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The research content in the thesis is the current problems which are renewed with a series of works of foreign authors; while in the country (in Vietnam), there are many published works related to methods and algorithms finding reducts of a decision table by different approaches. The objective of the thesis is to research some analyzed problems in range of relational databases.
MINISTRY OF EDUCATION AND TRAINING VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY - VU QUOC TUAN DISCOVERING FUNCTIONAL DEPENDENCIES AND RELAXED FUNCTIONAL DEPENDENCIES IN DATABASES Major: Math Fundamentals for Informatics Code: 46 01 10 SUMMARY OF MATHEMATICS DOCTORAL THESIS Ha Noi - 2019 This work is completed at: Graduate University of Science and Technology Vietnam Academy of Science and Technology Supervisor 1: Assoc Prof Dr Ho Thuan Supervisor 2: Assoc Prof Dr Nguyen Thanh Tung Reviewer 1: … Reviewer 2: … Reviewer 3: … This thesis will be officially presented in front of the Doctoral thesis Grading Committee, meeting at: Graduate University of Science and Technology Vietnam Academy of Science and Technology At hrs day month year This thesis is available at: Library of Graduate University of Science and Technology National Library of Vietnam INTRODUCTION Data dependencies play important roles in database design, data quality management and knowledge representation Dependencies in knowledge discovery are extracted from the existing data of the database This extraction process is called dependency discovery The aim of dependency discovery is to find important dependencies holding on the data of the database These discovered dependencies represent domain knowledge and can be used to verify database design and assess data quality Dependency discovery has attracted a lot of research interests from scientists since early 1980s At the present time, the problem of discovering data dependencies on big data sets becomes more important because these big data sets contain a lot of valuable knowledge Currently, with the development of digital devices, especially social networks and smart phone applications, the amount of data in the applications increases very quickly, these arise proplems in data storage, data management, especially the problem of knowledge discovery from those big data sets The problem of discovering FDs and RFDs in databases is one of important proplems of knowledge discovery Three typical types of data dependencies which are interested in discovering are FD, AFD and CFD AFD is an extension of FD, the "approximation" is based on a degree of satisfaction or an error measure; CFD is an extension of FD which aims to capture inconsistencies in data The research directions which solve the problem of RFD discovery in databases, firstly focus on FD discovery because FD is the separate case of all types of RFD, the research results about FD discovery can be adapted to discover other types of data dependencies (such as AFD) The general model of FD discovery problem includes steps: generating a search space of FDs, verifying the satisfaction of each FD, pruning the search space, outputting the set of satisfied FDs and reducing redundancies in this set of satisfied FDs In the FD discovery problem, the key discovery is the special case and is also an important problem in normalizing relational databases The time complexity of the FD discovery problem is polynomial in the number of tuples in the relation but is exponential in the number of attributes of that relation Therefore, for reducing the processing time, effective pruning rules should be developed Among the proposed pruning rules, it is important to prune keys, and if a key is discovered then it is possible to prune (delete) all sets containing the key in the search space However, the disadvantage of existing key pruning rules is to find keys on the entire set of attributes of the database (this is really a very difficult problem because the time complexity can be exponential in number of attributes of ) So is there any way to find keys in a proper subset