G. Grahne. "The Problem of Incomplete Information in Relational Databases". Springer-Verlag, Berlin, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... see [1, 4] Normalization for nested relations and objectoriented databases is studied in [23, 22, 27] Coding nested relations into flat ones, similar to our tree tuples, is done in [26, 29] We use FDs and relational algebra queries over incomplete relations using the techniques from [3, 7, 15, 19, 21]. XML constraints (mostly keys) have been studied in [5, 6, 12] these constraints do not use DTDs. XML constraints that takes DTDs into account are studied in [11] Finally, 2] considers normal forms for extended context free grammars similar to the Greibach normal form for CFGs; these, however, ....

....D, but: Proposition 1 [f t T(D) then treeD(t) D. We would like to describe XML trees in terms of the tuples they contain. For this, we need to select tupies containing the maximal amount of information. This is done via the usual notion of ordering on tu pies (and relations) with nulls, [7, 15, 16]. If we have two tree tuples tl,t2, we write tl U t2 if whenever tl.p is defined, then so is t2.p, and tl.p k implies tl.p = t2.p. As usual, tl Z t2 means tl U t2 and tl t2. Given two sets of true tuples, X and Y, we write X Z bYifVtlGX3t2 GYtl Zt2. Definition 6 (tuplesD) Given a DTD D and an ....

G. Grahne. The Problem of Incomplete Information in Relational Databases, Springer, Berlin, 1991.

....A G . Table 1 shows a sample c table with variables , and , where the particular form of the conditions is not detailed. When dependency constraints are concerned, an additional kind of conditions, called global conditions, needs to be introduced, as proposed in [4]. For simplicity, we do not consider dependency constraints here and can therefore ignore global conditions over conditional tables. We next define the result of evaluating c table # with respect to a valuation (mapping variables to actual values) denoted , as follows: A ....

....to each group. A sample representation for Table 3 is shown in Table 6. Again, the number of tuples in the resulting representation is bounded by the number of tuples in the original table. Table 6: Interval representation of 798; A (assuming ) A D [1,4] 1 [1,7] 3,12] 6 Conclusions In this paper, we have extended the theory of conditional tables with the definition of aggregate queries. This has been done carefully, so that we have been able to present a strong representation system with the desirable closure property that the ....

G. Grahne. The Problem of Incomplete Information in Relational Databases. Number 554 in Lecture Notes in Computer Science. Springer-Verlag, 1991.

....[22] is suitable for programming with approximations. In fact, a system based on this language [12] has been used in the problems of querying independent databases. 2 Preliminaries Partial objects and ordered sets Most models of partiality of data can be represented via orderings on values, e.g. [2, 33, 13, 9]. In [6, 19, 22] a general approach to the treatment of partial information in the context of ordered sets has been developed. Here we present the basics of that approach. First, elements of base types are ordered. For example, if there is only one null value , then the ordering is given by ....

G. Grahne, "The Problem of Incomplete Information in Relational Databases", Springer, Berlin, 1991.

....view instance I. In the example, the conditions simplify dramatically, but in general, this may result in rather gory tables. Note that, more generally, one could similarly introduce any total dependency [FV86, Var87] on the database by chasing [ASU79b, ASU79a] the conditional table as in [IJ84, Gra91] Observe also that from a practical viewpoint, this raises the issue of obtaining practical restrictions that prevent the conditions from becoming too complicated. 4 Open world assumption Figure 1 (a) gives an overview of the complexity of computing certain answers under OWA. Under OWA, the ....

G. Grahne. Problem of incomplete information in relational databases. Springer-Verlag, 1991.

....to the purely relational data model, we do not known of any way to autonomously compute the updates of the views with response to updates of exposed attributes in the base relations. 4 3 Conditional Tables We give a brief description of conditional tables largely based on the material from [13, 8, 1]. A condition is an expression built up by means of and from true, false, equality atoms of the forms x = y and x = c and inequality atoms of the forms x 6= y and x 6= c, where x; y are variables, and c is a constant. true, false, equality atoms and inequality atoms are called literals. The ....

.... represented; a system as such is called a strong representation system [13, 1] The relational operations on c tables are well de ned (see [13] page 780) for more rigorous de nitions) Note that global conditions play no role in the evaluation of queries against c tables; they remain invariant [8]. For join and intersection of c tables, one has to consider all pairs of c tuples from the two c tables to formalize the resulting local conditions. Selection is performed by adding new conjuncts to the local conditions. In Figure 2, examples of join and selection are illustrated. For projection, ....

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G. Grahne. The problem of incomplete information in relational databases. Springer-Verlag, 1991.

