## Horn clauses and database dependencies (1982)

### Cached

### Download Links

- [www.almaden.ibm.com]
- [researcher.watson.ibm.com]
- DBLP

### Other Repositories/Bibliography

Venue: | Journal of the ACM |

Citations: | 60 - 6 self |

### BibTeX

@ARTICLE{Fagin82hornclauses,

author = {Ronald Fagin},

title = {Horn clauses and database dependencies},

journal = {Journal of the ACM},

year = {1982},

volume = {29},

pages = {952--985}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Certain first-order sentences, called "dependencies, " about relations in a database are defined and studied. These dependencies seem to include all prewously defined dependencies as special cases A new concept is mtroduced, called "faithfulness (with respect to direct product), " which enables powerful results to be proved about the existence of "Armstrong relations " in the presence of these new dependencies. (An Armstrong relaUon is a relation that obeys precisely those dependencies that are the logical consequences of a given set of dependencies.) Results are also obtained about characterizing the class of projections of those relations that obey a given set of dependencies.

### Citations

598 |
A Mathematical Introduction to Logic
- Enderton
- 2002
(Show Context)
Citation Context ... of an atomic formula. Formulas (which can involve Boolean connectives and quantifiers) and sentences (formulas with no free variables) are defined as usual (see any standard textbook in logic, e.g., =-=[20]-=- or [49].) We sometimes abbreviate Yxl ... Vxn~, where each x, is universally quantified, by (Vxa ... x~)4,. Similarly, we sometimes abbreviate 3yl ... 3yrCk, where each y~ is existentially quantified... |

451 |
Model Theory
- Chang, Keisler
- 1990
(Show Context)
Citation Context ...tradiction. [] We close this section with some comments relating upward and downward faithfulness to concepts defined earlier in the literature. The following two definitions are standard (see, e.g., =-=[13]-=-). A sentence o is preserved under direct products if whenever (R,:i E I) is a family of relations such that o holds for every R,, then o holds for ®(R,:i E I). A sentence o is preserved under direct ... |

157 | E.F.: Relational Completeness of Data Base Sublanguages
- Codd
- 1972
(Show Context)
Citation Context ... not re.cursive [19, 55]. For this reason, various authors haves956 RONALD FAGIN defined syntactically defined subclasses of domain-independent formulas. These include Codd's range-separable formulas =-=[15]-=-, Nicolas' range-restricted formulas [40], Cooper's permissible formulas [16], UUman's safe formulas [54], and Demolombe's evaluable formulas [17]. Since each of these classes is syntactically defined... |

116 |
A Proof Procedure for Data Dependencies
- Beeri, Vardi
- 1984
(Show Context)
Citation Context ... A entry for the ith tuple of T, and if A is an attribute not in V, then the A entry for the ith row of the tableau is a new, distinct variable. Now apply the chase procedure (using ~) to the tableau =-=[8, 37, 46]-=- (actually, we are doing a slight generalization of the chase, since ~ may be infinite). The important point is that the chase procedure terminates with a finite tableau, since the dependencies in ~ a... |

113 |
Logic for Improving Integrity Checking in Relational Data Bases
- Nicolas
- 1982
(Show Context)
Citation Context ..., various authors haves956 RONALD FAGIN defined syntactically defined subclasses of domain-independent formulas. These include Codd's range-separable formulas [15], Nicolas' range-restricted formulas =-=[40]-=-, Cooper's permissible formulas [16], UUman's safe formulas [54], and Demolombe's evaluable formulas [17]. Since each of these classes is syntactically defined, each is recursive, unlike the full clas... |

110 | Y.: Testing Implications of Data Dependencies
- Maier, Mendelzon, et al.
- 1979
(Show Context)
Citation Context ... A entry for the ith tuple of T, and if A is an attribute not in V, then the A entry for the ith row of the tableau is a new, distinct variable. Now apply the chase procedure (using ~) to the tableau =-=[8, 37, 46]-=- (actually, we are doing a slight generalization of the chase, since ~ may be infinite). The important point is that the chase procedure terminates with a finite tableau, since the dependencies in ~ a... |

