Một số phụ thuộc logic mở rộng trong mô hình dữ liệu dạng khối TT TIENG ANH

MINISTRY OF EDUCATION AND TRAINING VIETNAM ACADEMY OF SCIENCE ANH TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY - Trinh Ngoc Truc SOME EXTENDED LOGICAL DEPENDENCIES IN THE DATABASE MODEL OF BLOCK FORM Major: Computer Science Code: 48 01 01 SUMMARY OF COMPUTER DOCTORAL THESIS Hanoi - 2021 The thesis was completed at Graduate University of Science and Technology - Vietnam Academy of Science and Technology Science instructor: Assoc Prof Dr Trinh Dinh Thang Dr Nguyen Nhu Son Reviewer 1: Reviewer 2: Reviewer 3: The thesis will be defended before the Academy-level PhD Thesis Evaluation Council, meeting at Graduate University of Science and Technology - Vietnam Academy of Science and Technology at o'clock, th, 2021 The thesis can be found at: - Graduate University of Science and Technology’s Library - National Library of Vietnam INTRODUCTION The urgency of the thesis The management, storage and exploitation of data to solve real-life problems are being studied by many scientists The relational data model proposed by E-Codd in 1970 has contributed to solving the problem of storage, data mining and describing data constraints through the concept of functional dependency (X  Y), but this model is not strong enough and has many limitations in storing and retrieving non-linear structured data Therefore, many domestic and international scientists are interested in researching to expand the relational data model to solve dynamic problems, problems with the non-linear structure Some of the proposed extensions are Data Cube, Data Cube, Multidimensional data model, Data Warehouse The finding of these data models has solved the inadequacies and difficulties in tracking the change of data over time The research area of the doctoral dissertation is about data dependencies of the attributes in the database model of block form This model has been built successefully in the theory and installation With id is an addition that allows tracking of data changes over time, location or distances Researches about data dependencies have been proposed: Functional dependencies, Multivalued dependencies, General Boolean dependencies Although the results have solved many problems of life, in practice there is still a data relationship whose attributes are not described on the block For the above reasons, the thesis chooses the project name "Some extended logic dependencies in the database model of block form" aiming to find new logic dependencies on the block to extend data relationships on the block The objective of the thesis - To find the deriving conjunction of positive Boolean formulas in database model of block form to find the smallest set of derived formulas of the attributes on the data block, contributing to the elimination of redundant attributes in the database design - To find multivalued positive Boolean dependencies, positive Boolean dependencies by groups, multivalued positive Boolean dependencies by groups in the database model of block form to find relationships of attributes, rules of attributes deriver multivalued or by groups, the multivalued by groups of attributes on the block which contributes to the extension of database dependencies on block Layout of the thesis The layout of the thesis consists of the introduction, the three chapters of content and the conclusions Chapter presents basic concepts related to the thesis Chapter presents the research results about derived conjunction of positive Boolean formulas and multivalued positive Boolean dependencies in the database model of block form Chapter presents the research results about Positive Boolean dependencies by groups and multivalued Positive Boolean dependencies by groups on the block CHAPTER SOME BASIC KNOWLEDGE 1.1 Database model of block form 1.1.1 The block, slice of the block The block: Let R = (id; A1, A2, , An) is a finite set of elements, where id is a non-empty finite index set, Ai (i=1 n) is the attribute For 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: t  r ( R )  t  t i : id  dom( Ai ) i 1 n Slice: Let R = (id; A1, A2, , An), r(R) is a block on R For each x id we denoted r(Rx) is a block with Rx =({x}; A1, A2, , An) such that: tx  r(Rx)  tx = {tix = ti } i=1 n , x Then r(Rx) is called a slice of the block r(R) at point x 1.1.3 Functional dependencies in the database model of block form Here, for