.... A family K of subsets of schema(R) is called a key set for a relation r over R, if 8t 1 ; t 2 2 r, 9K 2 K such that t 1 [K] and t 2 [K] are complete and t 1 [K] t 2 [K] Imielinski and Lipski [IMIE83] consider the more general framework of tuple and equality generating dependencies and Grahne [GRAH91] extends their work building on [IMIE84] Both Imielinski and Lipski, and Grahne deal mainly with marked nulls as opposed to the unmarked nulls 23 considered herein, and are interested in the problem of finding the most suitable representation for incomplete databases. The additivity problem does ....

G. Grahne, The Problem of Incomplete Information in Relational Databases, Lecture Notes in Computer Science, Volume 554. Berlin, Springer Verlag, 1991. 25

....handling partial information. This thesis can be subdivided into two. 1. In order to understand partial information in databases, we have to know exactly what it means. That is, we have to have a semantics for partial information. We develop a formalism, whose roots can be found in [Bis81, BJO91, Gra91, IL84, JLP92, Lib91, Vas79] and whose main idea is that partiality is represented via orderings on objects. 2. We are not interested in semantics per se; the semantics that we define will help us find the right programming constructs for query languages for partial information. Our approach is ....

....Sometimes it is helpful to restrict domains A to those in which every element x 2 A is bounded above by a maximal element xm x. The collection of maximal elements is denoted by A max , and the new semantic function then is [ x] max = x] A max . This semantic function was used in [AKG91, Gra91, IL84] Consistency in posets is another useful notion. Two elements x; y 2 A are called consistent if there exists z 2 A such that x; y z. In the case of records this means joinable as in [Zan84] i.e. they do not contradict each other) for example, Name: John; Dept: ni; Office: 176] and ....

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G. Grahne. The Problem of Incomplete Information in Relational Databases. Springer-Verlag, Berlin, 1991.

....out while he was visiting the Institut f ur Informationssysteme, TU Wien. 1 1 Introduction Disjunctive databases provide one of the major techniques for representing incomplete information a topic which has been receiving much attention in the areas of databases, logic programming and AI [14, 26, 11, 9]. Disjunctive databases can be regarded as sets of ground clauses 1 each of which contains at least one positive literal; so, for instance, incomplete information such as John su ers from disease d 1 or d 2 or d 3 or Nicky is either a man or a woman can be naturally encoded as follows: ....

....to each other by characterizing their expressiveness, i.e. the class of queries that they can express. These relationships are quite well understood [2, 7] The study of query languages for disjunctive databases is much less established. Work on querying incomplete databases is related; cf. [14, 1] for relations with null values, and [39] for the similar proviso of missing unique names axioms in logical databases. In these works, disjunctive information involving distinct relations as in (2) cannot be directly encoded. More importantly, these works consider only monotonic queries (in ....

[Article contains additional citation context not shown here]

G. Grahne. The Problem of Incomplete Information in Relational Databases, volume 554 of LNCS. Springer, 1991.

.... to representing partial information in databases that treats objects as elements of partially ordered sets, where the meaning of the order is being more informative , has proved to be very useful for analyzing incompleteness of information in the relational database model and its extensions [3, 5, 9, 12]. In particular, it has allowed a number of powerful tools from denotational semantics of programming languages to be used to analyze the phenomenon of incomplete information [1, 4, 7, 8] All these papers deal with set based databases. However, real database systems frequently use bags (also ....

....than C 2 if C 2 can be obtained from C 1 by a sequence of elementary updates that add information. This approach reduces the problem of choosing an ordering to the problem of formulating elementary updates. Such updates depend on certain assumptions on partial data. We consider two, following [5, 3]. The closed world assumption or CWA, says that only elementary updates that improve knowledge about objects in the database are allowed. That is, adding new objects is not allowed, unless they improve upon objects already in the database. In contrast, the open world assumption or OWA, allows both ....

G. Grahne. "The Problem of Incomplete Information in Relational Databases". Springer-Verlag, Berlin, 1991.

....[25] is suitable for programming with approximations. In fact, a system based on this language [15] has been used in the problems of querying independent databases. 2 Preliminaries 2. 1 Partial objects and ordered sets Most models of partiality of data can be represented via orderings on values [3, 16, 12]. In A general approach to the treatment of partial information in the context of ordered sets is developed in [6, 21, 25] Here we present the basics of that approach. First, elements of base types are ordered. For example, if there is only one null value , then the ordering is given by letting ....

