## Equational theories and database constraints (1985)

Venue: | |

Citations: | 5 - 0 self |

### BibTeX

@MISC{Cosmadakis85equationaltheories,

author = {Stavros Stylianos Cosmadakis},

title = {Equational theories and database constraints},

year = {1985}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

4092 |
Introduction to automata theory, languages, and computation. Menlo Park
- Hopcroft, Motwani, et al.
- 2000
(Show Context)
Citation Context ...ntents of STACK 1 (the top symbol is Xn); the string Ym'"Yl is the contents of STACK 2 (the top symbol is Ym)' The relation w=MW 2 (ID w yields 1D w2 via one step of M) is defined in the standard=-= way [50, 40]-=-. i is the reflexive, transitive closure of M' Let us now define a set S of word equations (over generators Qt_JFI) which capture the computation of M: 1. IfS(q,a)=(p,Pop 0, then aq=p is in S. If 8(q,... |

2541 |
The Design and Analysis of Computer Algorithms
- Aho, Hopcroft, et al.
- 1974
(Show Context)
Citation Context ... b to a in G I (cf. Rules 4,5). Thus, we can assign to each node u ofG I a number c(u) so that c(u)_c(a) (this can be done 55 by a topological sort of file dag of strongly connected components of G l =-=[2]-=-). Now consider a relation r consisting of tuplcs t k, k = 0,1,2 ..... where fir A m in % we have tk[A m] = k + (tam). Clearly r tisfies all u-llYs in Y. and violates AC_B. Moreover, r satisfies all F... |

1495 | The entity-relationship model - toward a uni view of data - Chen - 1976 |

509 |
Simple word problems in universal algebra
- Knuth, Bendix
- 1970
(Show Context)
Citation Context ...ached as implication problems involving FD's and IND's. Performance of Equational Theorem Provers An interesting practical question is how well theorem provers designed around the Knuth-Bendix method =-=[46]-=- perform on sets of equations obtained from database constraints. We have experimented with the w system [35, 49], which has been able to handle various non-trivial inferences of FD's and IND's. Howev... |

472 |
Optimal implementation of conjunctive queries in relational databases
- Chandra, Merlin
- 1977
(Show Context)
Citation Context ...ask associated with a class of dependencies. As a rule, algorithmic approaches to database schema design and query optimization are based on efficient solutions of the implication problem (see, e.g., =-=[12, 6, 3, 18, 62, 51]-=-). Evidenfiy, if we are concerned with applications then the finlie implication problem is the one which is most relevant. However, it tends to be much more difficult to deal with. Moreover, for the c... |

453 |
The Theory of Relational Databases
- Maier
- 1983
(Show Context)
Citation Context ...erful algebraic operators such as selection, projeclion and join) and an inlegrily part (constraints defining consistent database states, intended to capture the semantics of particular applications) =-=[62, 51]-=-. A relation is a table with columns named by attributes and with rows containing values from some domain, each row being a tuple. A database is a finite set of relations. A logical database or databa... |

288 |
Elements of the Theory of Computation
- Papadimitriou
- 1981
(Show Context)
Citation Context ... and flcrcfore I: does not imply a. Similarly ira is an IND. ! 2.4 Computations as Inferences It has been known, since at least Post's proof of the unsolvability of file word problem for Thue systems =-=[55, 50]-=-, that arbitrary computations can be simulated by inferences in semigroups. Using Corollary 2.3, we show that we can simulate computations by inferences of IND's and unary FD's. We thus obtain lower b... |

281 |
Principles of Database Systems
- Ullman
- 1983
(Show Context)
Citation Context ...area of database management. The model and its implementations have contributed significantly both to the increase of programmer productivity [23] and to file fundamental understanding of computation =-=[62]-=-. Among the advantages of the model, which account for its success, are [23]: 1. The sharp, clear boundary it provides between the conceptual and the physical aspects of database management. 2. Its si... |

266 | Extending the database relational model to capture more meaning
- Codd
- 1979
(Show Context)
Citation Context ...e relational model. This effort is motivated in large part by the success of the relational methodology and by the demands of specific application domains, in particular Office Automation (see, e.g., =-=[20, 24, 37, 42, 59, 61]-=-, which is by no means an exhaustive list). The approach generally taken is to appropriately enrich the integrity part by adding constraints which will enhance the expressive power of the model, while... |