of ? This question is one of basic motivations of this thesis After the set of data dependencies is discovered, this set can be very large and difficult to use because it contains unnecessary redundancies The important problem is how to eliminate (as much as possible) the redundancy in the set of discovered data dependencies This is also a problem interested in the thesis Another research direction in the thesis is to focus on discovering two typical types of RFD, namely AFD and CFD Both AFD and CFD have many applications and occurences in relational databases, especially CFD is also a powerful tool for dealing with data cleaning problems For AFD, the most important problem is to improve and develop techniques for computing approximate measures; For CFD, in addition to discovering them, the research about a unified hierarchy between CFD and other types of data dependencies is also a very interesting problem The research content in the thesis is the current problems which are renewed with a series of works of foreign authors; while in the country (in Vietnam), there are many published works related to methods and algorithms finding reducts of a decision table by different approaches The objective of the thesis is to research some analyzed problems in range of relational databases The main contents of the thesis are described as follows: Chapter An overview of relational data model, concepts of functional dependency, closure of a set of attributes, key for a relational schema, etc This chapter also focuses on RFD and the generalization of methods used for discovering FDs and RFDs Chapter The presentation of AFD and CFD (two typical types of RFD) and some related results Chapter The presentation of the closure computing algorithms of a set of attributes under a set of FDs, reducing the key finding problem of a relation schema and some related results Chapter The presentation of an effective preprocessing transformation for sets of FDs (to reduce redundancies in a given set of FDs) and some related results Chapter FUNCTIONAL DEPENDENCIES AND RELAXED FUNCTIONAL DEPENDENCIES IN THE RELATIONAL DATA MODEL 1.1 Recalling some basic notions A relation r on the set of attributes Ω = {A1, A2,…,An} r {(a1, a2,…,an) | Dom(Ai), i = 1, 2,…, n} where Dom(Ai) is the domain of Ai, i = 1, 2,…, n A relation schema S is an ordered pair S = , where Ω is a finite set of attributes, F is a set of FDs S can also denoted by S() 1.2 Functional dependency Functional dependency Given X, Y Then X Y if for all relations r over the relation schema S(), t1, t2 r such that t1[X] = t2[X] then t1[Y] = t2[Y] Armstrong's axioms For all X, Y, Z , we have: Q1 (Reflexivity): If Y X then X Y Q2 (Augmentation): If X Y then XZ YZ Q3 (Transitivity): If X Y and Y Z then X Z The closure of X under a set of FDs F, is the set X F : X F = {A (X A) F+} Keys for a relation schema Let S = be a relation schema and K We say that K is a key of S if the following two conditions are simultaneously satisfied: (i) (K ) F+ (ii) If K' K then (K' ) F+ If K only satisfies (i) then K is called a superkey 1.3 Relaxed functional dependency (RFD) 1.3.1 Approximate functional dependency (AFD) An AFD is a FD that almost holds To determine the degree of violation of X Y in a given relation r, an error measure, denoted e( X Y , r ) , shall be used Given an error threshold , We say that X Y is an AFD if and only if e( X Y , r ) 1.3.2 Metric functional dependency (MFD) Consider X Y in a given relation r A MFD is an extension of functional dependency by replacing the condition t1[Y] = t2[Y] with d(t1[Y], t2[Y]) ≤ , where d is a metric on Y, d: dom(Y) dom(Y) R and is a parameter 1.3.3 Conditional functional dependency (CFD) A CFD is a pair = (X Y, Tp), where X Y is a FD and Tp is a pattern tableau with all attributes in X and Y Intuitively, the pattern tableau Tp of refines the FD embedded in by enforcing the binding of semantically related data values 1.3.4 Fuzzy functional dependency (FFD) Let r be a relation on Ω = {A1, A2,…,An} and X, Y For each Ai Ω, the degree of equality of data values in Dom(Ai) is defined by the