106 |
Computational problems related to the design of normal form relational schemas
- Beeri, Bernstein
- 1979
(Show Context)
Citation Context ...he relation consisting of each pair of tuples from T obeys Z'. We begin with the first two tuples of Figure 11. It is easy to verify (by using, say, the FD membership algorithm of Beeri and Bernstein =-=[5]-=-) that if Y C {A, B, C, D} and AD ----> Y is an FD in Y/, then AD ---> Y is a trivial FD, that is, that Y _c (,4, D}. Since the first two tuples in Figure 11 agree precisely on AD, it follows that the... |

85 |
Inclusion dependencies and their interaction with functional dependencies
- Casanova, Fagin, et al.
- 1982
(Show Context)
Citation Context ... in which the assumption that the sentences are typed and unirelational is weakened. However, faithfulness is maintained. Our enlarged class of sentences includes the important inclusion dependencies =-=[12, 24]-=-, which can say, for example, that every manager is an employee. The models of interest are no longer simply relations, but instead databases, consisting of a number of relations. It turns out that Ar... |

76 |
On sentences which are true of direct unions of algebras
- Horn
- 1956
(Show Context)
Citation Context ...sion. Following Slagle and Koniver [51], let us cam an unquantified formula of the form ((a, A ... A a,) =* B), 0.6) where n _> 1 (i.e., where there is at least one A,), an implication. A Horn clause =-=[32]-=- is the disjunction of atomic and negation-atomic formulas, where at most one is atomic. The implication (1.6) is equivalent to the Horn clause ('~AI V "" V ~A. V B), which has exactly one atomic form... |

66 | The theory of joins in relational databases - Aho, Beeri, et al. - 1979 |

50 |
The implication problem for data dependencies
- Beeri, Vardi
- 1981
(Show Context)
Citation Context ...out to be identical [59]. This is evidence for the naturalnesssHorn Clauses and Database Dependencies 955 of the class. Yannakakis and Papadimitriou present a complete axiomatization. Beeri and Vardi =-=[10]-=- have defined tuple-generating dependencies and equality-generating dependencies, which, when they are restricted to be typed, together comprise our embedded implicational dependencies. (Beeri and Var... |

42 | On the structure of Armstrong relations for functional dependencies
- Beeri, Dowd, et al.
- 1984
(Show Context)
Citation Context ...and Melkanoff's approach is a partial solution, in the spirit of Query-by-Example [60], to the problem of helping a designer think of what dependencies should be included. Unfortunately, it turns out =-=[6]-=- that the time complexity of fmding an Armstrong relation, given a set of functional dependencies, is precisely exponential in the number of attributes. That is, there exists an exponential-time algor... |

27 |
Correction to ”An equivalence between relational database dependencies and a fragment of propositional logic
- Sagiv, Delobel, et al.
- 1987
(Show Context)
Citation Context ...nempty in the definition of upward faithful, but the assumption is important to us in the downward faithful case. We return to this point at the end of this section. Example 2.2. The "degenerate MVD" =-=[3, 48]-=-, (Vxyly2zlz2)((exylzl A exy2z2) =~ ((yl = yz) V (zl -- z2))) (2. I) is not upward faithful (although it is downward faithful, by Theorem 2.5 below). Thus, relations R1 and R2 in Figure 5 (where bl # ... |

21 | Armstrong databases
- Fagin
- 1982
(Show Context)
Citation Context ...cedures for constructing relations tend to produce an infinite relation. (Grant and Jacobs [29] describe such a chase-type procedure, which they describe in terms of deductions. For a discussion, see =-=[25]-=-.) 5. More Armstrong Relation Counterexamples In this section we present three amusing counterexamples about Armstrong relations. In each case we exhibit a set ~ of sentences and single sentences ol a... |

15 |
Template dependencies: A large class of dependencies in relational databases and its complete axiomatization
- Sadri, Ullman
- 1982
(Show Context)
Citation Context ...nd MVDs; Ginsburg and Zaiddan's implied dependencies [28], which generalize FDs; Sagiv and Walecka's subset dependencies [47], which generalize embedded MVDs; Sadri and Ullman's template dependencies =-=[46]-=-, which generalize embedded join dependencies; and Parker and Parsaye-Ghomi's generalized transitive dependencies [43], which generalize transitive dependencies. We remark that the last three kinds of... |

