnxsT: f(t)=0

and then CTBĐT fx also satisfies two conditions:

(iii) txiTx:ftxi=m

(iv)  txiBnTx:ftxi=0. Thence inferred: Tf,m = T và Tfx,m = Tx.Definition 3.3

Let R = (id; A1, A2,...,An ), r(R) is a block on R, U=i=1nid(i), each CTBĐT fMVL(U) is called a multivalued positive Boolean formula (CTBDĐT) if f(e) = 1, with i e is the unit value assignment. Here: e= 111...111

Example 3.2:

Let R={1,2},A1,A2,A3,U=1(1),1(2),1(3),2(1),2(2),2(3)  B = {0, 0.5, 1}. Then:

The formulars: 1(1)1(2)2(1)2(2),  1(1)1(2)2(1)2(2) are the CTBDĐT.

The formulars: 1(2)¬2(3),  ¬1(3)¬2(1) 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; A1, A2,...,An ), r(R) is a block on R, we denote di is the value domain of the attribute Ai (is also of index attribute x(i),xid, 1in. Then, for each value domain we consider mapping:  αi : di x di B satisfies the following conditions:

(i) Reflectivity: adi : αi (a,a)=1,

(ii) Symmetry:  a,bdi :αi(a,b)=αi(b,a),

(iii) Sufficiency:  mB,  a,bdi : αi(a,b)=m.

Thus, we see the mapping αiare the relationships above di 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.

Definition 3.5

Let R = (id; A1,A2,...,An ), r(R) is a block on R, u,vr, mappings αi define on each value domain di,  1in. we call αu,v is the value assignment:

α(u,v)=α1u.x(1),v.x(1),α2u.x(2),v.x(2),,αnu.x(n),v.x(n) xid

Then, for each block r, we denote the truth block of block r as Tr:

Tr={α(u,v)u,vr}

If block r contains at least a certain element k then: α(u,u)=1eTr.

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.

3.2 The multivalued positive Boolean dependencies on block

Definition 3.6

Let R=id;A1,A2,,An,r(R) is a block on R, U=i=1nid(i) , 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 TrTf,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:

r(F,m)fF:r(f,m)TrTF,m

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

Proposition 3.1

Let R = (id; A1,A2,...,An ), r(R) is a block on R, U=i=1nid(i) . Then:

i) If r is m-satisfying the multivalued positive Boolean dependency f: r(f,m) then rxfx,m,  xid

ii) If r is m-satisfying set of multivalued positive Boolean dependency F: r(F,m) then rxFx,m,xid

Proof

i) Under the assumption we have r(f,m)TrTf,mTrx=TrxTf,mx=Tfx,m,xid

So we have TrxTfx,m,xidrxfx,m,xid

ii) Under the assumption r(F,m)TrTF,mTrx=TrxTF,mx=TFx,m,xid

Therefore: TrxTFx,m,xidrxFx,m,xid

Proposition 3.2

Let R = (id; A1,A2,...,An ), r(R) is a block on R, U=i=1nid(i),  f=xidfx. Then:

i) If rxfx,m,xid then r is m-satisfying the multivalued positive Boolean dependency f: r(f,m) .

ii) If rxFx,m,xid then r m-satisfying set of multivalued positive Boolean dependency F: r(F,m).

Proof

i) Under the assumption we have: rxfx,m,xidTrxTfx,m,xidTrxTf,mx,xid

So we have: TrTf,mr(f,m) .

r is m-satisfying the multivalued positive Boolean dependency f.

ii) Under the assumption rxFx,m,xidTrxTFx,m,xidTrxTF,mx,xid

So we have: TrTF,mr(F,m) .

r m-satisfying set of multivalued positive Boolean dependency F.

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

Theorem 3.2

Let R = (id; A1,A2,...,An ), r(R) is a block on R, U=i=1nid(i),  f=xidfx. Khi đó:

i) rxfx,m,xidr is m-satisfying the multivalued positive Boolean dependency f: r(f,m) .

ii) rxFx,m,xidr m-satisfying set of multivalued positive Boolean dependency F: r(F,m).

