D. Suciu and V. Breazu-Tannen. A query language for NC. In Proceedings of 13th ACM Symposium on Principles of Database Systems, pages 167--178, Minneapolis, Minnesota, May 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... Theta : Theta fjt m jg, where t i s are arbitrary types. In other words, we are interested in queries that take a tuple of bags as an input and produce outputs that are tuples of bags. This restriction is often made when one captures a complexity class over relations or complex objects, cf. [19, 20]. Extension to scalar types can be achieved rather straightforwardly, for example, by using a function extracting an element from a singleton set. We use the standard encoding scheme such as the one in [1] Given a set of values A = fa 1 ; ang of the base type b such that a 1 : an ....

.... class of queries PTIME bag is captured by BQL( bfp; gen) Can we obtain similar characterizations for other complexity classes, for example, L bag , NL bag and NC bag For example, a characterization of NC queries over nested relations that uses divide and conquer recursion was given in [20]. Does a similar recursion mechanism (essentially the structural recursion on the union presentation, cf. 2] when added to BQL( gen) capture NC bag More generally, let C be a complexity class, and L a set language of the form NRA(p; that captures all C queries over sets. Here NRA is the ....

D. Suciu and V. Tannen. A query language for NC. In Proceedings of the 13th Symposium on Principles of Database Systems, Minneapolis, MN, 1994.

....and does not contain the parity test, then we would have solved Conjecture 1. It is known that AC 0 ae DLOGSPACE [5, 27] If NRL aggr had AC 0 data complexity, the same argument would solve at least Conjectures 1 and 2. However, while queries written in NRL have AC 0 data complexity [51], it is not hard to see that there are non AC 0 queries in NRL aggr since multiplication is not in AC 0 [5] As a more interesting example, recall that parity of cardinality is definable in NRL aggr if Q, unit and bool are the only base types. Note that this does not mean Conjecture 1 is ....

D. Suciu, V. Tannen, A query language for NC, in "Proceedings of 13th ACM Symposium on Principles of Database Systems," Minneapolis, Minnesota, May 1994.

....name pump in FAD [BBKV88] from which it is borrowed, under the name hom , in Machi1 avelli [OBB89] where it fit nicely into the language s type system. Called (a form of) transducer , it is part of SVP [PSV92] Some limitations of its theoretical expressive power were examined in [IPS91] In [SBT94] it plays a crucial role in a language that captures the complexity class NC . The most thoroughly investigated collection type is that of complex objects (see [Hul87, ABGG89, AK89] and the references therein) Variations on the theme are nested relations [JS82] and V relations [SAB 87, ....

Dan Suciu and Val Breazu-Tannen. A query language for NC. In Proceedings of 13th ACM Symposium on Principles of Database Systems, Minneapolis, Minnesota, May 1994. To appear. See also UPenn Technical Report MS-CIS-94-05.

....If it could be shown that the complexity of NRC aggr queries is in a class that is strictly lower than DLOGSPACE and does not con4 tain the parity test, then we would have solved conjecture 1. It is known that AC 0 ae DLOGSPACE [FSS84, Joh90] Queries written in NRC have AC 0 data complexity [ST94]. This inclusion implies that the parity test (is the cardinality of a set even ) and the transitive closure cannot be expressed in NRC because they can not be done within AC 0 [FSS84, Joh90] If NRC aggr had AC 0 data complexity, the same argument would solve at least conjectures 1 and 2. ....

D. Suciu and V. Tannen. A query language for NC. This volume.

....of RA. AC0 [FSS84] is the class of problems that can be solved on boolean circuits, with arbitrary fan in gates, of constant size and polynomially many processors. The AC0 upper bound offers potential for efficient parallel evaluation. RA enjoys an AC0 upper bound [AHV94] and so does NRA [ST94]. It is well known that there are simple functions that are not computable in AC0, such as multiplication and parity test [FSS84] It follows then from Theorem 3.1, that BALG is not in AC0. As a more interesting example of violation of the AC0 upper bound, we show that the parity of the ....

