A. Van Gelder. The well-founded semantics of aggregation. In PODS'92, pages 127--138.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....In an early paper [17] it was shown how to extend both relational algebra and calculus with aggregate constructs, but the resulting language did not correspond naturally to any reasonable logic. It is known how to integrate aggregation into datalog like languages (both recursive and nonrecursive) [30, 33], and various aspects of such aggregate languages were studied (e.g. query optimization [26] and handling constraints involving aggregation [31] At this point, let us give an example of a typical aggregate query that would be supported by all commercial versions of SQL, and use it to explain ....

A. Van Gelder. The well-founded semantics of aggregation. In PODS'92, pages 127--138.

....quite efficiently, as we have shown in our recent work [21] We show that partial order clauses help render clear and concise formulations to problems involving aggregate operations and recursion in database querying. This has been a topic of considerable interest in the literature recently [15, 26, 27, 29]. An aggregate operation is a function that maps a set to some value, e.g. the maximum or minimum in the set, the cardinality of this set, the summation of all its members, etc. In considering the problems with various semantic approaches, Van Gelder [29] notes that, for many problems in which ....

....in the literature recently [15, 26, 27, 29] An aggregate operation is a function that maps a set to some value, e.g. the maximum or minimum in the set, the cardinality of this set, the summation of all its members, etc. In considering the problems with various semantic approaches, Van Gelder [29] notes that, for many problems in which the use of aggregates has been proposed, the concept of subset is what is really necessary. Our proposed paradigm provides a natural and efficient realization of the concept of monotonic aggregation [26] The fact that aggregate operations are functions ....

[Article contains additional citation context not shown here]

A. Van Gelder, The Well-Founded Semantics of Aggregation, Proc. 11th Symp. on Principles of Database Systems, pp. 127-138, ACM Press, 1992.

....In an early paper [17] it was shown how to extend both relational algebra and calculus with aggregate constructs, but the resulting language did not correspond naturally to any reasonable logic. It is known how to integrate aggregation into datalog like languages (both recursive and nonrecursive) [30, 33], and various aspects of such aggregate languages were studied (e.g. query optimization [26] and handling constraints involving aggregation [31] At this point, let us give an example of a typical aggregate query that would be supported by all commercial versions of SQL, and use it to explain ....

A. Van Gelder. The well-founded semantics of aggregation. In PODS'92, pages 127-138.

....this topic in [OJP99,OJ99] and we have experimental evidence to make us think that our results, as described in this paper, will help to tackle this problem. The idea of expressing aggregation programs via logic programs with some form of negation as failure has been considered several times, see [VG92,OJ99]. The reason for this is that negation as failure is a well known non monotonic operation that has been investigated extensively in the logic programming eld during the last 20 years. The main idea is to consider a speci cation involving aggregation as a macro for a normal program. Note that the ....

....This approach has been used in the literature to explain new constructs; Sacca Zaniolo s LDL1 choice construct provides but one example of this nature ( SZ91] Here, the choice construct is modelled by taking the stable semantics as the underlying semantics of the translated normal program. In [VG92], the authors mainly investigated monotonic aggregation and used the WFS as the underlying semantics. We build on their approach, and expect to contribute to the understanding of the nature of aggregation, set grouping, negation as failure, and the relationship between them. To this end, our aimis ....

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Allen Van Gelder. The Well-Founded Semantics of aggregation. In Proc. of the 11th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, San Diego, USA, pages 127-138. ACM Press, 1992.

....The generalization is based on a simple but powerful idea of aggregation on three valued multisets. The use of three valued multisets makes our extended well founded semantics, which we call aggregate well founded semantics, more intuitive than the extension of well founded models by Van Gelder [Van92]. Our semantics and Van Gelder s semantics agree on many programs, and on others our semantics provides results that Van Gelder says are intuitive and desirable, but that his semantics does not provide. The extended valid semantics, which we call the aggregate valid semantics, generalizes the ....

....[VRS91] the three valued stable model semantics [Prz90] and the valid semantics [BRSS92a] define semantics for all logic programs with negation and set grouping. Recently, there have been several proposals to extend the semantics to take into account special properties of aggregate operations [GGZ91, RS92, Van92]. Beeri et al. BRSS92b] note that the valid semantics can be extended to take into account special properties of some aggregate operations such as min and max. We present a motivating example in Section 1.1, that illustrates the drawbacks of current semantics for nonstratified aggregation. In ....