For the set PTBDĐT F and PTBDĐT f:, m[0;1]:

We say F m-deduced f by the block and denoted F-m  f   if:   r : r(F,m) r(f,m).

We say F m-deduced f by the block contains no more than 2 elements and denoted F-2,m f  if:   r2 : r2(F,m)r2(f,m).

We have the following equivalent theorem:

Theorem 3.3

For the set PTBDĐT F and PTBDĐT f , R = (id; A1,A2,...,An ), r(R) is α block on R, m B. Then the following three propositions are equivalent:

(i) F |=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 F=m fTF,m  Tf,m(1). Let r be an arbitrary block and r(F,m), then by definition: TrTF,m(2) . From (1) and (2) we infer: TrTf,m , so we have: r(f,m).

(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 t=tx(1),tx(2),,tx(n) xid, tTF,m we need proof : tTf,m.

Indeed, if t = e then we have tTf,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: u=ux(1),ux(2),,ux(n)x  id, V=Vx(1),Vx(2),,Vx(n)xid satisfy α(u,v)=tmeanαiux(i),vx(i)=tx(i),1in,xid. The existence of the u and v elements as above is due to the properties of the mappings αi mentioned above. Thus r is a block with 2 elements and Tr={e,t}TF,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 TrTf,m (1).

From (1) we infer tTf,m.

Consequence 3.2

For the set PTBDĐT F and PTBDĐT f , R = (id; A1,A2,...,An ), r(R) is a block on R, m B. Then on rx the following three propositions are equivalent:

(i) Fx |=m fx (m-deduction by logic),

(ii) Fx |-m fx (m-deduction by slice rx),

(iii) Fx |-2,m fx (m-deduction by the slice r2x 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 m-equivalent theorem in the relational data model. Specifically, we have the following consequences:﻿Consequence 3.3

For the set PTBDĐT F and PTBDĐT f , R = (id; A1,A2,...,An ), r(R) is a block on R, m  B. Then if id = {x} then block r degenerates into relation and the following three propositions are equivalent:

(i) F |=m f (m-deduction by logic),

(ii) F |-m f (m-deduction by relation),

(iii) F |-2,m f (m-deduction by relation has no more than 2 elements).

Definition 3.7

Let R = (id; A1,A2,...,An ), r(R) is a block on R, U=i=1nid(i), m B. is a subset of PTBDĐT on U, we denoted ( ,m)+ is the set of all PTBDĐT are m-deduction from , in other words:

(Σ,m)+=gMVP(U)|Σ|=mg=gMVP(U)TΣ,mTg,m

Definition 3.8

Let R = (id; A1,A2,...,An ), r(R) is a block on R, U=i=1nid(i), m B, we denoted MBDĐT(r,m) is the set of all PTBDĐT m-right in r, In other words:

MBDĐT(r,m)={gMVP(U)r(g,m)}

So, we have:

gMBDĐT(r,m)gMVP(U)TrTg,m

Theorem 3.4

Let R = (id; A1,A2,...,An ), r(R) is a block on R, U=i=1nid (i), mB. Then we have:

(MBDĐT(r,m),m)+=MBDĐT(r,m)

By definition, we have:

(MBDĐT(r,m),m)+={gMVP(U)|MBDĐT(r,m)|=mg}

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

gMVP(U)|MBDĐT(r,m)|=m g={gMVP(U)|MBDĐT(r,m)|-m g}

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.

Consequence 3.4

Let R = (id; A1,A2,...,An ), r(R) is a block on R, U=i=1nid(i), mB . Then on rx we have:

MBDĐTrx,m,m+=MBDĐTrx,m,xid

Consequence 3.5

Let R = (id; A1,A2,...,An ), r(R) is a block on R, m B. Then we have, if id = {x} then block r degenerates into relation and we have in the relational data model:

(MBDĐT(r,m),m)+=MBDĐT(r,m)

Definition 3.9

Let R = (id; A1,A2,...,An ), r(R) is a block on R, U=i=1nid(i), mB, Σ is the subset of PTBDĐT on U. We say block r is m-representation set Σ if MBDĐT (r,m)  (Σ,m)+ and we say block r is m-tight representation set Σ if MBDĐT(r,m) = (Σ,m)+ .