13 | The inference problem for template dependencies
- Gurevich, Lewis
- 1982
(Show Context)
Citation Context ... problem for EMVDs is decidable, that is, whether, given a set ~ of EMVDs and a single EMVD o, it is the case that ~ logically implies o [46]. (However, we note that Vardi [57] and Gurevich and Lewis =-=[31]-=- have proven the undecidability of the decision problem for the more general class of template dependencies.) The existence of an Armstrong relation in the presence of EMVDs (which we shall prove) is ... |

11 |
A Method for Helping Discover the Dependencies of a Relation
- Silva, Melkanoff
- 1979
(Show Context)
Citation Context ...wo paragraphs why we asked the reader to discover for himseff that Gauss is the manager of two departments. We note an interesting "practical" application for Armstrong relations. Silva and Melkanoff =-=[50]-=- have developed a database design aid in which the database designer inputs a set of FDs and MVDs. The design aid then presents him with art Armstrong relation, that is, a "sample relation" that obeys... |

10 |
Armstrong databases for functional and inclusion dependencies
- Fagin, Vardi
- 1983
(Show Context)
Citation Context ...recisely those unirelational XEIDs involving R that are logical consequences of ZR. This follows from Theorem 7.2, just as Corollary 3.3 followed from Theorem 2.1. []s982 RONALD FAGIN Fagin and Vardi =-=[27]-=- show that if we restrict our attention to INDs and FDs, then there are not necessarily Armstrong databases. However, they show that if we restrict our attention to INDs and to "standard" FDs (FDs for... |

9 |
The decision problem for database dependencies
- Vardi
- 1981
(Show Context)
Citation Context ... of domain-independent formulas is equivalent to Kuhns' [36] class of definite formulas. Unfortunately, the class of domain-independent formulas (and of domainindependent sentences) is not re.cursive =-=[19, 55]-=-. For this reason, various authors haves956 RONALD FAGIN defined syntactically defined subclasses of domain-independent formulas. These include Codd's range-separable formulas [15], Nicolas' range-res... |

7 |
Decompositions and functional dependencies in relations
- Armstrong, Delobel
- 1980
(Show Context)
Citation Context ...nempty in the definition of upward faithful, but the assumption is important to us in the downward faithful case. We return to this point at the end of this section. Example 2.2. The "degenerate MVD" =-=[3, 48]-=-, (Vxyly2zlz2)((exylzl A exy2z2) =~ ((yl = yz) V (zl -- z2))) (2. I) is not upward faithful (although it is downward faithful, by Theorem 2.5 below). Thus, relations R1 and R2 in Figure 5 (where bl # ... |

5 |
Order Logic Formalization for Functional, Multivalued and Mutual Dependencies
- Nicolas, First
- 1978
(Show Context)
Citation Context ...umns of the same relation have the same name.) This FD holds for a relation R if every pair of tuples of R that agree in each of the columns A1 ..... A,, also agree in the B column. It is easy to see =-=[39]-=- that FDs can be represented as sentences in first-order logic. Assume, for example, that we are dealing with a 4-ary relation, where the first, second, third, and fourth columns are called, respectiv... |

2 |
Determining Correctness by Testing
- Brooks
- 1980
(Show Context)
Citation Context ...lly and that it is often fairly easy to verify in practice whether or not a given sentence is faithful with respect to direct product. Theorem 3.1 might well be useful in a number of contexts. Brooks =-=[11]-=- has noted an application of Theorem 3.1 in which a "model" is a set of test data about a computer program and a "sentence" is a characterization of the computations donesHorn Clauses and Database Dep... |

2 |
Generalized mutual dependencies and the decomposmon of database relations
- MENDELZON, MAmR
- 1979
(Show Context)
Citation Context ...ations of these dependencies have appeared: Nicolas' mutual dependencies [39], which say that a relation is the join of three of its projections; Mendelzon and Maier's generalized mutual dependencies =-=[38]-=-; Rissanen's [44] and Aho et al.'s [1] join dependencies, which generalize further to an arbitrary number of projections; Paradaens" transitive dependencies [41], which generalize both FDs and MVDs; G... |

2 |
Decomposition of relations: A comprehensive approach
- PARADAENS, JANNSENS
- 1978
(Show Context)
Citation Context ...to be typed, together comprise our embedded implicational dependencies. (Beeri and Vardi have defined both typed and untyped versions; the typed version they call many-sorted.) Paradaens and Janssens =-=[42]-=- have defined general dependencies, which are implicational (but not embedded implicational) dependencies. Also, Grant and Jacobs [29] have defined generalized dependency constraints, which are untype... |