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A. Van Gelder. The well-founded semantics of aggregation. In Proceedings of the ACM Symposium on Principles of Database Systems, pages 127--138, 1992.

....S. 2 In our prototype implementation and even in the running example, we make use of strati ed aggregation and extra logical constructs (computed functions or predicates also called built ins) This is not considered in the pure semantics, but can be integrated under certain conditions, see e.g. vG92] for well founded aggregation. De nition 3.18 [Transition Assignments] We de ne the mapping Log by Log(q( t ) f r j r 2 DefE [P IDB ] q)g for all ground DB atoms q( t ) 2 B and the mapping Upd by Upd(INS r( t ) f r( t )g Upd(DEL r( t ) fr( t )g for all ground EDB ....

A. van Gelder. The well-founded semantics of aggregation. In Proc. 11th ACM Symp. on Principles of Database Systems, pages 127-138, 1992.

.... 8 c; l; r; rc; n: room of lecture(l; r) capacity(r; rc) course(l; c) Card(fstjenrolled(st; c)g; n) rc n Other frequently occurring aggregates are summation, minimum, maximum, etc : Therefore aggregates are extensively studied in logic programming and deductive databases [16, 20, 17]. A natural choice of the alphabet al..so poses requirements on the level of problem solving capabilities of the used logic. Illustrated in the domain of university lecture scheduling, some basic concepts are lectures, time slots, rooms. Basic relations between these concepts describe when and ....

....we de ne an extension of ID logic with aggregates, sketch an extension of the solver of [9] to reason on aggregates and describe two computational experiments with the system for solving a sea battle puzzle and a scheduling problem. At the logical level, our work is based on Van Gelders s work [20] on aggregates in the context of well founded semantics. On the other hand the implementation and experiments extend previous work described in [10, 18] To the best of our knowledge, this paper is the rst report on solving satis ability and abductive problems in the context of (an extension of) ....

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Allen Van Gelder. The well-founded semantics of aggregation. In ACM, editor, PODS '92. Proceedings of the Eleventh ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, June 2-4, 1992, San Diego, CA, pages 127-148, New York, NY 10036, USA, 1992. ACM Press.

.... the usual Herbrand pre interpretation [Llo87] 2 In our prototype implementation and even in the running example, we make use of stratified aggregation and extra logical constructs (built ins) This is not considered in the pure semantics, but can be integrated under certain conditions, see e.g. [vG92] for well founded aggregation. We will now define the interpretation of update formulas. Note, that we define the semantics of quantifiers over a replacement of variables by ground terms. This is correct, because we use the Herbrand pre interpretation [Llo87] where every domain element can be ....

A. van Gelder. The well-founded semantics of aggregation. In Proc. 11th ACM Symp. on Principles of Database Systems, pages 127--138, 1992.

....In an early paper [25] it was shown how to extend both relational algebra and calculus with aggregate constructs, but the resulting language did not correspond naturally to any reasonable logic. It is known how to integrate aggregation into datalog like languages (both recursive and nonrecursive) [38, 41], and various aspects of such aggregate languages were studied (e.g. query optimization [34] handling constraints involving aggregation [39] query containment and rewriting [7, 17] interaction with functional programming constructs [6] At this point, let us give an example of a typical ....

A. Van Gelder. The well-founded semantics of aggregation. In ACM Principles of Database Systems 1992, pages 127--138.

....this topic in [OJP99,OJ99] and we have experimental evidence to make us think that our results, as described in this paper, will help to tackle this problem. The idea of expressing aggregation programs via logic programs with some form of negation as failure has been considered several times, see [VG92,OJ99]. The reason for this is that negation as failure is a well known non monotonic operation that has been investigated extensively in the logic programming field during the last 20 years. The main idea is to consider a specification involving aggregation as a macro for a normal program. Note that ....

....This approach has been used in the literature to explain new constructs; Sacca Zaniolo s LDL1 choice construct provides but one example of this nature ( SZ91] Here, the choice construct is modelled by taking the stable semantics as the underlying semantics of the translated normal program. In [VG92], the authors mainly investigated monotonic aggregation and used the WFS as the underlying semantics. We build on their approach, and expect to contribute to the understanding of the nature of aggregation, set grouping, negation as failure, and the relationship between them. To this end, our aimis ....