fuzzy tolerance relation Ri Given a parameter (0 ≤ ≤ 1), we say that two tuples t1[X] and t2[X] are equal with the degree , denoted t1[X] E() t2[X], if Rk(t1[Ak], t2[Ak]) for all Ak X Then, X Y is called a FFD with the degree if t1, t2 r, t1[X] E() t2[X] t1[Y] E() t2[Y] 1.3.5 Differential dependency (DD) DD extends the equality relation (=) in FD X Y The conditions t1[X] = t2[X] and t1[Y] = t2[Y], in turn, are replaced by the conditions which t1, t2 satisfies differential functions L and R In fact, the differential functions use metric distances to extend the equality relation used in FD FD is a special case of DD if L[t1[X], t2[X]) = and R[t1[Y], t2[Y]) = In addition, DD is also an extension of MFD if L[t1[X], t2[X]) = and R[t1[Y], t2[Y]) ≤ 1.3.6 Other types of RFDs There are many other types of RFDs Starting from reality applications, each type of RFDs is the result of extending (relaxing) the equality relation in the traditional FD concept by a certain way 1.4 FD Discovery Top-down methods These methods generate candidate FDs following an attribute lattice, test their satisfaction, and then use the satisfied FDs to prune candidate FDs at lower levels of the lattice to reduce the search space An important prolem is how to check if a candidate FD is satisfied? Two specific methods were used: the partition method (algorithms: TANE, FD_Mine) and the free-set method (algorithm: FUN) Bottom-up methods Different from the top-down methods above, bottom-up methods compare the tuples of the relation to find agreesets or difference-sets These sets are then used to derive FDs satisfied by the relation The feature of these mothods is that they not check candidate FDs against the relation for satisfaction, but check candidate FDs against the computed agree-sets and differencesets Two typical algorithms using these methods are Dep-Miner and FastFDs The worst case time complexity of the FD discovery problem is exponential in the number of attributes of There are some topics relating to FD discovery, such as sampling, maintenance of discovered FDs, key discovery, 1.5 RFD Discovery 1.5.1 AFD Discovery To test the satisfaction of AFDs, FD discovery methods can be adapted to discover AFDs by adding a certain approximate measure to these methods 1.5.2 CFD Discovery On the discovery of CFDs, challenges are from two areas Like in classical FDs, the number of candidate embedded FDs for possible CFDs is exponential At the same time, the discovery of the optimal tableau for an embedded FD is NP-C Three typical algorithms for CFD discovery are CFDMiner, CTANE and FastCFD 1.6 Summary of chapter This chapter presents an overview of FD and RFD in the relational data model The dependency discovery problem has an exponential search space on the number of attributes involved in the data The FD discovery methods can be adapted to discover RFDs For example, an error measure can be used in a FD discovery algorithm for finding AFDs Some algorithms are proposed for discovering FDs and RFDs Chapter APPROXIMATE FUNCTIONAL DEPENDENCIES AND CONDITIONAL FUNCTIONAL DEPENDENCIES 2.1 About some results relating to FD and AFD This section shows relationships for the results in two works of two groups of authors (([Y Huhtala et al., 1999] and [S King et al., 2003]) and proves some important lemmas as the foundation to discover FD and AFD (these lemmas have not been proven) 2.1.1 Partitions For t r and X , let us denote: [t]X = {u r | t[X] = u[X]}and X = {[t]X | t r} The product of two partitions X and Y, denoted by X Y The number of equivalence classes in X is denoted by |X | 2.1.2 Some results The following theorems of [S.King et al., 2003]) are really some lemmas of [Y Huhtala et al., 1999], these lemmas have been proven in detail in the thesis Theorem 2.1 FD X A holds if and only if X refines A Theorem 2.2 FD X A holds if and only if |X| = |X{A}| Theorem 2.3 FD X A holds if and only if g3(X) = g3(X {A}) Theorem 2.4 We have X Y = X Y Theorem 2.5 For B X and X - {B} B Then, if X A then X {B} A If X is a superkey then X - {B} is also a superkey Theorem 2.6 C+(X) = {A R | B X, X - {A, B} B