simlicity, we use the notation: x(i) = (x; Ai ); id(i) = {x(i) | x  id}, and x(i) (x  id, i = n) is called an index attribute of block scheme R   id ; A1 , A2 , , An  Let r(R) is a block on R and X , Y  n id ( i ) , XY is a notation i 1 of function dependency A block r satifies X  Y if:  t1, t2  R such that t1(X) = t2(X) t1(Y) = t2(Y) 1.2 Boolean algebra 1.2.1 Boolean formulas Let U = {x1, x2, , xn} ís a finite set of Boolean variable, B is the set of Boolean values, B = {0, 1} Then the Boolean formulas (CTB) also known as logic formulas are constructed as follows: (i) For each value 0/1 in B is a CTB (ii) For each variable in U is a CTB (iii) If a is a Boolean formula then (a) is a CTB (iv) If a and b are CTB then a  b, a  b,  a and a  b are CTB (v) Only formulas created by rules from (i) - (iv) are CTB We denote L(U) is a set of CTB building on set of variables U 1.2.4 Positive Boolean formulas Formula f  L(U) is called Boolean formula (CTBD) if f(e)=1 with e is the unit value assignment: e = (1, 1, , 1), We denoted P(U) as the set of all Boolean formulas on U 1.2.5 Multivalued Boolean formula 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: (i) For each value on B is a CTBĐT (ii) For each variable on P is a CTBĐT (iii) For function Ib, b  B is a CTBĐT (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 We denote MVL(P) as a set of CTBĐT building on the set variables P   x1 , x2 , , xn  and set values B = b1 , b2 , , bk  include k values on [0;1], k  1.2.8 Multivalued positive Boolean formula Formula f  MVL(P) is called multivalued positive Boolean formula (CTBDĐT) if f(e) = with e is the unit value assignment Notation MVP(P) is set for all multivalued positive Boolean formulas on P 1.3 Positive Boolean dependencies in the database model of block form 1.3.1 The truth block of the data block Let R  (id ; A1 , A2 , , An ), r(R) is a block on R; u, v  r ( R) We have  (u, v) is the value assignment:  (u, v)  (1 (u.x(1) , v.x(1) ), (u.x (2) , v.x (2) , , n (u.x ( n ) , v.x ( n ) ) xid where: i (u.x(i ) , v.x(i ) )  u.x (i )  v.x ( i )  i (u.x(i ) , v.x (i ) )  if opposite, i=1 n ; x  id Then, with each block r(R) we notation truth block of data block r is Tr   (u, v) | u, v  r ( R) 1.3.2 Boolean dependencies on the block Let R = (id; A1,A2, ,An), r(R) is a block on R, we call each positive Boolean formula in MVL(P) is a positive Boolean dependency (PTBD) on the block Block r satifies positive Boolean dependency f and denoted r(f) if Tr  T f 1.4 Conclusion chapter Chapter of the thesis uses some basic concepts as a foundation for the research process, including: Theory of the database model of block form Theory of Boolean Algebra The basic concepts presented in chapter are an important foundation, enough to accomplish the objectives set out in the thesis CHAPTER DERIVED CONJUNCTION AND MULTIVALUED POSITIVE BOOLEAN DEPENDENCIES IN THE DATABASE MODEL OF BLOCK FORM 2.1 Ask the problem The derived formula X  Y, where X and Y are the logical conjunctions of finite variables In the class of positive Boolean dependencies, the derived formulas are the class of functional dependencies proposed by E-Codd in 1970, from function dependency concept, apply concepts of algebra and logic then the logical dependencies are represented as a conjunction of derived formulas In the database model of block form, the research to represent the logical formulas that represented derived conjunction has not been proposed Therefore, in this chapter, the thesis proposes concepts and proof necessary and sufficient conditions for a logical formula to be represented as derived conjunction of logic formulas and install experimentally algorithm find derived conjunction according to the given truth block The obtained results are an important premise to study the data dependencies on the block presented in chapters and In the next part of chapter 2, the thesis proposes a multivalued positive Boolean dependency on the block, the results of this study will solve a series of problems as finding relationships of attributes through the comparison expansion of pairs of elements The results presented in this chapter are published in articles (i), (ii), (iv) the list of research works of the author 2.2 Derived Conjunction in the database model of block form 2.2.1 Derived formula on the block scheme Definition 2.1: Let R   id ; A1 , A2 , , An  , the derived formula on the blocking scheme is a