206 |
Universality of data retrieval languages
- Aho, Ullman
- 1979
(Show Context)
Citation Context ...gebraic properties of ', the PI) C=AoB is equivalent to C=CoA-B A A-B=C-A,B, and therefore it is expressed by the set {CAB, AB--C}. However, because of Example 5.2 above it should come as no surprise =-=[4]-=- that the PD C=A+B cannot be expressed by any set of EID's: Theorem 5.1: Let %=ABC; the PD C--A+B cannot be expressed by any set of first-order sentences. Proof: I.et Y. be a set of first-order senten... |

192 | Database Description with SDM: A Semantic Database Model
- Hammer, Mcleod
- 1981
(Show Context)
Citation Context ...e relational model. This effort is motivated in large part by the success of the relational methodology and by the demands of specific application domains, in particular Office Automation (see, e.g., =-=[20, 24, 37, 42, 59, 61]-=-, which is by no means an exhaustive list). The approach generally taken is to appropriately enrich the integrity part by adding constraints which will enhance the expressive power of the model, while... |

187 |
On the structure of abstract algebras
- Birkhoff
- 1935
(Show Context)
Citation Context ...1=,2 deduce ,2 = 1, J?om 1 =s2 ands2 =s3 deduce 1 =s3, from *k = rf, k = 1 ..... m, deduce 0rl...rm: 0ri...r/n (ARrrY(O)=m), from r t: *2 deduce p(rl)= p(r2) (p is any substitution). Proposition 2.2: =-=[14, 41]-=- El=r=r'iff El'-r=r: ! Proofs in the above system can also be viewed as reduction sequences, as follows [41]: Whenever El=r=,', there is a sequence of terms r 0 ..... r m such that r 0 is r, r m is r'... |

174 |
On the desirability of acyclic database schemes
- Beeri, Fagin, et al.
- 1983
(Show Context)
Citation Context ...ise consislency, i.e. every pair of the database relations is required to have a universal relation. This condition is easy to test and maintain, as described in l0 numerous works on the subject (see =-=[8]-=- fi)r a review). In fact, if tile database scheme is acyclic [8] then pairwise consistency #lp/ies tile existence of a universal instance. Most of the theoretical work on dependencies is done in the c... |

122 |
A proof procedure for data dependencies
- Beeri, Vardi
- 1984
(Show Context)
Citation Context ...s as sentences in first-order logic, namely as Horn claua'es [34] (see Section 5.1 of this thesis for some examples). Closely related to this approach is a particular proof procedure, tile chase; see =-=[52, 11, 62, 51]-=- for its wide applicability (proof procedures for general dependencies also appear in [10, 68, 57]). It has been observed that tile chase is a special case of a classical theorem proving technique, na... |

118 |
Computational problems related to the design of normal form relational schemes
- BEERI, BERNSTEIN
- 1979
(Show Context)
Citation Context ...ask associated with a class of dependencies. As a rule, algorithmic approaches to database schema design and query optimization are based on efficient solutions of the implication problem (see, e.g., =-=[12, 6, 3, 18, 62, 51]-=-). Evidenfiy, if we are concerned with applications then the finlie implication problem is the one which is most relevant. However, it tends to be much more difficult to deal with. Moreover, for the c... |

116 | Testing implication of data dependencies
- Maier, Sagiv
- 1979
(Show Context)
Citation Context ...s as sentences in first-order logic, namely as Horn claua'es [34] (see Section 5.1 of this thesis for some examples). Closely related to this approach is a particular proof procedure, tile chase; see =-=[52, 11, 62, 51]-=- for its wide applicability (proof procedures for general dependencies also appear in [10, 68, 57]). It has been observed that tile chase is a special case of a classical theorem proving technique, na... |

115 |
Testing containment of conjunctive queries under functional and inclusion dependencies
- Johnson, Klug
- 1984
(Show Context)
Citation Context ...e relational model has been recognized as realistic and desirable (because of their conceptual simplicity and expressive power), they have become only recently the object of theoretical investigation =-=[16, 43, 54, 19, 58, 17, 44, 48, 26]-=-. General questions relating to the implicalion lroblem for IND's and [:l)'s have been studied in [16, 54, 19]. A rather surprising result [54, 19] is that the combination of IND's with F1)'s is as po... |