does not hold} Theorem 2.7 For A X and X - {A} A FD X - {A} A is minimal iff for all B X, we have A C+(X - {B}) discovering FD (testing satisfaction) and AFD (computing measures TRUTHr(X Y), g1(X Y, r), g2(X Y, r)) 2.3 Conditional Functional Dependencies (CFD) Definition A CFD on a relation schema R is a pair = (X Y, Tp), where X Y is a standard FD (referred to as the FD embedded in ) and Tp is a tableau with all attributes in X Y (referred to as the pattern tableau of ), where for each A in X or Y and each tuple t Tp, t[A] is either a constant "a" in the domain Dom(A) of A or an unnamed variable "" Semantics The pattern tableau Tp of CFD = (X Y, Tp) defines tuples (in the relation) which satisfy FD X Y Intuitively, the pattern tableau Tp of refines the standard FD embedded in by enforcing the binding of semantically related data values The consistency problem for CFDs is NP-complete The inference system is sound and complete for implication of CFDs The proposed algorithms for discovering CFD are CFDMiner, CTANE and FastCFD 2.4 About a unified hierarchy for FDs, CFDs and ARs The work of [R.Medina et al., 2009] is interesting and original The authors have shown a hierarchy between FDs, CFDs and ARs: FDs are the union of CFDs while CFDs are the union of ARs The hierarchy between FDs, CFDs, and ARs has many benefits: algorithms for discovering ARs can be adapted to discover many other types of data dependencies and further generate a reducted set of dependencies The contents below are some remarks and preliminary results after researching the work of [R.Medina et al., 2009]: 10 Remark 2.1 It is different from most authors researching into CFDs, [R.Medina et al., 2009] have extended all t p Tp , these pattern tuples are now defined on the whole set Attr(R), where tp[A] = if A X Y Remark 2.2 Instead of matching of a tuple t r with a tuple Tp (tp is now defined on Attr(R)), we match t(X) with tp(X), t(Y) with tp(Y) More formally, t(X) and tp(X) (respectively t(Y) and tp(Y)) are matching if A X: t(X)[A] = tp(X)[A] = a Dom(A) or t(X)[A] = a and tp(X)[A] = Remark 2.3 Consider a pattern tuple defining a fragment relation of [R.Medina et al., 2009] as follows: rt p = {t r | t} (*) It is clear that the formula (*) is incorrect The reason is that in almost cases, (*) returns the empty set In fact, in case contains at least one component , then there exists not t r such that t In the opposite case (tp does not contain the component ) and X Y Attr(R), we have tp[A] = and t[A] = a for A X Y Therefore, there does not exist t r such that t So, rt p , defined by (*) returns the non-empty result only when X Y = Attr(R) and coineides with a certain tuple t in r Hence, the expression (*) must be changed to rt p = {t r | t(X Y) tp(X Y)} [R.Medina et al., 2009] used the following definitions: X-complete property A relation r is said to be X-complete if and only if t1, t2 r we have t1[X] = t2[X] X-complete pattern: (X, r) = {t r} X-complete horizontal decomposition: RX(r) = {r' r | r' is X-complete} 11 Set of X-patterns: (X, r) = {(X, r') | r' RX(r)} Closure operator: (X, r) = {A Attr(R) | tp (X, r), tp[A] } Remark 2.4 Let r be an X-complete relation and r' r Then (X, r') = {t r'} Returning to the definition of on two tuples t1, t2 r Here, using the order relation a for any constant a of an attribute to compute t1 t2, we can have unnecessary difficulties Essentially, we have to compare the respective components of two tuples t1 and t2 to know whether they are aqual or not Therefore, , instead of the operator , it is better to use the simple operator defined as follows: For all t1, t2 r, t2 = t such that A Attr(R), t[ A] t1[ A] if t1[ A] t2 [ A] t1 t[ A] if t1[ A] t2 [ A] The following proposition which expresses the relationship between (X, r) and X F proved in detail in the thesis Proposition Let r be a relation defined over the set of attributes Attr(R), X Attr(R), and r satisfies the set of FDs F Then (X, r) = {A Attr(R) | tp (X, r), tp[A] } = X F = {A Attr(R) | (X A) F+} 2.5 Summary of chapter This chapter presents some results relating to FDs and AFDs, the matrix method