D. Suciu, Fixpoints and bounded fixpoints for complex objects, in "Proceedings of 4th International Workshop on Database Programming Languages," Manhattan, New York, August 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....languages is to strike a reasonable balance between expressiveness and tractability. We use the term tractability to mean polynomial time computable. The focus of this paper is on studying tractable iteration mechanisms for bags. Such mechanisms have been developed in the context of set languages [11, 12, 14, 19, 21]. Most typically, an inflationary fixpoint construct is used for flat relations (sets of tuples) It was shown by Vardi [21] and by Immerman [14] that the relational algebra, when augmented with the inflationary fixpoint construct, can express all polynomial time queries over sets in the presence ....

....of nesting can be constructed. For instance, the powerset operator is definable via an inflationary fixpoint operator. Thus, several techniques have been developed in order to restrict the fixpoint operator. In [12] no operation creating additional levels of nesting can be iterated over; and in [19] a bound for the result of the fixpoint operator is precomputed. Both approaches give us precisely the PTIME queries over nested sets when a total order on the domain of atomic elements is available (this follows from the results in the papers cited above and in [13] It is shown in [10] that ....

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D. Suciu. Fixpoints and bounded fixpoints for complex objects. In Proceedings of the 4th Workshop on Database Programming Languages, Springer Verlag, 1994.

....2 . 2 The conservative extension property was first studied by Paredaens and Van Gucht [46] and later by Van den Bussche [55] They proved that NRC(eq) has it when the input and output are restricted to flat relations. It was then extended by Wong [59] to any input and output. More recently, Suciu [50] managed to prove the remarkable theorem that NRC(eq; bfix ) note the absence of natural numbers, has the conservative extension property when input and output are restricted to flat relations. Here bfix is a bounded version of the fixpoint operator. When added to first order logic, it yields a ....

D. Suciu, Fixpoints and bounded fixpoints for complex objects, in "Proceedings of 4th International Workshop on Database Programming Languages," Manhattan, New York, August 1993.

....the level of nesting of sets is restricted to 1, the language expresses precisely the PTIME queries over flat relations. However, when the restriction on the level of nesting is removed, the power of the language increases to the power of all primitive recursive functions. More recently, Suciu [17] has shown that, in the presence of ordering, the nested algebra plus a bounded fixpoint operator expresses all PTIME queries over nested relations, which he extended with Gyssens and Van Gucht [7] to a more syntactic characterization. Although set based models are well suited for reasoning about ....

D. Suciu, Fixpoints and bounded fixpoints for complex objects, Proceedings of the Fourth International Workshop on Database Programming Languages, 1993.

....13 It is also possible to define a safe version of the uniform calculus and establish its equivalence to the safe uniform algebra. We do this in the full paper. Proof (sketch) The safe UA can easily be seen to be a sublanguage of the nested relational algebra [25] In a recent paper, Suciu [23] established that nested relational algebra expressions can be evaluated in AC 0 . The simulation of the relational algebra in safe UA uses techniques similar to the ones used to define derived UA operators such as , Gamma, Theta, etc. Remark 4.6 (On the meta data querying power of safe UA. ....

....direction for future research is to adapt relational query languages with iteration constructs (such as datalog, FO IFP, and FO PFP) to uniform query languages with corresponding iteration mechanisms. Technically, to limit the expressiveness of such languages one can use the ideas of Suciu [23] related to bounded fixpoints in nested relational query languages. Such query languages will enable the formulation of meta data queries which need to be expressed in terms of the path structure of the meta data space. An example of such a query is Are Bob and Alice related in the database ....

D. Suciu. Fixpoints and bounded fixpoints for complex objects. In Proc. DBPL 1993, pages 263--281.

....(s; b) 2 Cg 2 Conservative extension property was first studied by Paredaens and Van Gucht [29] and later by Van den Bussche [10] They proved that NRC(eq) has it when input and output are restricted to flat relations. It was then extended by Wong [38] to any input and output. More recently, Suciu [32] managed to prove the remarkable theorem that NRC(eq; bfix) note the absence of natural numbers, has the conservative extension property when input and output are restricted to flat relations. The results presented in this section show that, with very little extra, conservative extension property ....