formula of the form f : X  Y ; X , Y  n id (i ) , i 1 where X, Y are the conjunctions of the index attributes contained in it Assuming f : X  Y is a derived formula on n id ( i ) , then i 1 with for each value assignment v  B if ( Set (v)  X )  ( Set (v)  Y ) n m we have: f (v)  if and only Definition 2.2: Let R   id ; A1 , A2 , An  , V is the set of value n assignments on id ( i ) Assuming u , v V , the multiplication i 1 operation of u and v, demoted u & v , which is determined as follows: If and then u  (ux(1) , ux(2) , , ux( n ) ) v  (vx(1) , vx(2) , , vx( n) ) u & v   u x(1)  v1x , u x(2)  vx2 , u x( n )  vxn  xid 2.2.2 Properties of the close set family and the truth block Definition 2.3: The set of V value assignment is closed with multiplication & if V contains the product of every pair of elements in it, that is: u, v V :u & v V Proposition 2.1: Let R   id ; A1 , A2 , An  , the derived formula on the block scheme f : X  Y ; X ,Y  n id (i ) i 1 Then, Tf contains assignments unit e, zero z and closed with multiplication & 2.3.3 The properties of derived conjunction and the truth block Let F is a set of derived formulas, F   f1 , f ,, f n  , We conventionally F is logical conjunction of the set of member derived formulas F  f1  f   f n and F is called derived conjunction 2.3 The algorithm build deriving conjunction Let a binary block T, size each element is n  m  m | id | , contains the unit-value assignment e, zero-value assignment z and closed with multiplication & Then, the algorithm XDF build deriving conjunction F on R   id ; A1 , A2 , An  , get T is truth block 2.3.1 The XDF algorithm Input: Binary block T  B nm , contains e, z and closed with multiplication & Output: Derived conjunction F on R, satisfy the condition TF  T Method: Begin XDF F :  ; For each x  id Fx :  ; For each u  B n \ Tx X : Set (u ) ; Y set (v) \ X ; vTx set ( v )  X Fx : Fx  { X  Y }; endfor; 10 F : F  Fx ; 11 endfor; 12 return F; 13 End XDF Theorem 2.1: Let binary block T  B nm , contains e, z and closed with multiplication &, XDF algorithm calculates truth set of derived formula F get T is the truth block We denoted h is as the number of lines of block T, and k is the number of lines of block Bn×m\T Then we see that the XDF algorithm builds k derived formulas To build each formula, we must perform h traversals, h intersections of two sets, and one difference of two sets Each set operation performed on n elements of the slice at xid requires complexity n Finally, to regroup to F we need m unions To summarizing, we have the complexity of the XDF algorithm is O(hkmn) 2.3.2 The algorithm XDF-S Input: The binary block same level T  Bn×m, contains e, z and closed with multiplication & Output: Derived conjunction F on R, satisfy the condition TF = T Method: Begin XDF-S F :=  ; Lấy x  id thực hiện: Fx := ; For each u Bn \ Tx X := Set(u); Y set (v) \ X ; vTx Set ( v )  X Fx := Fx  {X  Y}; endfor; F Fx ; xid 10 11 return F; End XDF-S Therefore, in this case, the complexity of the XDF-S algorithm is O(hkn) From the properties of the derived conjunction on the block scheme we see: The truth block of every derived conjunction contains two assignments, the unit-value assignment e and the zero-value assignment z Thus, not every binary block is a truth block of a derived conjunction on the block scheme From the theorem on the correctness of the XDF algorithm, we derive the following necessary and sufficient n id ( i ) to be a truth block of a conditions for a value block of T on i 1 derived conjunction on the block scheme n id ( i ) is a truth block of the Theorem 2.2: The binary block T on i 1 derived conjunction on the block scheme R   id ; A1 , A2 , , An  if and only if T contains two assignments, the unit-value assignment e and the zero-value assignment z and closed with multiplication & n id ( i ) can be represented by a Theorem 2.3: Logical formula f on i 1 derived conjunction on the block scheme R   id ; A1 , A2 , , An  if and only if the truth block of f contains two assignments, the unitvalue assignment e and the zero-value assignment z and closed with 11 2.4.4 The multivalued positive Boolean dependencies on block Definiton 2.9: Let R   id ; A1 , A2 , , An  , r(R) is a block on R, U n id ( i ) , we call each multivalued positive Boolean formula in MVP(U) i 1 is a multivalued positive Boolean dependency (PTBDĐT) on block We say, block r m-satisfying the mutilvalued positive