for discovering FDs and AFDs and some preliminary results relating to a unified hierarchy for FDs, CFDs and ARs FD, AFD and CFD are three important types of data dependencies Researching and continuing to solve problems relating to these three types is a new and very interesting direction The main results of this chapter are published in the works [CT1, CT2, CT8, CT9] 12 Chapter THE CLOSURE COMPUTING ALGORITHMS AND REDUCING THE KEY FINDING PROBLEM OF A RELATION SCHEMA 3.1 The closure computing algorithms 3.1.1 The closure concept Let F be a set of FDs defining over and X We have: X F = {A (X A) F+} We use the symbol X+ instead of X F when F is clear from the context 3.1.2 Some closure computing algorithms This section mentions some closure computing algorithms The main content is the improvement of Mora et al's algorithm The experimental results show that Mora at al's algorithm are more efficient than other algorithms However, the correctness of this algorithm is not proved Furthermore, its disadvantage is that each time F is browsed, all FDs with the left and right sides contained in Xnew will still be checked on the left side to compute the new value of Xnew (this takes unnecessary time because the Xnew is essentially unchanged The improved algorithm avoids the disadvantage because all FDs with the right side contained in Xnew are removed before computing the clorure The correctness of Mora at al's algorithm (and the improved algorithm) is proved in detail in the thesis Furthermore, the improved algorithm is showed to be more efficient than Mora et al's one 13 The algorithm of Mora et al Input: , F, X Output: X+ begin Xnew = X; repeat Xold = Xnew; for each Y Z F if Y Xnew then Xnew = Xnew Z; (I) F = F - {Y Z}; elseif Z Xnew then F = F - {Y Z}; (II) else F = F - {Y Z}; (III) F = F {Y-Xnew Z-Xnew}; end if; end for each; until ((Xnew = Xold) or (|F| = 0)); return(Xnew); end; The improved algorithm Input: , F, X Output: X+ begin Xnew = X; repeat Xold = Xnew; for each Y Z F if (Z Xnew) then F = F - {Y Z} (I) else if (Y Xnew) then (II) Xnew = Xnew Z; F = F - {Y Z} else (III) F = F - {Y Z }; F= F {Y-XnewZ-Xnew}; end if; end for each; until (Xnew = Xold) or (|F| = 0); return(Xnew); end; 3.2 Reducing the key finding problem of a relation schema 3.2.1 Some known results Let S = be a relation schema, where = {A1 , A2, , An} and F = {L1 R1, , Lm Rm | Li, Ri , i = 1, ,m} Let us denote: m m i 1 i1 L Li , R Ri , S = {Kj | Kj is a key of S}, G K j S Kj , H K j and H = \ H K j S Theorem 3.1 (Ho Thuan and Le Van Bao, 1985) Let S = be a relation schema Then: - If X is a key of S then \ R X ( \ R) (L R) (1) - We have G = \ R R \ L H 14 The concepts core and body have been proposed by P Cordero et al in 2013: Let S = be a relation schema Then, core and body of S are defined as follows: core(, F) = \ Ri and ( Li Ri )F body(, F) = Li [ \ core(, F)+] ( Li Ri )F It is easily seen that core(, F) = \ R and body(, F) = L [ \ ( \ R)+] Theorem 3.2 (Mora et al, 2011) Let S = be a relation schema and K is a (minimal) key of S Then, we have: core K (core body), it means \ R K ( \ R) [L [ \ ( \ R)+] ] (2) It is clear that (2) is a necessary condition for which K is a key of S 3.2.2 An improved form of the necessary condition (1) Based on (1) and the familiar semantics of FD in the relational model, we have the following theorem: Theorem 3.3 Let S = be a relation schema Then \ R K ( \ R) [(L R) \ (R ( \ R)+ )], K S (3) It is clear that (3) is a improved form of (1) 3.2.3 Comparing the necessary conditions Theorem 3.4 Two necessary conditions (2) and (3) are actually equivalent expressions Theorem 3.5 (Ho Thuan et al, 1996) Let S = be a relation schema and K be a key of S Then: ( \ R) K ( \ R) [(L R) \ ( \ R)+] (4) The following theorem shows the relationship between (2) and 15 (4) Theorem 3.6 The necessary condition (2) is actually coincides with the one (4) The theorems 3.3, 3.4 and 3.6 are proved in detail in the thesis Intuitively, the purpose of the key finding problem is to find all keys K of S = and we always know that all keys K are contained in Ω If we find keys in