D. Suciu, Fixpoints and bounded fixpoints for complex objects, Technical Report MS-CIS-9332 /L&C 58, University of Pennsylvania, 1993.

....that implement external functions (point 4) 7.2 Further results on structural recursion and collection types Since the appearance of the papers on which this work was based [7, 9] a substantial body of related research has appeared. Following the conservative extension result of [63] [52] shows that by adding a bounded fixed point construct to R( cond) gives us, at relational types, inflationary datalog. In [34, 35] it is shown that nesting at intermediate types does not add expressiveness in presence of aggregate functions and certain generic queries. Other results on ....

D. Suciu, Fixpoints and bounded fixpoints for complex objects, in "Proceedings of 4th International Workshop on Database Programming Languages," 263--281, Manhattan, New York, August 1993.

....that NRC( powerset) is not conservative over flat input and output. This failure of conservativity for NRC( powerset) was generalized to all input and output heights by Grumbach and Vianu [6] In contrast, our result shows that conservativity can be repaired with very little extra. Suciu [21] showed that NRC( bfix) is conservative over flat relations. His result is remarkable in that it did not need any arithmetic nor order. Furthermore, it is also valid when bounded fixpoint is replaced by bounded partial fixpoint operator. Our result uses arithmetic but holds for bounded fixpoint ....

....failure of conservative extension for NRC( powerset) with respect to flat relations. The latter generalized this result to any i and o. The corollary above showed that the failure at higher heights can be repaired by augmenting NRC( powerset) with a summation operator. More recently, Suciu [21] showed, using a technique related to that of Van den Bussche [5] that NRC( bfix) i;o;h = NRC( bfix) i;o;h 1 for i = o = 1. This is remarkable because he did not need any arithmetic operation. The corollary above showed that the conservativity of bounded fixpoint can be extended to all input ....

Dan Suciu. Fixpoints and bounded fixpoints for complex objects. This volume.

....and different from y, so h A (x; y) false. The cases h B ( x) y) true and h B ( x) y) undefined are similar. All queries in the nested relational algebra NRA( Sigma) 10] even when extended with fixpoints or bounded fixpoints (NRA( Sigma) fix and NRA( Sigma) bfix, see [26]) are Sigma queries. Here is a brief description of theses languages. First they contain the primitives: p : d p c p for p 2 Sigma, i : 1 Theta 2 i the projections (i = 1; 2) j : oe foeg the singleton (oe(x) fxg) ffoegg foeg flatten ( x) S y2x y) f g ....

....objects. We immediately have: Proposition 3 Any rm computable function is a Sigma query. Proof Obvious 2 Proposition 4 The class of rm computable functions contains all functions expressible in NRA( Sigma) and is closed under composition, pairing, fixpoints and bounded fixpoints (see [26]) Proof Any function in NRA( Sigma) can be computed in one step. It is easy to see that the rm computable functions are closed under composition and pairing. fix(f) is also easily computed, by iteration. 2 Proposition 5 The rm computable functions are closed under map. Proof Suppose f : oe ....

D. Suciu, Fixpoints and Bounded Fixpoints for Complex Objects, MS-CIS93, Univ. of Pennsylvania.

....9] and its connection with structural recursion and complex object algebras was studied in [9] That comprehension syntax at relational types gives us a language equivalent to the relational algebra was shown in [25, 35] even if nesting is used in intermediate results. Another result of this kind [27] shows that by adding a bounded fixed point construct to comprehension syntax gives us, again at relational types, inflationary datalog, and in [19, 21] it is shown that nesting at intermediate types does not add expressiveness in presence of aggregate functions and certain generic queries. Other ....

D. Suciu. Fixpoints and Bounded Fixpoints for Complex Objects. In Proceedings of 4th International Workshop on Database Programming Languages, Manhattan, New York, 1993.

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D. Suciu, Fixpoints and bounded fixpoints for complex objects, in "Proceedings of 4th International Workshop on Database Programming Languages," Manhattan, New York, August 1993.

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