Boolean dependency f and denoted r(f,m) if Tr  T f ,m The block r m-satisfying set of mutilvalued positive Boolean dependency F and denoted r(F,m) if r satisfies all PTBDĐT f F: r  F , m  f  F : r  f , m   Tr  TF ,m If r  f , m  then we say PTBDĐT f m-right in the block r Theorem 2.5: Let set of PTBDĐT F and one PTBDĐT f, r(R) is a 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 two elements) In this case id  {x} , then the block degenerates into a relation and the above m-equivalence theorem becomes the m-equivalent theorem in the relation database model 2.4.5 Closure of set mutilvalued positive Boolean dependencies Denifition 2.10: Let R   id ; A1 , A2 , , An  , r(R) is a block on R, U n id ( i ) , mB ,  is a subset of PTBDĐT on U, we denoted i 1 (,m)+ is set of all PTBDĐT are m-dedution from : (, m)  {g  MVP U  |  m g}  {g  MVP U  | T,m  Tg ,m } Denifition 2.11: Let R   id ; A1 , A2 , , An  , r(R) is a block on R, U n id ( i ) , mB , we denoted MBDĐT(r,m) is set of all PTBDĐT i 1 m-right in r, another way: 12 MBDĐT  r , m   g  MVP U  | r  g , m Thus, we have: g  MBDĐT  r , m   g  MVP U   Tr  Tg ,m Theorem 2.6: Let r(R) is a block on R, U  n id ( i ) , mB Then we i 1 have:  MBDĐT  r , m  , m   MBDĐT  r , m   2.4.6 Representation and tight representation set of multivalued positive Boolean dependencies n Denitation 2.12: Let r(R) is a block on R, U  id ( i ) , mB ,  is i 1 subset of PTBDĐT on U We call r is m-representation set  if MBDĐT  r , m   (, m) and r is m-tight repesentation set  if MBDĐT  r , m   (, m) If r is m-tight representation set PTBDĐT  then we say r is block m-Armstrong of set PTBDĐT  Theorem 2.7: Let R   id ; A1 , A2 , , An  , mB Then, for each block r(R) different from the empty set on R we have: Tr  TMBDĐT  r ,m,m Theorem 2.8: Let R   id ; A1 , A2 , , An  , U  n id ( i ) , m  B ,  is i 1 the subset of PTBDĐT on U Then, with every block r(R) different from the empty on R we have: r is m-tight representation set of PTBDĐT  if and only if Tr  T,m 2.5 Experimental install of the problem of finding a multivalued positive Boolean dependency on block - Target: Verify the computer installation to find multivalued positive Boolean dependencies on block with specific data - Tools and test environment: Tools to set up programs: PHP programming language, Javascript Test environment: PC Computer configuration: Intel(R) Core™ i7 2.5Ghz, RAM 8G, Windows 10 OS - Test run data: Sales data of bread, butter and milk of Vinmart+ Supermarket in Xuan Hoa Ward, Phuc Yen City, Vinh 13 Phuc Province The imported item dataset with the number of seasonal customer pairs is as follows: Summer Spring Winter 435 435 435 (customer pairs) (customer pairs) (customer pairs) Set of value B ={0, 0.3 0.7, 1} Requirement: Check with the above customer data that there exists a multivalued positive Boolean dependency on the block f: Bread 0.7 Butter  Milk? what are the buying trends of customers according to the season? - Result: Find the multivalued positive Boolean dependency on the block: Bread 0.7 Butter  Milk The results are shown in the following charts: Practical significance: Customers tend to: In the summer, they buy more bread with milk than with butter Spring buys more bread with butter than with milk Winter purchases are the same as spring, but at a higher rate than spring 2.6 Conclusion chapter From the proposed concepts of the derived formula on the block diagram, the thesis has shown that the truth block of each 14 derived formula f contains the unit value assignments e, not z and closed with the multiply &, so the derived conjunction F has similar properties The thesis shows the relationship between the closed set family and the truth block of the blocking scheme, and successfully builds and installs the derived conjunction algorithm F that accepts a given binary block as the truth block In the next part, the thesis proposes a new logical dependency, which is a multivalued positive Boolean dependency on the block, the results of which are to solve a series of problems of finding data constraints on blocks through the expansion of comparison and prove the equivalence theorem three types of the derived formula on block (deduction by logic, deduction by block, deduction by block with no more than p elements) In the case of the index set id = {x}, the block degenerates into a relation, these results coincide with the published results in the relational