the universe set Ω then it is not effective because Ω is the largest superkey containing all keys Therefore, it is important to find a superkey Z (as few attributes as possible) containing all keys of S such that Z Ω If we find a such set Z then the finding keys in Z instead of finding ones in Ω will be simpler The necessary conditions (1), (2), (3) and (4) are represented again below, they show the general structure of all keys K of S, the righthand sets are superkeys containing all keys of S \ R K ( \ R) (L R) \ R K ( \ R) [L [ \ ( \ R)+] ] \ R K ( \ R) [(L R) \ (R ( \ R)+ )] \ R K ( \ R) [(L R) \ ( \ R)+] In the thesis, we show that the right-hand set of (2) is better than the one of (1) and prove that (2), (3) and (4) are equivalent expressions As analyzed, we expect that the right-hand sets have as few attributes as possible (the smaller the better) This is obviously related to reducing the key finding problem Indeed, suppose that we have Z (a proper subset of ) which contains all keys of S = Then, the reducing for the key finding problem of S is performed through the following steps: Step Creating the schema S' = where 16 ' = Z \ ( \ R) and F' = {Li ' Ri ' | (Li Ri) F, i = 1, 2, , m} Step Finding S ' by a certain algorithm Step Finding S = {( \ R) K | K S ' } 3.3 Summary of Chapter In experiments, Mora et al's closure computing algorithm is more effective than other ones However, this algorithm still has some limitations and its correctness has not been proven We prove the correctness of the algorithm and improve it The improved algorithm is more efficient than Mora et al's one because it has the replacement of FDs with simpler FDs in the computing process ; especially in many cases, the closure computing process and set F are much simpler because all FDs with the right side contained in Xnew are removed before building the closure For the reduction of the key finding problem, based on the familiar semantics of FD in the relational data model, we improve the necessary condition (1) to obtain the necessary condition (3), and prove that the ones (2), (3) and (4) are actually equivalent expressions Finding a necessary condition which is better than (2), (3) or (4) to further reduce the key finding problem is a very interesting problem The main results of this chapter are published in the works [CT3, CT4, CT6, CT7] 17 Chapter ABOUT AN EFICIENT PREPROCESSING TRANSFORMATION FOR SETS OF FUNCTIONAL DEPENDENCIES 4.1 Introduction Let r be a relation defined on Each assertion as the form XY, where X, Y , is called a FD in r We say r satisfies XY if for all t1, t2 r such that t1[X] = t2[X] then t1[Y] = t2[Y] 4.2 Redundancy in a set of FDs Given a set of FDs F, the symbol F (Z W) for indicating Z W to be derived from F through Armstrong's inference rules Consider FD f = XY F: We say that f is redundant in F if F \{ f } | f We say that f is l-redundant in F if exists Z , Z X such that (F \{ f }) {(X Z)Y} | f We say that f is r-redundant in F if exists U , U Y such that (F \{ f }) { X(Y U)} | f We say that F have redundancy if it has an element which is redundant or l-redundant or r-redundant 4.3 An eficient preprocessing transformation for sets of FDs Mora et al have designed an efficient preprocessing transformation based on the substitution paradigm to eliminate redundancy in a set of FDs The basis and correctness of this preprocessing transformation is theorem 4.1 In this section, we show an unacceptable error in the proof of theorem 4.1 and give a new, correct and simpler proof for that theorem; These are confirmed by Mora 18 4.3.1 The Paredaens Logic The Par language: Par = {XY | X, Y 2 and X } The Paredaens Logic (LFD) is the logic given by the pair (Par, SPar) where SPar as an axiom schema AxPar: if Y X and the following inference rules: Trans XY, Y Z |SPar XZ Augm XY |SPar XXY Union XY, X Z Comp XY, W Z |SPar XWYZ Inters XY, X Z |SPar XY Z |SPar gAug (Transitivity Rule) (Augmentation rule) XYZ where Y Z Reduc XY |SPar XY Z where Y Z Frag XYZ |SPar XY |S Par XY (Union Rule) (Composition Rule) (Intersection Rule) (Reduction Rule) (Fragmentation Rule) XY |SPar UV where X U V XY (Generalized Augmentation Rule) gTrans XY, Z U |SPar VW where Z XY, X V and W UV (Generalized Transitivity Rule) Theorem 4.1 (P.Cordero et al.): For XY, UV LFD such that X Y = (a) If X U then {XY, UV} S Par {XY, (U Y)(V Y)} (1) (b) If X U and X UV then {XY, UV} S Par {XY, U(V Y)} (2) The goodness and novelty of theorem 4.1 is that it gives two important substitution rules It is clear that no axiomatic system for 19 FDs has the above rules which can detect and remove redundancy effectively in a set of FDs The proof of the theorem 4.1 is performed by P.Cordero et al Consider the proof of (b) in direction of P.Cordero et al as follows: U X (AxFD) XY (Hypothesis) U Y (1, 2, Transitivity) U (V Y) (Hypothesis) U VY (3, 4, Union) U V (2, 5, Generalized Augmentation) Next, consider step 1: U X (AxFD) This assertion is obviously wrong because we have hypotheses X U, X UV, X Y = in (b) of theorem 4.1 4.3.2 A new proof for theorem 4.1 To simplify the proof of theorem 4.1, we use the system of three Armstrong's axioms A1, A2, A3 and the following two rules: - If X Y and U V then XU YV (Union Rule) - If X Y then X Z for all Z Y (Decomposition Rule) Proof (a) Since X U so X Y U Y Since X Y = so X Y = X U Y Therefore, we have: (U Y) X (A1) XY (Hypothesis) (U Y) Y (1, 2, A3) (U Y) (U Y) (A1) (U Y) UY (3, 4, Union) (U Y) U (5, Decomposition) U V (Hypothesis) 20 (6, 7, A3) (U Y) V (U Y) (V Y) (8, Decomposition) (a) XY (Hypothesis) (UY)(V Y) (Hypothesis) U X (A1, since X U) U Y (3, 1, A3) U VY (2, 4, Union) U V (5, Decomposition) (b) U V (Hypothesis) U (V Y) (1, Decomposition) (b) XY (Hypothesis) U (V Y) (Hypothesis) U U(V Y) (2, A2) U(V Y) (UV Y) (A1) since U (V Y) (U Y) (V Y) and (U Y) (V Y) = UV Y U (UV Y) (3, 4, A3) (UV Y) X (A1) since X UV and X Y = so X = (X Y) UV Y (UV Y) Y (6, 1, A3) U Y (5, 7, A3) U UVY (5, 8, A2) 10 U V (9, Decomposition) In the new proof of theorem 4.1, the proof of (a) is essentially the same as the original proof The difference is that the way to explain the reasoning steps Moreover, in the new proof, we use Armstrong's axioms, so the steps is simpler and clearer To overcome the error in proof of (b) of Theorem 4.1, our proof of (b) is completely new It makes theorem 4.1 (a very good theorem as the basis for the preprocessing transformation which effectively eliminates redundancies in a given set of FDs) be stable and usable In practice, in many cases, for simplicity, we can use the following substitution rule: If X U, X V and X Y = then {X Y, U V} {X Y, U (V Y)} (3) The experimental results on many sets including different quantities and sizes of FDs indicate the percentages of application of the substitution rules are very high and are increased substantially with the complexity of the sets of FDs - In 28,25% of FDs sets, it is not necessary to apply the transitivity rule and the preprocessing transformation eliminates redundancy efficiently - The size of FDs sets has been reduced by 52,89% 21 - When the number of attributes is increased, the number of cases in which it is not necessary to apply the transitivity rule (A3) is also increased This demonstrates that the substitution rules are specially adequate for dealing with large databases schemas - The percentages of application of the substitution rules are independent from the number of attributes and from the length of the FD 4.4 Summary of Chapter Redundancy makes it increase unnecessary sizes for storing data, which causes data inconsistencies and reduces the efficiency in the management and exploitation of database systems The preprocessing transformation for eliminating redundancy in sets of FDs is new and very effective The basis of this preprocessing transformation is the theorem 4.1 Unfortunately, the proof of (b) of theorem 4.1 is wrong and unacceptable In this chapter, we give a new proof for theorem 4.1 and a simple substitution rule which is easy to apply in practice It makes theorem 4.1 be stable and usable Creating new substitution rules for preprocessing sets of FDs is also an interesting research direction The main results of this chapter