100 | Equations and rewrite rules: a survey
- Huet, Oppen
- 1980
(Show Context)
Citation Context ...rom equational logic. This is a fragment of first-order logic which has attracted a lot of attention, because of its relevance to areas such as applicative languages, interpreters and data types (see =-=[41]-=- for a survey). However, it does not seem to have been noticed by the database theory community, since a constant effort has been made to minimize the role of equality in dependencies (multivalued dep... |

92 |
Inclusion dependencies and their interaction with functional dependencies
- Casanova, Fagin, et al.
- 1984
(Show Context)
Citation Context ...e want to add to the semantics of our relations that a MANAGI!R entry in relation R must also appear as a MANAGER entry in relation S. This constraint is formally captured by the inclusion dependency =-=[16]-=- R:MANAGl!RCS:MANAGER. In general, an inclusion dependency (1ND) is a statement of the form R:A1...AmC_2_S:Bi...B m. Such a statement is satisfied by a database instance iff whenever a tuple with entr... |

72 |
Dependency Structures of Database Relationship
- Armstrong
- 1974
(Show Context)
Citation Context ... odd). It turns out that this is the only non-trivial interaction: by turning the above observation into a set of inference rules (one for each odd m) and including the usual inference rules for FlYs =-=[5]-=- and IND's [16], we obtain a complele axiomatization for FD's and unary IND's in the finite case (Theorem 4.1). The completeness proof is rather long and it involves an intricate construction of a fin... |

66 | Horn clauses and database dependencies
- Fagin
- 1982
(Show Context)
Citation Context ...with unrestricted implication. The problem of dependency implication can be approached in a very general setting by formulating dependencies as sentences in first-order logic, namely as Horn claua'es =-=[34]-=- (see Section 5.1 of this thesis for some examples). Closely related to this approach is a particular proof procedure, tile chase; see [52, 11, 62, 51] for its wide applicability (proof procedures for... |

60 |
Complexity of finitely presented algebras
- Kozen
- 1977
(Show Context)
Citation Context ...ch was derived from an efficient algorithm for congruence closure [31]. Also, the best algorithm for implication of FI)'s [6] can be seen directly (as we observe) as a special case of an algorithm of =-=[47]-=- for the generator problem in finilely l)resenled algebras. We use the methods of equational logic to formulate and study hnplication problems involving Vl)'s and lNI)'s. Wc also use equations to defi... |

58 |
Synthesizing third normal form relations from functional dependencies, Trans
- Bernstein
- 1976
(Show Context)
Citation Context ...ask associated with a class of dependencies. As a rule, algorithmic approaches to database schema design and query optimization are based on efficient solutions of the implication problem (see, e.g., =-=[12, 6, 3, 18, 62, 51]-=-). Evidenfiy, if we are concerned with applications then the finlie implication problem is the one which is most relevant. However, it tends to be much more difficult to deal with. Moreover, for the c... |

57 |
The power of natural semijoins
- Bernstein, Goodman
- 1981
(Show Context)
Citation Context ...ery can be processed with a minimum amount of data communication between sites. Some work has already been done on characterizing database schemes and queries for which such preprocessing is possible =-=[8, 13]-=-. An interesting research direction is to extend these results to allow for the presence of FD's (conceivably we will be able to preprocess more queries if the database is constrained to satisfy a set... |

55 |
The Implication Problem for Functional and Inclusion Dependencies is Undecidable
- Chandra, Vardi
- 1985
(Show Context)
Citation Context ...e relational model has been recognized as realistic and desirable (because of their conceptual simplicity and expressive power), they have become only recently the object of theoretical investigation =-=[16, 43, 54, 19, 58, 17, 44, 48, 26]-=-. General questions relating to the implicalion lroblem for IND's and [:l)'s have been studied in [16, 54, 19]. A rather surprising result [54, 19] is that the combination of IND's with F1)'s is as po... |

53 | The format model: a theory of database organization - Hull, Yap - 1984 |

51 |
Data Models
- Tsichritzis, Lochovsky
- 1982
(Show Context)
Citation Context ...e relational model. This effort is motivated in large part by the success of the relational methodology and by the demands of specific application domains, in particular Office Automation (see, e.g., =-=[20, 24, 37, 42, 59, 61]-=-, which is by no means an exhaustive list). The approach generally taken is to appropriately enrich the integrity part by adding constraints which will enhance the expressive power of the model, while... |