data model Basic concepts in chapter 1; The results obtained in chapter are the background knowledge for we to continue to propose new logical dependencies presented in chapter CHƯƠNG POSITIVE BOOLEAN DEPENDENCIES BY GROUPS ANDMULTIVALUEDPOSITIVEBOOLEANDEPENDENCIES BY GROUPS IN THE DATABASE MODEL OF BLOCK FORM 3.1 Ask the problem In the fact, comparing two elements is not only comparing by equality, the general positive Boolean dependencies and the multivalued positive Boolean dependencies on the block have extended the comparison of pairs of elements according to many different criteria and with different levels The problem is, if each time comparing, instead of comparing pairs of elements, we compare p elements (p ≥ 2), if in p elements (not necessarily different) there 15 are two elements that are the same then it is possible to conclude whether there is a data dependency or not or compare p elements of which they are similar by level In this chapter, the thesis focuses on researching and proposing the concept of positive Boolean dependency by groups and multivalued positive Boolean by groups in the database model of block form to solve the above problem The results presented in this chapter are published in articles (iii), (v) the list of research works of the author 3.2 Positive Boolean dependencies by goups on the block 3.2.1 The assignments operation Definition 3.1: Let block schemes R   id ; A1 , A2 , , An  , r(R) is a block on R, we convention that each domain di of the attribute Ai (also of the index attribute x(i), xid), 1 i  n, contains at least p elements (p  2) Then, for each value domain di, we consider the mapping: i:(di) p B satisfies the following properties: (i) Reflectivity: a  (di) p: i(a) = 1, if in a contains at least two identical components (ii) Commutation: a  (di) p: i(a) = i(a’), where a' is a permutation of a (iii) Sufficiency:  a  (di) p: i(a) = 3.2.2 The truth block by groups of the data block Denifition 3.2: Let R = (id; A1,A2, ,An ), r(R) is a block on R, for each value domain di of attributes Ai (also of the index attribute x(i), xid,  i  n, contains at least p elements (p  2), i is an evaluation on groups containing p value of x(i), xid,  i  n For each group of p element: u1, u2, , up arbitrary (not necessarily distinguish) on the block, we call ( u1, u2, , up) is the value assignment: (u1, u2, , up) = (tx1, tx2, , txn), with txi = i(u1.x(i), u2.x(i),, , up.x(i)), xid, 1 i  n 16 Then, for each block r, we denote the truth block by groups of block r as Tr: Tr = {(u1, u2, , up) | uj  r, 1 j  p} 3.2.3 The positive Boolean dependencies by groups of a data block Definition 3.3: Let R = (id; A1,A2, ,An ), r(R) is a block on R, each value domain di of attribute Ai (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) value of x(i), x id, 1 i  n We call each positive Boolean formula in P(U), with U n id ( i ) is a positive Boolean dependency by groups i 1 We say block r is satisfying the positive Boolean dependency by groups (PTBDTNB) f, or f-right in block r and denoted r(f) if Tr  Tf The block r satisfying PTBDTNB F and denoted r(F) if r satisfies all PTBDTNB f in F: r(F)   f F: r(f)  Tr  TF if r(F) then we say set of PTBDTNB F is right in the block r Let set of PTBDTNB F and one PTBDTNB f: - We say F deduce f by the block with groups and denoted F├ f if: r: r(F)  r(f) - We say F deduce f by the block with groups, block contains no more than p elements and denoted F ├ p f if: rp : rp(F)  rp(f) Theorem 3.1: Let R = (id; A1,A2, ,An ), r(R) is a block on R, for each value domain di of attributes Ai (is also of index attribute x(i), x id,  i  n), contains at least p elements, i is an evaluation on groups containing p (p  2) value of x(i), xid, 1 i  n, the set PTBDTNB F and one PTBDTNB f Then the following three propositions are equivalent: (i) F ╞ f (deduction by logic), (ii) F ├ f (deduction in groups by block), 17 (iii) F ├p f (deduction in groups by block has no more than p elements) In the case id = {x}, then the block degenerates into a relation and the above equivalence theorem becomes the equivalent theorem in the relational data model 3.2.4 Closure of set positive Boolean dependencies by groups Definition 3.5: Let R = (id; A1,A2, ,An ), r(R) is a block on R, for each value domain di of attributes Ai (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) value of x(i), xid, 1 i  n Then, we denoted NBD(r) is set