are published in the work [CT5] 22 CONCLUSIONS The thesis presents an overview of FD and RFD in the relational data model, researching into the closure computing algorithm of a set of attributes and the reduction for the key finding problem of a relation schema, researching into an efficient preprocessing transformation for sets of FDs, researching into AFDs and CFDs The results of the thesis are summarized as follows: - Some results about FDs, AFDs (showing the relationships between results of the two works, proving some lemmas, finding some new relationships between approximate measures which are usually used for discovering AFDs, proposing a new approach using matrix to discover FDs and AFDs) and some preliminary results about a unified hierarchy for FDs, CFDs and ARs (the appropriate adjustment for the expression which defines a fragment relation; the improvement for the pattern intersection operator; the proof of the relationship (X, r) = X F - Proposing an improved algorithm for computing the closure of a set of attributes under a set of FDs - For the reduction of the key finding (discovering) problem, based on the familiar semantics of FDs in the relational model, we improve a necessary condition (for which a subset of attributes is a key for a relational schema) and prove that the three necessary conditions are actually equivalent expressions - Showing a serious error in the proof of a theorem which is the basis of an efficient preprocessing transformation in order to eliminate redundancies in a set of FDs Moreover, we give a new, correct and simpler proof for that theorem, as well as a simple substitution rule which is easy to apply in practice 23 LIST OF WORKS HAS BEEN PUBLISHED [CT1] Vu Quoc Tuan, About some results related to approximate functional dependencies in relational databases, Proceedings of XVI National Conference: Some Selected Issues of Information Technology and Communication - Da Nang, 11-2013, pp 352354 [CT2] Vu Quoc Tuan, Vu Chinh Thuy, Matrix method for discovering functional dependencies in a database, Journal of Military Science and Technology, No.34, 12-2014, pp 73-80 [CT3] Ho Thuan, Vu Quoc Tuan, A new algorithm for computing the closure of a set of attributes under a set of functional dependencies, Journal of Military Science and Technology, No.45,10-2016, pp 109-117 [CT4] Vu Quoc Tuan, Ho Thuan, Some results for reducing the key finding problem, Journal of Military Science and Technology, No.47, 02-2017, pp 102-107 [CT5] Vu Quoc Tuan, Ho Thuan, About an efficient preprocessing transformation for functional dependencies sets, Journal of Military Science and Technology, No.50, 08-2017, pp.162-170 [CT6] Vu Quoc Tuan, Ho Thuan, Some results for the closure computing algorithm and reducing the key finding problem of a relation schema, Journal of Research and Development on Information and Communication Technology, Vol V-2, No.18 (38), 12-2017, pp 12-18 [CT7] Vu Quoc Tuan, Ho Thuan, Some results related to reducing the key finding problem of a relation schema, Journal of Military Science and Technology, No.54, 04-2018, pp 157-162 [CT8] Vu Quoc Tuan, Some results about relaxed functional dependencies, Journal of Cybernetics and Information Technologies, Bulgarian (Submitted) [CT9] Vu Quoc Tuan, Some improvements about a unified hierarchy for functional dependencies, conditional functional dependencies and association rules, Journal of Military Science and Technology, Vietnam, Special Issue - No.60A, 05-2019, pp 87-94 24 ... possible) containing all keys of S such that Z Ω If we find a such set Z then the finding keys in Z instead of finding ones in Ω will be simpler The necessary conditions (1), (2), (3) and (4) are... generating a search space of FDs, verifying the satisfaction of each FD, pruning the search space, outputting the set of satisfied FDs and reducing redundancies in this set of satisfied FDs In the... model, researching into the closure computing algorithm of a set of attributes and the reduction for the key finding problem of a relation schema, researching into an efficient preprocessing transformation