2 |
Finding resolution proofs using duplicate goals m AND/OR trees
- SLAGLe, KONIVR
- 1971
(Show Context)
Citation Context ... the arity of the relations is not, whereas in the notation of (1.1), the reverse is true. We hope that this double manner of thinking about FDs does not cause confusion. Following Slagle and Koniver =-=[51]-=-, let us cam an unquantified formula of the form ((a, A ... A a,) =* B), 0.6) where n _> 1 (i.e., where there is at least one A,), an implication. A Horn clause [32] is the disjunction of atomic and n... |

1 |
Dependency structures of database relationships
- ARMSXRONG
- 1974
(Show Context)
Citation Context .... As958 RONALD FAGIN FIGURE 4 dora (A) = {0,1} dora (A) = {0,1,2} domain mapping is a mapping that associates to each attribute A in U a set dom(A), called the domain of A. In the spirit of Armstrong =-=[2]-=- and of Aho et al. [1], we define a tuple to be a function that maps each attribute A into a member of dom(A). We call the value associated with the attribute A the A entry of the tuple. If the attrib... |

1 |
A complete ax~omatization for functional and multivalued dependencies in database relaUons
- BEERI, FAOIN, et al.
- 1977
(Show Context)
Citation Context ...ppens to hold in some snapshot of the database but may fail in other snapshots. An Armstrong relation is precisely one that obeys every specified dependency and no accidental dependency. Beeri et al. =-=[7]-=- generalized Armstrong's result to allow not just FDs but also MVDs. That is, they showed that if ~ is a set of FDs and MVDs and X* is the set of all FDs and MVDs that are logical consequences of X, t... |

1 |
Further normahzation of the database relational model
- CODD
- 1971
(Show Context)
Citation Context ...nterest for relational databases. For historical reasons, such sentences are usually called dependencies. The first dependency introduced and studied was the functional dependency, or FD, due to Codd =-=[14]-=-. As an example, consider the relation in Figure 1, with three columns: EMP (which represents employees), DEPT (which represents departments), and MGR (which represents managers). The relation in Figu... |

1 |
On the expressive power of query languages
- COOPER
- 1980
(Show Context)
Citation Context ...GIN defined syntactically defined subclasses of domain-independent formulas. These include Codd's range-separable formulas [15], Nicolas' range-restricted formulas [40], Cooper's permissible formulas =-=[16]-=-, UUman's safe formulas [54], and Demolombe's evaluable formulas [17]. Since each of these classes is syntactically defined, each is recursive, unlike the full class of domain-independent formulas. Ea... |

1 |
A syntactical characterization of a subset of definite and safe formulas
- DeMoLOe
- 1981
(Show Context)
Citation Context ...rmulas. These include Codd's range-separable formulas [15], Nicolas' range-restricted formulas [40], Cooper's permissible formulas [16], UUman's safe formulas [54], and Demolombe's evaluable formulas =-=[17]-=-. Since each of these classes is syntactically defined, each is recursive, unlike the full class of domain-independent formulas. Each of these classes is a proper subset of the class of domain-indepen... |

1 |
On the characterization of "valid" formulas for database querying
- DEMOLOMBE, NICOLAS
- 1981
(Show Context)
Citation Context ...hese classes is a proper subset of the class of domain-independent formulas, since each of these classes is recursive, while the class of all domain-independent formulas is not. Demolombe and Nicolas =-=[18]-=- show that each of Ullman's safe formulas is a domainindependent formula, and, conversely, that for each domain-independent formula there is an equivalent safe formula. In line with dependency traditi... |

1 |
The recursive unsolvabllity of the decision problem for the class of definite formulas
- PAOLi
- 1969
(Show Context)
Citation Context ... of domain-independent formulas is equivalent to Kuhns' [36] class of definite formulas. Unfortunately, the class of domain-independent formulas (and of domainindependent sentences) is not re.cursive =-=[19, 55]-=-. For this reason, various authors haves956 RONALD FAGIN defined syntactically defined subclasses of domain-independent formulas. These include Codd's range-separable formulas [15], Nicolas' range-res... |