PTBDTNB satisfies in r, meaning: NBD(r) = {f | fP(U), r(f)} Proposition 3.1: Let R = (id; A1,A2, ,An ), r(R) is a block on R, for each value domain di of attribute Ai (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 x(i), xid, 1 i  n Then, we have: (NBD(r))+ = NBD(r) Proposition 3.2: Let R = (id; A1,A2, ,An ), r(R) is a block on R, for each value domain di of attributes Ai (is also of index attributes 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 Then, we have: Tr = TNBD(r) 3.2.5 Representation and tight representation set of positive Boolean dependencies by groups Definition 3.6: Let R = (id; A1,A2, ,An ), r(R) is a block on R, for each value domain di of attributes Ai (is also of index attributes x(i), x  id,  i  n), contains at least p elements, i is an evaluation on groups containing p (p  2) value of x(i), x id, 1 i  n 18 We say r representation set PTBDTNB  if NDB(r)   + and r tight representation set PTBDTNB  if NDB(r) =  + Theorem 3.2: Let R = (id; A1,A2, ,An ), r(R) is a block on R, for each value domain di of attributes Ai (is also of index attributes x(i), x  id,1  i  n), contains at least p elements, i is an evaluation on groups containing p (p  2) value of x(i), x  id,  i  n Then r tight representation set PTBDTNB  if and only if Tr = T 3.3 Multivalued positive Boolean dependencies by groups on block 3.3.1 The assignment operation Definition 3.7: Let R = (id; A1,A2, ,An ), r(R) is a block on R, for each value domain di of attributes Ai (is also of index attributes x(i), x  id, 1 i  n, contains at least p (p  2) elements Then, for each value domain di we consider the mapping: i:(di) p B satisfies the following properties: - Reflectivity: a  (di) p: i(a) = 1, if in a contains at least two identical components - Commutation: a  (di) p: i(a) = i(a’), where a’ is a permutation of a - Sufficiency: m B,  a (di) p: i(a) = m 3.3.2 The multivalued truth block by groups of the data block Definition 3.8: Let R = (id; A1,A2, ,An ), r(R) is a block on R, for each value domain di of attributes Ai (is also of index attributes x(i), x  id, 1 i  n, contains at least p (p  2) elements, i is an evaluation on groups containing p (p  2) values of x(i), x  id, 1 i  n For each group p elements: u1, u2, , up arbitrary (not necessarily distinguish) on the block, we call (u1, u2, , up) is the value assignment: (u1, u2, , up) = (tx1, tx2, , txn) with txi = i(u1.x(i), u2.x(i), , up.x(i)), x  id,  i  n Then, with each block r we denoted the multivalued truth block by groups of block r is Tr: 19 Tr = {(u1, u2, , up) | uj  r,  j  p} 3.3.3 The multivalued positive Boolean dependencies by groups of the data block Definition 3.9: Let R = (id; A1,A2, ,An ), r(R) is a block on R, for each value domain di of attributes Ai (is also of index attributes x(i), x  id, 1 i  n, contains at least p (p  2) elements, i is an evaluation on groups containing p (p  2) values of x(i), x  id, 1 i  n With evaluations i on the value domain of index attribute x(i), xid, 1 i  n, then a multivalued positive Boolean dependency by groups is a multivalued positive Boolean formula in MVP(U) with U  n id ( i ) i 1 Let value m  B, we say block r is m-satisfying by groups the multivalued positive Boolean dependency by groups (PTBDĐTTNB) f and denoted r(f,m) Tr  Tf, m The block r is m-satisfying by groups set PTBDĐTTNB F and denoted r(F,m) if block r is m-satisfying by groups all dependencies f in F: r(F,m)   f F: r(f,m)  Tr  TF,m If r(F,m) we say set PTBDĐTTNB F is m-right by groups in the block r Let set PTBDĐTTNB F and one PTBDĐTTNB f: - We call F is m-deduced f by block with groups and denoted F├(m) f if: r: r(F,m)  r(f,m) - We have F is m-deduced f by block with groups, block contains no more than p elements rp and denoted F├ p(m) f if:  rp : rp(F,m)  rp(f,m) Theorem 3.3: Let R = (id; A1,A2, ,An ), ), r(R) is a block on R, for each value domain di of attributes Ai (is also of index attributes x(i), x  id, 1 i  n, contains at least p (p  2) elements, i is an evaluation 20 on groups containing p (p  2) values of x(i), x  id, 1 i  n, set PTBDĐTTNB F and PTBDĐTTNB f Then the following three propositions are equivalent: (i) F╞ (m) f (deduction by logic), (ii) F├ (m) f (deduction by block), (iii) F├ p (m) f (deduction in groups by block has 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 3.3.4 Closure of set multivalued positive Boolean dependencies by groups Definition 3.11: Let R = (id; A1,A2, ,An ), r(R) is a block on R, U n id ( i ) , m  B, for each value domain di of attributs Ai (is also of i 1 index attribute x(i), x  id,1  i  n), contains at least p (p  2) elements, i are evaluations on groups containing p value of each value domain di, 1 i  n When, we denoted NMBDĐT(r, m) are set PTBDĐTTNB msatisfis in block r, means: NMBDĐT(r,m) = {f | fMVP(U), r(f,m)} Proposition 3.3: Let R = (id; A1,A2, ,An ), r(R) is a block on R, U n id ( i ) , m  B, for each value domain di of attribute Ai (is also i 1 of the index attribute x(i), x  id,1  i  n), contains at least p (p  2) elements, are evaluations on groups containing p value of each value domain di, 1 i  n Then, we have: (NMBDĐT(r,m),m)+ = NMBDĐT(r,m) Proposition 3.4: Let R = (id; A1,A2, ,An ), r(R) is a block on R, U n i 1 id ( i ) , m  B, for each value domain di of attribute Ai (is also of 21 index attribute x(i), x  id,1  i  n), contains at least p (p  2) elements, i are evaluations on groups containing p value of each value domain di, 1 i  n Then, we have: Tr = T(NMBDĐT(r,m),m) 3.3.5 Representation and tight representation set of multivalued positive Boolean dependencies by groups Denifition 3.12: Let R = (id; A1,A2, ,An ), r(R) is a block on R, U n id ( i ) , m  B, for each value domain di of attribute Ai (is also of i 1 index attribute x(i), x  id,1  i  n), contains at least p (p  2) elements, i are evaluations on groups containing p value of each value domain di, 1 i  n We say, r is m-representation set PTBDĐTTNB if NMBDĐT(r,m)  (,m)+ and r is m-tight representation set PTBDĐTTNB  if NMBDĐT(r,m) = (,m)+ Theorem 3.5: Let R = (id; A1,A2, ,An ), r(R) is a block on R, U n id ( i ) , m  B, for each value domain di of attribute Ai (is also i 1 of the index attribute x(i), x  id,1  i  n), contains at least p (p  2) elements, i are evaluations on groups containing p value of each value domain di, 1 i  n Then, r is m-tight representation set PTBDĐTTNB  if and only if Tr = T,m 3.4 Experimental install of the problem of finding a positive Boolean dependency by groups and multivalued positive Boolean dependency by groups on block - Target: Verify the computer installation to find positive Boolean dependencies and multivalued positive Boolean dependencies by groups on block with specific data - Tools and test environment: Tools to set up programs: PHP programming language, Javascript Test environment: PC Computer configuration: Intel(R) Core™ i7 2.5Ghz, RAM 8G, Windows 10 OS 22 - Test run data: Sales data of bread, butter and milk of Vinmart+ Supermarket in Xuan Hoa Ward, Phuc Yen City, Vinh Phuc Province The imported item dataset with the number of seasonal customer groups is as follows: Summer Spring Winter 455 455 455 (Customer groups) (Customer groups) (Customer groups) Value set B ={0, 0.3 0.7, 1} Groups p = Requirement: Check with the above customer data that there exists a positive Boolean dependency by groups and multivalued positive Boolean dependency by groups on the block with level m = 0.7? What are the buying trends of customers according to the season? Results of running the program found: - Positive Boolean dependencies by groups on block: Bread p Butter  Milk - Multivalued positive Boolean dependencies by groups on block: Bread p,0.7 Butter  Milk The results are shown in the following charts: Practical implication: The customer group trends: In the summer, buy bread with milk more than buy with butter Spring buys more bread with butter than with milk Winter tends to buy like spring, but at a higher rate than spring 3.5 Conclusion chapter 23 In the next part, the thesis proposes a new logical dependency, which is a multivalued positive Boolean dependency on the block, the results of which are to solve a series of problems of finding data constraints on blocks through the expansion of comparison and prove the equivalence theorem three types of the derived formula on block (deduction by logic, deduction by block, deduction by block with no more than p elements) In the case of the index set id = {x}, the block degenerates into a relation, these results coincide with the published results in the relational data model In this chapter, by expanding the comparisons on p elements, the thesis proposes two new logical dependency, which are positive Boolean dependency by groups and multivalued positive Boolean dependency by groups on the block, at the same time, state and prove