G. Grahne, "The Problem of Incomplete Information in Relational Databases", Springer, Berlin, 1991.

....3: Relational operations on C tables R A B C 0 1 x y z 1 2 0 v R C 2 R 1 R A B C con 0 1 x x = 2 2 0 v v = 2 C=2R A B C con 0 1 x x = 2 2 0 v v = 2 The above de nitions of the relational operations generalize straightforward to the cases with more than two C tables. As shown in [18] (Theorem 5.1) the time taken to evaluate the operations on a set of C tables is bounded by a polynomial of the maximal size of the C tables. Given a set I c of C tables and a query q, we denote the result of the query on the set I c of C tables by q(I c ) It has been proven (as Theorem 7.1 in ....

....include dependencies, the additional information provided by the dependencies can be encoded in C tables as well. Given a C table T c , the completion of a set F of full dependencies on the C table, denoted compF (T c ) can be represented by another C table, using an extended chase procedure [18]. We omit the presentation of the detail of the procedure. It is worth mentioning that the completion of dependencies on a C table introduces only changes to the global condition of the C table. For example, consider a functional dependency AB C over relation R in Table 3. The completion ....

G. Grahne. The Problem of Incomplete Information in Relational Databases. Springer-Verlag, 1991.

....out while he was visiting the Institut fur Informationssysteme, TU Wien. 1 Introduction Disjunctive databases provide one of the major techniques for representing incomplete information a topic which has been receiving much attention in the areas of databases, logic programming and AI [14, 26, 11, 9]. Disjunctive databases can be regarded as sets of ground clauses 1 each of which contains at least one positive literal; so, for instance, incomplete information such as John suffers from disease d 1 or d 2 or d 3 or Nicky is either a man or a woman can be naturally encoded as follows: ....

....to each other by characterizing their expressiveness, i.e. the class of queries that they can express. These relationships are quite well understood [2, 7] The study of query languages for disjunctive databases is much less established. Work on querying incomplete databases is related; cf. [14, 1] for relations with null values, and [39] for the similar proviso of missing unique names axioms in logical databases. In these works, disjunctive information involving distinct relations as in (2) cannot be directly encoded. More importantly, these works consider only monotonic queries (in ....

[Article contains additional citation context not shown here]

G. Grahne. The Problem of Incomplete Information in Relational Databases, volume 554 of LNCS. Springer, 1991.

....constraint store which is used to represent all we know about these Skolem constants. Essentially this proposal is a combination of constraint databases (without indefinite information) as defined in (Kanellakis, Kuper, Revesz 1990) and the marked nulls proposal of (Imielinski Lipski 1984; Grahne 1991). Example 1 The following is an indefinite L LIN constraint database: f 1 2 ; 1 3 ; 3 4 g; f (8x=D) 8t 1 ; t 2 =Q) x = mary t 1 = 1 t 2 = 2 ) j walk(x; t 1 ; t 2 ) 8x=D) 8t 3 ; t 4 =Q) x = mary t 3 = 3 t 4 = 4 ) j paper(x; t 1 ; t 2 ) g ) This ....

.... constant in the database is logarithmic in the size of the database (Kanellakis, Kuper, Revesz 1990) Evaluating possibility queries over indefinite constraint databases can be NP hard even when we only have equality and inequality constraints between atomic values (Abiteboul, Kanellakis, Grahne 1991) (similarly evaluating certainty queries is co NP hard) It is therefore important to seek tractable instances of query evaluation. In this paper we start with the assumption that we have classes of constraints with some nice computational and closure properties. Under these assumptions, we show ....

Grahne, G. 1991. The Problem of Incomplete Information in Relational Databases, vol. 554 of LNCS. Springer Verlag.

....OE(t) with each tuple t of T . OE(t) is called a local condition. A c tuple is a tuple t of T together with local condition OE(t) denoted (t; OE(t) When dependency constraints are concerned, an additional kind of conditions, called global conditions, needs to be introduced, as proposed in [2]. For simplicity, we do not consider dependency constraints and therefore ignore the global conditions of conditional tables. Table 1. A conditional table (c table) A C con t c x = u s c y = u t c x = v s c y = v To interpret information in a c table is to map it to relation instances by ....

G. Grahne. The problem of incomplete information in relational databases. Springer-Verlag, 1991.

....all others put together. Rejecting a nice idea of Peter Buneman to call it the kitchen sink powerdomain, I shall use the salad powerdomain instead. Let me first motivate the use of the edible powerdomains. Since late 70s, many researchers tried to understand partial information in databases; [1, 14, 9, 23, 6, 5, 17] name just a few of the many books and papers in this area. In [5, 6] Buneman and others proposed to recast the main principles of relational databases in domain theory. The ordering on objects was interpreted as partiality, i.e. x y means that x is more partial than y, or y is more informative ....