1 |
FuncUonal dependencies in a relational database and propositional logic. IBM
- FAOIN
- 1977
(Show Context)
Citation Context ...mstrong did not explicitly state or prove the existence of what we call an Armstrong relation. Instead, he proved a result that implies both the completeness of a certain set of axioms about FDs (see =-=[22]-=-) and the existence of an Armstrong relation. Let us consider an example. Let Y~ be the set (EMP --, DEPT, DEPT --* MGR), containing two FDs. Then Z* contains the FDs in Z, along with, for example, th... |

1 |
Tools for template dependencles
- FAGIN, MAIER, et al.
(Show Context)
Citation Context ...ula on the right-hand side of the implication and in which this atomic formula is a relational formula (i.e., s -- 1 in (1.7), and also B, in (1.7) is a relational formula). We note that Fagin et al. =-=[26]-=- develop a number of techniques, counterexamples, and results about TDs. In Section 2 we introduce the concept of "faithfulness" (with respect to direct product) and show that IDs and EIDs are faithfu... |

1 |
Properties of functional-dependency families
- GINSaURO, ZIDDAN
- 1982
(Show Context)
Citation Context ... dependencies, which generalize further to an arbitrary number of projections; Paradaens" transitive dependencies [41], which generalize both FDs and MVDs; Ginsburg and Zaiddan's implied dependencies =-=[28]-=-, which generalize FDs; Sagiv and Walecka's subset dependencies [47], which generalize embedded MVDs; Sadri and Ullman's template dependencies [46], which generalize embedded join dependencies; and Pa... |

1 | On the family of generalized dependency constraints - GRAWr, JACOes - 1982 |

1 |
Implicauonal dependency and finite specdication
- HULL
- 1981
(Show Context)
Citation Context ...considering FD classes and their projections one is "forced" into considering more general dependencies, such as IDs. Apparently, Sadri [45] obtained Theorems 6.1-6.3 independently. We note that Hull =-=[33]-=- showed that the join of ID classes is an ID class. Thus the collection of ID classes is closed under projection and join. Hull [33] has given an example of an FD class with a projection 9"such that 9... |

1 |
Some applications of infinitely long formulas
- KEtslmR
- 1965
(Show Context)
Citation Context ...from what we just showed that 0 is not preserved under direct factors. However, 0 is downward faithful. So being preserved under direct factors and being downward faithful are not equivalent. Keisler =-=[35]-=- gave a complicated characterization of sentences that are preserved under direct factors. His class and our class of EIDs have in common only tautologies.sHorn Clauses and Database Dependencies 967 E... |

1 |
Answering questions by computer: A logical study
- KUHS
- 1967
(Show Context)
Citation Context ...this latter property domain independence. It is possible to define domain independence not only for sentences, but also for formulas. This class of domain-independent formulas is equivalent to Kuhns' =-=[36]-=- class of definite formulas. Unfortunately, the class of domain-independent formulas (and of domainindependent sentences) is not re.cursive [19, 55]. For this reason, various authors haves956 RONALD F... |

1 |
Transitive dependencies in a database scheme
- PAtADAENS
- 1979
(Show Context)
Citation Context ...aier's generalized mutual dependencies [38]; Rissanen's [44] and Aho et al.'s [1] join dependencies, which generalize further to an arbitrary number of projections; Paradaens" transitive dependencies =-=[41]-=-, which generalize both FDs and MVDs; Ginsburg and Zaiddan's implied dependencies [28], which generalize FDs; Sagiv and Walecka's subset dependencies [47], which generalize embedded MVDs; Sadri and Ul... |

1 |
Inference involving embedded multtvalued dependencies and transitive dependencies
- unknown authors
- 1980
(Show Context)
Citation Context ...es [47], which generalize embedded MVDs; Sadri and Ullman's template dependencies [46], which generalize embedded join dependencies; and Parker and Parsaye-Ghomi's generalized transitive dependencies =-=[43]-=-, which generalize transitive dependencies. We remark that the last three kinds of dependencies mentioned were introduced to deal with the issue of a complete axiomatization: subset dependencies were ... |

1 |
Theory of relations for databases---A tutorial survey
- Risst, J
(Show Context)
Citation Context ...ependencies have appeared: Nicolas' mutual dependencies [39], which say that a relation is the join of three of its projections; Mendelzon and Maier's generalized mutual dependencies [38]; Rissanen's =-=[44]-=- and Aho et al.'s [1] join dependencies, which generalize further to an arbitrary number of projections; Paradaens" transitive dependencies [41], which generalize both FDs and MVDs; Ginsburg and Zaidd... |