[Article contains additional citation context not shown here]

Allen Van Gelder. The Well-Founded Semantics of aggregation. In Proc. of the 11th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, San Diego, USA, pages 127--138. ACM Press, 1992.

....problems, and to incorporate new logic programming paradigms such as constrained logic programming and concurrent logic programming in the nonmonotonic framework. Extensions of the languages by allowing more complex data such as sets and aggregates are very important in database applications [KS91, BRSS92, Gel92a]. Even more questions remain for noncategorical programs, i.e. logic programs with multiple answer sets or disjunctive logic programs. In this case one of the most important problems seems to be the lack of clear procedural interpretation of rules of a program. Such interpretation of definite ....

A. Van Gelder. The well-founded semantics of aggregation. In Proceedings of Principles of Database Systems, pages 127--138, 1992.

....the program. The problem is to reconciliate the evaluation procedures for optimization goals with the declarative semantics of CLP, and its properties of compositionality. 1 There has been work on aggregation in deductive databases, the recursive use of aggregation [20] well founded aggregation [24] and the bottom up evaluation of such programs. In presence of function symbols the undertaken canonical model approach to the declarative semantics of deductive databases is not viable however as such semantics then become highly undecidable, and can be used only for restricted classes of CLP ....

A. Van Gelder, "The well-founded semantics of aggregation", ACM Symposium on Principles of Database Systems, PODS'92, 1992.

....no uninterpreted function symbols. Preference datalog programs can be thought of as adding to datalog [89] the capability of expressing criteria for choosing among alternative solutions to a query. This can be viewed as a generalization of extremevalue aggregate operations in deductive databases [91]. In both the grammar and database contexts, a bottom up evaluation mechanism seems more natural, and we therefore propose to investigate extensions of bottom up evaluation techniques for definite clause programs [72] The rest of this proposal is organized as follows. Section 2 discusses related ....

....that place. There may be mistakes in the output of the OCR and we may have to perform relaxation to obtain a reasonable solutions. 4.1. 3 Generalized Aggregation and Constraints in Databases Semantics for aggregate operations has been a topic of great interest in the deductive database community [48, 77, 84, 83, 85, 91]. Our current formulation provides semantics for only the extreme value aggregates such as min and max. However, the natural specification of many database applications, such as the bill of materials problem , the company controls problem , etc. 43, 71, 77] require recursion through inductive ....

A. van Gelder. The Well-Founded Semantics of Aggregation. In Proc. ACM Symp. on Principles of Database Systems, pages 127--138, 1992.

....of extreme value aggregate operations (such as min and max) in deductive databases. A program with such aggregate operations has an equivalent first order formulation using negation. There has been a considerable interest recent years in providing a satisfactory semantics for aggregate operations [3, 11, 17, 21, 24]. Ganguly et al. [3] considered first order aggregates and showed that under certain monotonicity conditions, the firstorder equivalent program has a total well founded model [25] that can be computed using a greedy fixed point procedure. Kemp and Stuckey [11] examined programs with recursion ....

A. Van Gelder. The Well-founded Semantics of Aggregation. In Proc. 11th ACM Symposium on Principles of Database Systems, pages 127--138, 1992.

....but powerful idea of aggregation on three valued multisets. The use of three valued multisets makes our extended well founded semantics, which we call aggregate wellfounded semantics, easier to understand and more intuitive, in our opinion, than the extension of well founded models by Van Gelder [14]. 1 Introduction In recent years there has been much interest in defining semantics for deductive databases logic programs that use negation and aggregation. We call such programs extended logic programs. Early approaches restricted the class of programs to those that are stratified in some ....

....for programs with general (non stratified) negation. Semantics that allow non stratified aggregation were proposed in [2, 5] but they leave too many facts undefined. Recently, there have been several proposals to define semantics that take into account special properties of aggregate operations [4, 13, 14]. Most of the above semantics are three valued, i.e. in models based on these semantics, facts may be true, false or undefined. The collections (i.e. sets multisets) of values to be aggregated upon are constructed by a groupby operator; the above semantics for aggregation assumed that the ....

[Article contains additional citation context not shown here]

A. Van Gelder. The well-founded semantics of aggregation. In Procs. of the ACM Symp. on Principles of Database Systems, pages 127--138, 1992.