the equivalence of the three types of the derived formula on block (deduction by logic, deduction by block, deduction by block with no more than p elements) The obtained results have solved a series of problems of finding data constraints on blocks by expanding the comparison on p elements In the case of the index set id = {x}, the block degenerates into a relation, these results coincide with the published results in the relational data model CONCLUSION The main results of the thesis: - Proposing concepts of derived formulas, derived conjunction on blocks, properties of closed sets, and truth blocks in the database model of block form; Proposing an algorithm to find the derived conjunction on the block through the truth block; Proposing multivalued positive Boolean dependencies on a block The obtained results have solved the problem of finding data constraints on the block through matching pairs of elements reflecting the certainty or confidence of the threshold m 24 - Proposing positive Boolean dependencies by groups and multivalued positive Boolean dependencies by groups on the block The obtained results contribute to the method of finding new logical dependencies on the block, where instead of comparing each pair of elements, we compare with p elements Thus, this extension significantly reduces the number of comparisons of elements in the block scheme - In case the index set consists of only one element, the block degenerates into a relation, the above results coincide with the results studied by the authors in the relational data model Future research of the thesis: - Further research to develop new types of data dependencies on attributes (as also of index attributes), relationships between types of extended logical dependencies on blocks , expand research in the particular case of the set of functional dependencies F such as the set of functional dependencies Fh, the set of functional dependencies Fhx - Extend data dependencies across multiple slices NEW FINDINGS OF THE DOCTORAL DISSERTATION - New findings of the deriving conjunction of positive Boolean formulas in the database model of block form to find the smallest set of deriving formulas of the attributes on the data block, contributing to the elimination of redundant attributes in the database design - New findings of multivalued positive Boolean dependencies, positive Boolean dependencies by groups, multivalued positive Boolean dependencies by groups in the database model of block form to find relationships of attributes, rules of attributes deriver multivalued or by groups, the multivalued by groups of attributes on the block to add extended relationships LIST OF WORKS OF AUTHOR [i] Trinh Dinh Thang, Tran Minh Tuyen, Trinh Ngoc Truc, The derived formula in the database model of block form, The 8th National Conference on Fundamental and Applied IT (FAIR), Hanoi 7/2015, 103-110 [ii] Trinh Dinh Thang, Tran Minh Tuyen, Trinh Ngoc Truc, Multivalued positive Boolean dependencies in the database model of block form, The 9th National Conference on Fundamental and Applied IT (FAIR), Hanoi 8/2016, 602-609 [iii] Trinh Dinh Thang, Tran Minh Tuyen, Trinh Ngoc Truc, Positive Boolean dependencies by groups in the database model of block form, The 11th National Conference on Fundamental and Applied IT (FAIR) (FAIR), Hanoi 8/2018, 446-452 [iv] Trinh Dinh Thang, Tran Minh Tuyen, Trinh Ngoc Truc, Pham Thi Phuong, Some properties of multivalued positive Boolean dependencies in the database model of block form, Indian journal of Science and Technology, 2020, 13(25):2509-2519 [v] Trinh Dinh Thang, Trinh Ngoc Truc, Tran Minh Tuyen, Nguyen Nhu Son, Multivalued positive Boolean dependencies by group in the database model of block form, International Journal of Advanced Research in Computer Science, 2020, 11(4):6-12 ... attributes are not described on the block For the above reasons, the thesis chooses the project name "Some extended logic dependencies in the database model of block form" aiming to find new logic. .. database model of block form to find relationships of attributes, rules of attributes deriver multivalued or by groups, the multivalued by groups of attributes on the block which contributes to the... concepts of algebra and logic then the logical dependencies are represented as a conjunction of derived formulas In the database model of block form, the research to represent the logical formulas that