46 |
A sophisticate’s introduction to database normalization theory
- Beeri, Bernstein, et al.
- 1978
(Show Context)
Citation Context ...ntations have been proposed and analyzed, assuming that the only integrity constraints of the database schema are FD's. The general idea is that the database schema should be in a certain normal form =-=[22, 7, 62, 51]-=-, i.e. certain restrictive conditions should be satisfied by the FD's of the schema and their logical consequences. Given a universe % of attributes and a finite set Z of FD's, one can construct a dat... |

45 |
Equivalence of relational expressions
- Aho, Sagiv, et al.
- 1979
(Show Context)
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42 |
Database Abstractions: Aggregation
- Smith, Smith
- 1977
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38 |
Functional and inclusion dependencies: A graph theoretic approach
- Cosmadakis, Kanellakis
- 1986
(Show Context)
Citation Context ...e relational model has been recognized as realistic and desirable (because of their conceptual simplicity and expressive power), they have become only recently the object of theoretical investigation =-=[16, 43, 54, 19, 58, 17, 44, 48, 26]-=-. General questions relating to the implicalion lroblem for IND's and [:l)'s have been studied in [16, 54, 19]. A rather surprising result [54, 19] is that the combination of IND's with F1)'s is as po... |

37 | Towards a Sound View Integration Methodology - Casanova, Vidal - 1983 |

35 |
Formal systems for tuple and equality generating dependencies
- BEERI, VARDi
- 1984
(Show Context)
Citation Context ...some examples). Closely related to this approach is a particular proof procedure, tile chase; see [52, 11, 62, 51] for its wide applicability (proof procedures for general dependencies also appear in =-=[10, 68, 57]-=-). It has been observed that tile chase is a special case of a classical theorem proving technique, namely resolutiou [10, 11]. The chase provides straightforward algorithms for implication of classes... |

35 | Referential integrity - Date - 1986 |

34 |
The implication problem for functional and inclusion dependencies
- Mitchell
- 1983
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26 |
Testing satisfaction of functional dependencies
- Honeyman
- 1982
(Show Context)
Citation Context ...problem of testing consislemT J38, 64] of a database with a set of PD's. Using our polynomial-time algorithm for ilnplication, we show that it can be reduced to testing consistency with a set of FlYs =-=[38]-=-. It fi)llows that the problem can be solved in polynomial time (Theorem 5.4). 1.7 Credits The research reported in this thesis was done in close collaboration with Paris C. Kanellakis, and has been d... |

22 |
Free lattices
- Whitman
- 1941
(Show Context)
Citation Context ... 5. if ridP or r_(id q then ridp+ q(The intended meaning of id is that Pidq iff every lattice satisfies pq, no matter how the A's in % are interpreted). 69 The relation id iS reflexive and transitive =-=[28,65]-=-. Also, if pl_ Pt + P2--idql +q2 and plop2_idql'q2 . Now define =id as fi)llows: P=idq iffboth Pidq and q----idP. The relation = id is an equivaletce relation, and in particular it is a congruence: i.... |

16 |
Algebraic dependencies
- Yannakakis, Papadimitriou
- 1982
(Show Context)
Citation Context ...some examples). Closely related to this approach is a particular proof procedure, tile chase; see [52, 11, 62, 51] for its wide applicability (proof procedures for general dependencies also appear in =-=[10, 68, 57]-=-). It has been observed that tile chase is a special case of a classical theorem proving technique, namely resolutiou [10, 11]. The chase provides straightforward algorithms for implication of classes... |

15 |
Template dependencies: A large class of dependencies in relational databases and its complete axiomatization
- Sadri, Ullman
- 1982
(Show Context)
Citation Context ...some examples). Closely related to this approach is a particular proof procedure, tile chase; see [52, 11, 62, 51] for its wide applicability (proof procedures for general dependencies also appear in =-=[10, 68, 57]-=-). It has been observed that tile chase is a special case of a classical theorem proving technique, namely resolutiou [10, 11]. The chase provides straightforward algorithms for implication of classes... |