....symbol f . For f , ext(f) has a similar definition. Definition [OJ97] Given a program P , we define ext( P) S f2 head(P ) ext(f) and transl(P) P [ ext(P ) Definition [OJ97] The semantics of a program P is defined as WFS (transl(P ) The following two examples are taken from [Van92]. In both cases our approach defines the correct intended model. Example from [Van92] Min Max game) p(X) f(X) final(X) p(X) q(Y) edgep(X,Y) q(X) f(X) final(X) q(X) p(Y) edgeq(X,Y) This program is considered nonmonotonic due to the interaction of with . It models a two player ....

....P , we define ext( P) S f2 head(P ) ext(f) and transl(P) P [ ext(P ) Definition [OJ97] The semantics of a program P is defined as WFS (transl(P ) The following two examples are taken from [Van92] In both cases our approach defines the correct intended model. Example from [Van92] (Min Max game) p(X) f(X) final(X) p(X) q(Y) edgep(X,Y) q(X) f(X) final(X) q(X) p(Y) edgeq(X,Y) This program is considered nonmonotonic due to the interaction of with . It models a two player game defined on a bipartite graph represented by edgep, edgeq. The function f ....

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A. Van Gelder, "The Well-Founded Semantics of Aggregation", Proc. 9th ACM Symp. on Principles of Database Systems, 1990, pp. 205-217. This article was processed using the L a T E X macro package with LLNCS style

....problems, and to incorporate new logic programming paradigms such as constrained logic programming and concurrent logic programming in the nonmonotonic framework. Extensions of the languages by allowing more complex data such as sets and aggregates are very important in database applications [KS91, BRSS92, Gel92a]. Even more questions remain for noncategorical programs, i.e. logic programs with multiple answer sets or disjunctive logic programs. In this case one of the most important problems seems to be the lack of clear procedural interpretation of rules of a program. Such interpretation of definite ....

A. Van Gelder. The well-founded semantics of aggregation. In Proceedings of Principles of Database Systems, pages 127--138, 1992.

.... Then, having derived all atoms with stage value J a single pass through the rules of Q ordered according to the primed sorting computes all the atoms with stage value s(J) Therefore, if p 0 is the k th predicate name in the primed sorting, let T k denote the immediate consequence operator [13, 18] for the recursive rules in Q defining p. The composite consequence operator Gamma Q , will be defined as follows: Gamma Q (I) Tn (Tn Gamma1 : T 1 (I) where I denotes subset of Q s Herbrand Base BQ , and n 1. In Example 1, Gamma Q is the composition of the immediate consequence ....

....S 1 and S 2 multi . For instance, the definition of min is single(min; X; X) multi(min; Y1; Z1; Z) if(Y1 Z1 then Z = Y1 else Z = Z1) Therefore, we now need to provide a formal semantics for such extensible aggregates: whereas previous approaches have focused on a fixed set of primitives [10, 19, 13]. Here we propose a semantics that is simple, general and conducive to efficient implementation, as demonstrated by the LDL system [2] Our approach builds on the choice construct, where formal logic based semantics, usability in programming and computational efficiency have successfully ....

[Article contains additional citation context not shown here]

A. Van Gelder. The Well-Founded Semantics of Aggregation. In Proceedings of the Eleventh ACM Symposium on Principles of Database Systems, pages 127--138, 1992.

....semantics to aggregation. Theorem 6.4 carries over to Kemp and Stuckey s extension of the well founded semantics to aggregation. It follows from [KS91] that the valid semantics is equivalent to the perfect model semantics programs that are aggregate stratified or group stratified. Van Gelder [Van92], Ganguly, Greco and Zaniolo [GGZ91] and Ross and Sagiv [RS91] consider how to assign semantics to programs with specific kinds of aggregation. For many programs, by making use of special properties of specific aggregate operations they are able to assign true or false to facts that Kemp and ....

A. Van Gelder. The well-founded semantics of aggregation. In Proceedings of the ACM Symposium on Principles of Database Systems, 1992. (To appear.).

....in database querying. An aggregate operation is a function that maps a set to some value, e.g. the maximum or minimum in the set, the cardinality of this set, the summation of all its members, etc. This has been a topic of considerable interest in the deductive database literature recently [MPR90, KS91, RS92, Van92, GSZ93, SSRB93]. In considering the problems with various semantic approaches, Van Gelder notes that, for many applications in which the use of aggregates has been proposed, the concept of subset is what is really necessary [Van92] Moreover, for many problems requiring aggregate operations to be performed, the ....

