S. Grumbach and C. Tollu. Query languages with counters. In Proc. 4th Int. Conf. Database Theory, LNCS 646, 124--139. Springer-Verlag, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....because they involve finding these symmetries. On the other hand, there is a large class of natural queries requiring symmetry based choices that can be accomplished in polynomial time. For instance, it was shown in the full version of [17] that the polynomial time counting queries (e.g. [9, 10]) can be expressed efficiently by uniformly semi deterministic IQL witness programs. In view of the importance of this class of queries, it is the purpose of the present paper to characterize a minimal class of semi deterministic queries in which the polynomial time counting queries can be ....

.... This class is a two sorted extension of fixpoint logic 8 with counting terms of the form countfy j Phi(x; y)g, which can be used as arguments of numerical functions computable in polynomial time assuming unary notation for numbers, and was studied by Gradel and Otto [9] and Grumbach and Tollu [10]. In the context of object creation, we can alternatively define the counting queries without having to introduce a separate sort for the natural numbers, as a natural number in unary notation is an ordered list of objects. Using this representation, polynomial time functions on natural numbers ....

S. Grumbach and C. Tollu. Query languages with counters. In Proc. 4th Int. Conf. Database Theory, LNCS 646, 124--139. Springer-Verlag, 1992.

....[7] that there is no efficient RA new loop program expressing the parity query. 5.2 Counting semi deterministically We still have to specify precisely what is the class of counting queries. A precisely defined extension of fixpoint logic with counting capabilities has been introduced in [12, 13]. This approach, however, involves a two sorted extension of the relational signature with a sort for natural numbers to express the results of the counting operations. In our object creating framework, however, an essentially equivalent counting mechanism can be introduced without at the same ....

....interchangeable with respect to the whole intermediate result of the computation at that moment. The class of counting queries includes a wide variety of useful queries. In particular, it can be verified that every query expressible in the extension 28 of fixpoint logic with counting defined in [12, 13] is a counting query in our sense. Rather than proving this claim formally (which is tedious but straightforward) we illustrate it in the remainder of this section. For example, we can express all global boolean counting queries, such as the parity query. This class of queries could be defined ....

S. Grumbach and C. Tollu. Query languages with counters. In J. Biskup and R. Hull, editors, Database Theory---ICDT'92, volume 646 of Lecture Notes in Computer Science, pages 124--139. Springer-Verlag, 1992.

....often intractable simply because they involve finding these symmetries. On the other hand, there is a large class of natural queries requiring symmetry based choices that can be accomplished in polynomial time. For instance, it was shown in [22] that the the polynomial time counting queries (e.g. [14]) can be expressed efficiently by uniformly semi deterministic IQL witness programs. In view of the importance of this class of queries, it is the purpose of the present paper to characterize a minimal class of semi deterministic queries in which the polynomial time counting queries can be ....

....The obvious implementation of the above program runs in low polynomial time. In pure CQL, however, the query Q cannot even be computed in polynomial space [6] The above query Q is but one example of a large class of polynomial time counting queries. This class was defined by Grumbach and Tollu [14] as a two sorted extension of fixpoint logic 5 with counting terms of the form countf y j Phi( x; y)g, which can be used as arguments of numerical functions computable in polynomial time assuming unary notation for numbers. In the context of object creation, we can alternatively define the ....

S. Grumbach and C. Tollu. Query languages with counters. In Proc. 4th Int. Conf. Database Theory, in Lecture Notes in Computer Science, 646, 124--139. SpringerVerlag, 1992.

....it can be shown using a result on generic complexity [8] that there is no efficient IQL program expressing the parity query. 5.2 Counting semi deterministically We still have to specify precisely what is the class of counting queries. To this end, we will adopt the class of queries introduced in [14], where a very general polynomial time extension of first order logic (and fixpoint logic) with counting capabilities was defined. However, since the query language formalism used there is somewhat different from the one used in the present paper, we will define our own counting operation (denoted ....

....defined. However, since the query language formalism used there is somewhat different from the one used in the present paper, we will define our own counting operation (denoted count ) which better fits the present formalism, but still captures the essence of the counting mechanism considered in [14]. This operation is introduced in the following example: Example 5.1 Assume given a binary relation r, interpreted as a parentchild relation. Suppose we want to count the number of children of each parent. A uniform, generic way to do this would be to construct for each parent a list of new ....

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S. Grumbach and C. Tollu. Query languages with counters. In J. Biskup and R. Hull, editors, Database Theory---ICDT'92, volume 646 of Lecture Notes in Computer Science, pages 124--139. Springer-Verlag, 1992.

....of (resp. is x, of length r (resp. y, of length s) then I r;s x; y( x) y) is a sentence of FO (I r;s ) Its semantics is given by: ffl A j= I r;s x; y( x) y) iff j A j=j A j, where A = fa 2 A r j A j= a)g and A = fa 2 A s j A j= a)g. It is shown in [GT92] that, for any fixed relational schema, there is a sentence of FO (I r;s ) that is not expressible in terms of Hartig quantifiers of type hr 0 ; s 0 i, with maxfr 0 ; s 0 g maxfr; sg. Let I stand for the set of Hartig quantifiers of any type. We next establish our main theorem on the ....

S. Grumbach and C. Tollu. Query languages with counters. In J. Biskup and R. Hull, editors: Proc. 4th Int. Conf. on Database Theory, Lecture Notes in Computer Science, Vol. 646 (Springer-Verlag, Berlin, 1992) 124--139.

....k u . We then turn to the asymptotic probabilities of yet another restriction of the counting logics, where free variables are disallowed in counting expressions. To our knowledge, the only results, on the asymptotic probabilities of extensions of first order logic with counting can be found in [Kny90, GT92, FGT93]. It is proved in [Kny90] that, for a rational r such that 0 r 1, if the asymptotic probability of a formula (x) is different from r, sentences of the form: there is at least a fraction r of the elements of the domain satisfying (x) have a 0 1 law. The restriction on r is crucial. Indeed, ....

....all the PTIME computable ones [Gur83] 2.3 Fixpoint Logic with Counters We now explore the possibility to extend FO both with counters and an operator enabling to define new relations by monotone induction up to a fixpoint. Fixpoint extensions of FO with counting have been investigated in [GT92], but the following presentation owes much to [Ott92] In sharp contrast to the case of the family fFO C k g 1k , we shall no longer have to introduce polyadic counters: indeed, thanks to the fixpoint operator and the possibility to use arbitrarily deep nestings of counters, counters of ....

[Article contains additional citation context not shown here]

S. Grumbach and C. Tollu. Query languages with counters. In J. Biskup and R. Hull, editors, Proc. Int. Conf. on Database Theory, pages 124--139, Berlin, 1992. Springer Verlag. LNCS 646.

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