11 |
Partition semantics for relations
- Cosmadakis, Kanellakis, et al.
- 1999
(Show Context)
Citation Context ...this result appeared in [44]. Theorem 4.2 was obtained jointly, and appeared in [25]. Theorem 5.3 was obtained jointly, but the author of this thesis was the main contributor; this result appeared in =-=[27]-=-. Theorems 5.1, 5.2, 5.4 were obtained jointly, and appeared also in [27]. The extension to general dependencies outlined in the concluding chapter is due to file author of this thesis. 14 Chapter Two... |

11 |
Every finite lattice can be embedded into a finite partition lattice, Algebra Universalis 10
- Pudlák, T˚uma
- 1980
(Show Context)
Citation Context ... the uniform word problem for (finite) lattices (Lcmma 5.1). This follows from two deep results of lattice theory, namely that (finite) equivalence relations can represent arbitrary (finite) lattices =-=[66, 56]-=-. Using techniques from universal algebra [36, 47] and lattice theory [28], we show that these word problems are equivalent and they can be solved in polynomial time (Theorem 5.3). 13 Finally, we exam... |

11 |
Lattices, equivalence relations, and subgroups
- Whitman
- 1946
(Show Context)
Citation Context ... the uniform word problem for (finite) lattices (Lcmma 5.1). This follows from two deep results of lattice theory, namely that (finite) equivalence relations can represent arbitrary (finite) lattices =-=[66, 56]-=-. Using techniques from universal algebra [36, 47] and lattice theory [28], we show that these word problems are equivalent and they can be solved in polynomial time (Theorem 5.3). 13 Finally, we exam... |

11 | Design of relational views over network schemas
- Zaniolo
- 1979
(Show Context)
Citation Context ........ Brn appears in relation S. Inclusion dependencies make it possible to selectively define what data must be duplicated in what relations and thus they provide a valuable tool for database design =-=[24, 59, 69]-=-. The central notion of refi'renlial mtegriO' [24, 29] can be expressed using IND's. Together with FITs, INl)'s form the basis of the structural model of [67]. l)cscriptions of logical databases writt... |

9 | Unary inclusion dependencies have polynomial time inference problems - Kanellakis, Cosmadakis, et al. - 1983 |

7 |
Testing the universal instance assumption
- Honeyman, Ladner, et al.
- 1980
(Show Context)
Citation Context ...universal instance and the problem of adjusting the database relations to maintain the existence of a universal instance as the database is updated. Both of these problems are known to be NP-complete =-=[39]-=-. An alternative, weaker condition we may impose on a multi-relational database is pairwise consislency, i.e. every pair of the database relations is required to have a universal relation. This condit... |

7 |
Entity-Relationship Views over Uninterpreted Enterprise Schemas
- Klug
- 1980
(Show Context)
Citation Context ...)'s form the basis of the structural model of [67]. l)cscriptions of logical databases written in a variety of languages can be translated into a common language which uses relations, Fl)'s and IND's =-=[45]-=-. Inclusion dependencies have also been employed to map an entity-relationship schema to the relational model [20]. We mention in passing that IND's have been commonly known in Artificial Intelligence... |

6 |
Inclusion Dependencies and the Universal Instance
- Sciore
- 1983
(Show Context)
Citation Context |

5 | Reve: a term rewriting system generator with failureresistant Knuth-Bendix - Forgaard, Guttag - 1984 |

4 |
A Formal Treatment of Imperfect Information in Database Management
- Vassiliou
- 1980
(Show Context)
Citation Context ...on w over ql is a weak instance for diff every tuplc of relation R[U] of d appears in the projection of w on U. Weak instances have been proposed as a way to model incomplete information in databases =-=[38, 64]-=-. Given a database d and a set of FD's E, we can test ifd has a weak instance satisfying E in polynomial time [38]. We now show how this test can be generalized to arbitrary PD's. First, we replace E ... |

4 |
A Structural Model for Database Systems
- Wiederhold, El-Masri
- 1979
(Show Context)
Citation Context ...valuable tool for database design [24, 59, 69]. The central notion of refi'renlial mtegriO' [24, 29] can be expressed using IND's. Together with FITs, INl)'s form the basis of the structural model of =-=[67]-=-. l)cscriptions of logical databases written in a variety of languages can be translated into a common language which uses relations, Fl)'s and IND's [45]. Inclusion dependencies have also been employ... |