1 |
Subset dependencies and a completeness result for a subclass of embedded multivalued dependencies
- SolV, WALECr
- 1982
(Show Context)
Citation Context ...ojections; Paradaens" transitive dependencies [41], which generalize both FDs and MVDs; Ginsburg and Zaiddan's implied dependencies [28], which generalize FDs; Sagiv and Walecka's subset dependencies =-=[47]-=-, which generalize embedded MVDs; Sadri and Ullman's template dependencies [46], which generalize embedded join dependencies; and Parker and Parsaye-Ghomi's generalized transitive dependencies [43], w... |

1 |
Mathematical Logic
- SHOENVlELD
- 1967
(Show Context)
Citation Context ...tomic formula. Formulas (which can involve Boolean connectives and quantifiers) and sentences (formulas with no free variables) are defined as usual (see any standard textbook in logic, e.g., [20] or =-=[49]-=-.) We sometimes abbreviate Yxl ... Vxn~, where each x, is universally quantified, by (Vxa ... x~)4,. Similarly, we sometimes abbreviate 3yl ... 3yrCk, where each y~ is existentially quantified, by (3y... |

1 |
Private commumcation
- STATMAN
(Show Context)
Citation Context ...B. Then ~ ~ (ol V 02); for, if S is a nonempty relation, then it obeys m, and if S is an empty relation, then it obeys 02. However, Z ~ 0.1 and ~ ~ 0.2. [] Example 5.3. This example is due to Statman =-=[52]-=-. We show that if we deal with sentences that look like IDs, except that they are not typed, then there is not necessarily an Armstrong relation. Let 00, Ol, and o2 be the following three sentences re... |

1 |
Contributions to the theory of Models I. Neded. Akad. Wetensch
- TARSKI
- 1980
(Show Context)
Citation Context ... given relation by considering only the tuples, and not the underlying domains (this property we have called domain independence). Our definition of "relation" is in the spirit of Tarski's definition =-=[53]-=- of "model," in that domains are explictly considered. Our definition is analogous to considering a graph as a set of nodes, along with a set of edges, whereas the usual definition (of a relation as s... |

1 |
Private commumcaUon 57 VARDI, M.Y. The tmplication and t'mite unplicatlon problems for typed template dependencies
- VAOI
- 1982
(Show Context)
Citation Context ... 4.2 by using only TDs. In particular, they show that there is a set Z of two TDs such that there iss974 RONALD FAGIN no finite relation that obeys precisely those TDs o for which ~ ~, o. Also, Vardi =-=[56]-=- has shown that there is an EID ~- such that the set of EIDs o where 9 ~r,n o is not recursive. This result implies that there is no finite Armstrong relation for % since we could test whether or not ... |

1 |
Private communication
- YANNAKArdS
(Show Context)
Citation Context ... an q + 2)-tuple path from a to b, then there is an i-tuple path from a to b." For example, ~'3 is (VabXlX2X3X4)((Paxl A PX2Xl A Px~,x3 A Px4x3 A Px4b) =* (3yayz)(eaya A Pyzy~ A P.,v,zb)). Yannakakis =-=[58]-=- introduced the EIDs ~', to show that there are an infinite number of nonequivalent EIDs. We now make use of these EIDs to show that Theorem 4. I is false if the assumption that 5ais f'mite is dropped... |

1 |
Algebraic dependencies. Proe. 21st
- XrOS, PAeADIMITRXOU
- 1980
(Show Context)
Citation Context ...ependency of one of the types listed above is equivalent to a i'mite set of our implicational (or embedded implicational) dependencies, which we define soon. We note that Yannakakis and Papadimitriou =-=[59]-=- have, independently of the author, dei'med "algebraic dependencies," which, on the surface, look very different from our embedded implicational dependencies. Somewhat surprisingly, the class of algeb... |

1 |
pp 431--438
- Proc
- 1975
(Show Context)
Citation Context ...he Armstrong relation and thinking about what it said, he simply noticed that the FD MGR ---, DEPT failed. Thus Silva and Melkanoff's approach is a partial solution, in the spirit of Query-by-Example =-=[60]-=-, to the problem of helping a designer think of what dependencies should be included. Unfortunately, it turns out [6] that the time complexity of fmding an Armstrong relation, given a set of functiona... |