G. Grahne, "The Problem of Incomplete Information in Relational Databases", SpringerVerlag, Berlin, 1991.

....an encoding of approximation constructs with or sets [14, 17] and explain how the language for or sets from [17] is suitable for programming with approximations. 2 Preliminaries Partial objects and ordered sets. Most models of partiality of data can be represented via orderings on values, e.g. [1, 13, 9]. In [4, 15, 17] a general approach to the treatment of partial information in the context of ordered sets has been developed. Here we present the basics of that approach. First, elements of base types are ordered. For example, if there are three kinds of nulls no information ni, existing ....

G. Grahne. "The Problem of Incomplete Information in Relational Databases". Springer, Berlin, 1991.

....(OE(t) T and v(t) 6= jg (1) We call v(Tc) the state of Tc with respect to v. For a set Ic of C tables, v(Ic) fv(Tc) j Tc 2 Icg. rep(Tc) denotes the collection of all possible states of Tc . We do not review the definitions of relational operators on C tables. Interested readers may consult [6, 4] or [13] Table 2 illustrates a C table representing a relational view A;C (T V 1 1 T V 2 ) where T V 1 and T V 2 are naive tables. Table 2. T V 1 A B t x s y T V 2 B C u z v r A;C (T V 1 1 T V 2 ) A C con t z x = u s z y = u t r x = v s r y = v Conditional table can be ....

....do with the definitions of the relational operators. The definitions are not affected by the introduction of j. The time space taken to evaluate the operations on a set of C tables is bounded by a Belief Revision Nonmonotonic Reasoning 34 H. Shu polynomial in the maximal size of the C tables ([4], Theorem 5.1) 4 The Basic Ideas In this work, we consider relational views defined in terms of relational operators projections, selections and joins involving only equalities. For simplicity, we consider only update operations that deal with single tuples in a relation. A binary CSP is a CSP ....

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G. Grahne, The Problem of Incomplete Information in Relational Databases, Springer-Verlag, 1991.

....incomplete database instances or simply incomplete databases. More formally, an incomplete database is of the form D[ S, where S ae CWA(D) There has been several different approaches to represent incomplete informations which includes null values[Gra77, Zan84] Codd tables, and conditional tables [Gra91, IL84a, IL84b] in Databases, use of disjunctions, classical negations and abductions in extensions of logic programming(For a survey on this see [BG94] As before we are interested in ancestor and non ancestor pairs. But this time with respect to the incomplete database S. The function Q and our earlier ....

G. Grahne. The problem of incomplete information in relational databases. Springer Verlag, 1991.

....FTP sites, and files accessible through the Internet. 1 What do we mean by incomplete information In this paper, we focus on incomplete but correct information about the state of the external world (see Section 2. 1 for a formal description) In contrast to work on relational database theory [23], we do not assume that all objects in the external world are known in advance; agents constantly encounter new objects. In addition, we do not assume that the world is static; agents constantly sense (or cause) changes to the world. In Section 5, we consider the implications of changes that the ....

G. Grahne. The problem of incomplete information in relational databases. In Lecture Notes in Computer Science, volume 554. Springer Verlag, New York, 1991.

....include dependencies, The additional information provided by the dependencies can be encoded in C tables as well. Given a C table T c , the completion of a set F of full dependencies on the Ctable, denoted compF (T c ) can be represented by another C table, using an extended chase procedure [4]. We omit the presentation of the detail of the procedure. It is worth mentioning that the completion of dependencies on a C table introduces only changes to the global condition of the C table. For example, consider a functional dependency AB C over relation R in Table 3. The completion ....

G. Grahne. The Problem of Incomplete Information in Relational Databases. Springer-Verlag, 1991.

....R, and develop a temporal logic based on events [118] Another: to generalize the notion of a temporal database. We describe the latter solution as it has been predominant in the database area. To deal with incomplete temporal information, several authors proposed to apply the classic framework of [57, 54] to timestamp databases. 57] modelled incomplete information in the context of the relational data model using marked nulls placeholders standing for 1 A rule stating that entities should not be multiplied needlessly, which is interpreted to mean that the simplest of two or more competing ....

....appear in different columns and different rows of a table. Moreover, every row has a (quantifier free) local condition associated with it that contains some nulls of this row. Finally, the entire table has a (quantifier free) global condition relating nulls in different rows (this was proposed in [54]) It should be clear now that there is a close correspondence between local conditions in tables and quantifier free timestamp formulas. Koubarakis [75, 77, 76, 78] pursued this correspondence in full generality in the context of the temporal domains Q and Z. In his approach, there may be one or ....