.... in the deductive database literature recently [MPR90, KS91, RS92, Van92, GSZ93, SSRB93] In considering the problems with various semantic approaches, Van Gelder notes that, for many applications in which the use of aggregates has been proposed, the concept of subset is what is really necessary [Van92]. Moreover, for many problems requiring aggregate operations to be performed, the concept of monotonic functions is necessary [RS92] We have recently shown that the subset logic paradigm and its generalization (i.e. replacing subset clauses by partial order clauses) are particularly well suited ....

A. Van Gelder, The Well-Founded Semantics of Aggregation, Proc. 11th PODS, pp. 127-138, ACM Press, 1992.

.... predicates defined using aggregation, but required other forms of stratification to ensure that no fact depended on itself through aggregation (e.g. 13] However, there are many useful queries that cannot be easily (if at all) expressed using these semantics, and more recent semantics such as [7, 9, 14, 15, 16, 17] have relaxed or removed stratification requirements. In particular, the monotonic semantics of Ross and Sagiv [14] provides an intuitive meaning for a large class of programs that are not handled by the various stratified semantics. The semantics is a natural extension of the traditional least ....

A. Van Gelder. The well-founded semantics of aggregation. In Proceedings of the ACM Symposium on Principles of Database Systems, pages 127--138, 1992.

....of extreme value aggregate operations (such as min and max) in deductive databases. A program with such aggregate operations has an equivalent first order formulation using negation. There has been a considerable interest recent years in providing a satisfactory semantics for aggregate operations [2, 7, 12, 14, 16]. The most noteworthy semantic difference in our approach is that the ordering among worlds, which is determined from the preferences, explicitly conveys the ordering among solutions, and this ordering is crucial to providing the semantics of relaxation queries. The idea of hard and soft ....

A. Van Gelder. The Well-founded Semantics of Aggregation. In Proc. 11th ACM Symposium on Principles of Database Systems, pages 127--138, 1992.

....facts in which the multiset of Y values is incomplete without affecting the final answer to the user s query. Monotonic programs, where a derivation using an incomplete set does not affect the final set of facts computed, were discussed in [CN89, CM90, MPR90] Ross and Sagiv [RS92] and Van Gelder [Van92] examine broader classes of such programs. A generalization of the well founded model semantics that deals with such programs is presented in [SSRB93] 4 The construct is a generalization of SQL s group by construct. It is defined to generate a nested set of values in LDL, where it was ....

A. Van Gelder. The well-founded semantics of aggregation. In Proceedings of the ACM Symposium on Principles of Database Systems, pages 127--138, 1992.

.... predicates defined using aggregation, but required other forms of stratification to ensure that no fact depended on itself through aggregation (e.g. 13] However, there are many useful queries that cannot be easily (if at all) expressed using these semantics, and more recent semantics such as [7, 9, 14, 15, 16, 17] have relaxed or removed stratification requirements. In particular, the monotonic semantics of Ross and Sagiv [14] provides an intuitive meaning for a large class of programs that are not handled by the various stratified semantics. The semantics is a natural extension of the traditional least ....

A. Van Gelder. The well-founded semantics of aggregation. In Proceedings of the ACM Symposium on Principles of Database Systems, pages 127--138, 1992.

....both min and max; min and max are each monotonic but with respect to opposite orderings. The choice of such an intermediate semantics is beyond the scope of this paper. 5. 6 Van Gelder s Well Founded Semantics Recently, Van Gelder has given a different extension of the well founded semantics [18]. Van Gelder s extension is based on the definition of the well founded model using the alternating fixpoint [19] Van Gelder s extension of the well founded semantics can simulate Kemp and Stuckey s extension, while being less undefined on a number of important examples. In [18] Van Gelder ....

....semantics [18] Van Gelder s extension is based on the definition of the well founded model using the alternating fixpoint [19] Van Gelder s extension of the well founded semantics can simulate Kemp and Stuckey s extension, while being less undefined on a number of important examples. In [18], Van Gelder provides a translation of monotonic programs into his formalism. This translation often (but not always, as illustrated in Example 5.1 below) yields a program whose semantics agrees with ours in the sense that Van Gelder s well founded model contains the atom p if and only if p is in ....

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A. Van Gelder. The well-founded semantics of aggregation. In Proc. Eleventh ACM Symposium on Principles of Database Systems, 1992.

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