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; A1,A2,...,An ), m  B. Then, with every block r(R) different from the empty set on R we have:

Tr=TMBDĐT(r,m),m

﻿Proof

Suppose gMBDĐT(r,m)r is m-satisfying gTrTg,m. From Tr and value m, According to theorem 3.1 we find a multivalued boolean formula f satisfying conditions: f(e) = 1 và Tf,m = Tr. So: eTr=Tf,m infer f is one CTBDĐT and and more due Tr=Tf,mr is m-satisfying f, mean: f  MBDĐT (r,m).

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

gMBDĐT(r,m)TrTg,mTrgFTg,m

fMBDĐT(r,m):Tr=Tf,mTrgFTg,m

From (3) and (4) we infer:

Tf=gFTg,m¯=TF,m

Thus: Tt=TMBDĐT(r,m),m .

Consequence 3.6

Let R = (id; A1,A2,...,An ), r(R) is a block on R, m B . Then, on the slice rx we have:

TTx=TMBDĐT (r,m), m , xid

Consequence 3.7

Let R = (id; A1,A2,...,An ), r(R) is a block on R, m B . Then we have, if id = {x} then block r degenerates into relation and we have in the relational data model: Tr = TMBDĐT(r,m),m.

Theorem 3.6

Let R = (id; A1,A2,...,An ), U=i=1nid(i), mB, Σ  is the subset of PTBDĐT on U. Then, with every block r(R) is otherwise empty on R we have: r is m-tight representation set PTBDĐT Σ if and only if Tr=TΣ,m .

Proof

By definition, we have: r is m-tight representation set ΣMBDĐT(r,m)=(Σ,m)+  MBDĐT(r,m)m  .

Other way:

MBDĐT(r,m)m ΣTMBDĐT(r,m),m=TΣ,m

Apply the results of theorem 3.4 we obtain:

Tr=TMBDĐT(r,m),m

So from (5) and (6) we infer: Tr = TS,m.

Therefore: r is m-tight representation set ΣTx=TΣ,m.

Consequence 3.8

Let R = (id; A1,A2,...,An ), U=i=1nid(i), mB, Σ is the subset of PTBDĐT on U. Then we have, if id = {x} then block r degenerated into relation and we have in the relational data model: every relation r is different from the empty set on R is m-tight representation set PTBDĐT Σ if and only if Tr = T,m.

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

We denoted Σx=Σi=1nx(i)

Theorem 3.7

Let R = (id; A,A,...,A), U=i=1nid(i),mB,Σ is the subset of PTBDĐT on if U, Σ=xidΣx,  Σx Then, with every block r(R) is otherwise empty on R we have: r is m-tight representation set PTBDĐT Σ if and only if rx is m-tight representation set Σx,xid .

Proof

) Suppose r is m-tight representation set PTBDĐT Σ we need proof rx is m-tight representation set Σx,xid.

Indeed, under the assumption we have: r is m-tight representation set PTBDĐT Σ, using the results of theorem 3.6 we have: Tr=TΣ,m .

Thence inferred: Trx=TΣ,mx,xid.

Which we have: Trx=Trx=TΣ,mx=TΣx,m,xidTrx=TΣx,mrxΣx,m,xid.

So rx is m-tight representation set Σx,xid.

) Suppose rx is m-tight representation set Σx,xid we need proof r is m-tight representation set Σ.

Indeed, under the assumption rx is m-tight representation set Σx,xidTrx=TΣx,m,xid.

Inferred: Trx=Trx=TΣx,m=TΣ,mx,  xid

Which we have: Tr=x ∈id Trx,TΣ,m=x∈id TΣx,mTr=TΣ,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 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.

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).

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