[Article contains additional citation context not shown here]

G. Grahne. The Problem of Incomplete Information in Relational Databases. SpringerVerlag, LNCS 554, 1991.

....database, especially if there are infinitely many of them. One rather wants a more abstract and compact representation. Several approaches for representing incomplete information in databases have been investigated in the literature. The most well known tool for this purpose are null values (e.g. [1,4,6,3]) In this paper a new technique for modelling indefinite information is investigated. It works with a data model based on algebraic datatypes, like the ones used in modern functional programming languages such as ML and Haskell. A (definite) value of an algebraic datatype is a tree and an ....

G. Grahne. The Problem of Incomplete Information in Relational Databases. Number 554 in LNCS. Springer Verlag, 1991.

....is indeed in the view instance I . In the example, the conditions simplify dramatically, but in general, this may result in rather gory tables. Note that, more generally, one could similarly introduce any total dependency [17, 39] on the database by chasing [4, 3] the conditional table as in [20, 18]. Observe also that from a practical viewpoint, this raises the issue of obtaining practical restrictions that prevent the conditions from becoming too complicated. 4 Open world assumption Figure 1 gives an overview of the complexity of computing certain answers under OWA. Under OWA, the problem ....

G. Grahne. Problem of incomplete information in relational databases. Springer-Verlag, 1991.

....which some form of generalized closed world reasoning is polynomial. The complexity of closed world reasoning in relational databases has been studied by many authors, like for example Vardi in [155] The problems addressed are outside the scope of this survey, and we refer to the book by Grahne [67] for a general overview. Several algorithms for inferencing under circumscription have been proposed: Przymusinski [122] Ginsberg [61] Inoue and Helft [73] Bossu and Siegel [19] Olivetti [110] Bell et al. 8, 107] Each of them presupposes languages of limited expressiveness. The algorithms ....

G. Grahne. The Problem of Incomplete Information in Relational Databases, volume 554 of Lecture Notes in Computer Science. Springer-Verlag, 1991.

....FTP sites, and files accessible through the Internet. 1 What do we mean by incomplete information In this paper, we focus on incomplete but correct information about the state of the external world (see Section 2. 1 for a formal description) In contrast to work on relational database theory [26], we do not assume that all objects in the external world are known in advance; agents constantly encounter new objects. In addition, we do not assume that the world is static; agents constantly sense (or cause) changes to the world. However, we do assume that agents are informed about changes to ....

G. Grahne. The problem of incomplete information in relational databases. In Lecture Notes in Computer Science, volume 554. Springer Verlag, New York, 1991.

.... to representation of partial information in databases that treats objects as elements of partially ordered sets, where the meaning of the order is being more informative, has proved to be very useful for analyzing incompleteness of information in the relational database model and its extensions [3, 5, 8, 11]. In particular, it has allowed a number of powerful tools from denotational semantics of programming languages to be used to analyze the phenomenon of incomplete information [1, 4, 6, 7] All of these papers deal with set based databases. However, real database systems frequently use multisets ....

....than C 2 if C 2 can be obtained from C 1 by a sequence of elementary updates that add information. This approach reduces the problem of choosing an ordering to the problem of formulating elementary updates. Such updates depend on certain assumptions on partial data. We consider two, following [5, 3]. The closed world assumption or CWA, says that the only elementary updates allowed are those that improve our knowledge about objects in the database. That is, adding new objects is not allowed, unless they improve upon objects already in the database. In contrast, the open world assumption or ....

G. Grahne. "The Problem of Incomplete Information in Relational Databases". Springer-Verlag, Berlin, 1991.

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Grahne G. The Problem of Incomplete Information in Relational Databases. Lecture Notes in Computer Science, vol. 554. Springer-Verlag, Berlin 1991.

....to Phi is NP complete. Lecture Notes in Computer Science c flSpringer Verlag 15 5 Related Work Many works deal with finite representations of infinite sets of databases, both in the constraint database literature [KKR90,vdM93] and in the literature on uncertain and indefinite databases [AKG91,Gra91,IL84,Men84]. We will not go into details; for surveys of these fields see [Kan95] and [vdM98] The work of Motro [Mot97] is close in spirit and motivation to ours; however, Motro assumes the existence of a real world global database instead of modeling the uncertainty in the sources by associating a set of ....

Grahne G. The Problem of Incomplete Information in Relational Databases. Lecture Notes in Computer Science, vol. 554. Springer-Verlag, Berlin 1991.

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G. Grahne. "The Problem of Incomplete Information in Relational Databases". Springer-Verlag, Berlin, 1991.

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G. Grahne, The Problem of Incomplete Information in Relational Databases, Springer-Verlag, Berlin, 1991.

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