D. Suciu and V. Tannen. A query language for NC. In Proc. 13th ACM Symp. on Principles of Database Systems, 1994.

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D. Suciu and V. Breazu-Tannen. A query language for NC. In Proceedings of 13th ACM Symposium on Principles of Database Systems, pages 167--178, Minneapolis, Minnesota, May 1994.

....syntax of functional programming. Moreover Trinder and Wadler [71] showed that an extension of comprehension can implement the (flat) relational calculus. Trinder and Watt [70, 76] have also sought after a uniform algebra for several different collection types. This was followed by work at Penn [14, 15, 77, 17, 48, 47, 68, 67] and elsewhere [29] NESL NESL is a powerful, general purpose parallel functional languages designed by Guy Blelloch [8] which comes with a high level definition of the parallel complexity. It is compiled into a Vector Random Access Machine (VRAM) by flattening nested parallelism and using ....

....design and analysis of parallel algorithms [42] Our optimal algorithm for monotone communication is for the LogP model. Our previous work Some results in this work have been previously announced in [66] and [64] Moreover, this work can be seen as a materialization of the theoretical result in [65, 67]. Indeed, we give there a characterization of NC in terms of a query language for sets featuring the monad constructs seen also here and divide and conquer recursion (dcr) and it is easy to see that dcr admits an NC implementation with map recursion. 1.3 Overview This paper is organized as ....

Dan Suciu and Val Tannen. A query language for NC. Journal of Computer and System Sciences, 55(2):299--321, October 1997.

....design and analysis of parallel algorithms [42] Our optimal algorithm for monotone communication is for the LogP model. Our previous work Some results in this work have been previously announced in [66] and [64] Moreover, this work can be seen as a materialization of the theoretical result in [65, 67]. Indeed, we give there a characterization of NC in terms of a query language for sets featuring the monad constructs seen also here and divide and conquer recursion (dcr) and it is easy to see that dcr admits an NC implementation with map recursion. 1.3 Overview This paper is organized as ....

Dan Suciu and Val Breazu-Tannen. A query language for NC. In Proceedings of 13th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, pages 167--178, Minneapolis, Minnesota, May 1994.

....is complete for NLOGSPACE, w.r.t. first order reductions [Imm87] hence, if transitive closure were expressible in NRA, then NRA = NLOGSPACE. By generalizing Immerman s result that FO AC 0 [Imm89, BIS90] we can prove that the complexity of NRA (without powerset) is included in AC 0 (see [SBT94] for more details) Proposition 4.3 NRA AC 0 . On the other hand, using some result by Furst, Saxe and Sipser, and independent by Ajitai, one can show that AC 0 6= NLOGSPACE (see [BIS90] hence transitive closure cannot be expressed in NRA. Trying to reason along the same lines for the ....

Dan Suciu and Val Breazu-Tannen. A query language for NC. In Proceedings of 13th ACM Symposium on Principles of Database Systems, pages 167--178, Minneapolis, Minnesota, May 1994. See also UPenn Technical Report MS-CIS94 -05.

....it is easy to prove the following: Proposition 4.2 All queries in NRA( Sigma) fix are ef domain independent and continuous. Also, queries expressed with other forms of iterations, like loop of [18] the structural recursions sru; sri of [7, 8] and the divide and conquer recursion dcr of [24] are also ef domain independent and continuous. We take the above proposition as evidence that the notion of ef domain independence is more appropriate for queries with external functions than the notions of em domain independence or boundeddepth domain independence. 5 Computable Queries A ....

Dan Suciu and Val Breazu-Tannen. A query language for NC. In Proceedings of 13th ACM Symposium on Principles of Database Systems, pages 167--178, Minneapolis, Minnesota, May 1994. See also UPenn Technical Report MS-CIS-94-05.

....featuring structural recursion (together with a few basic constructs such as the ability to manipulate pairs) asking whether equational conditions like associativity, left idempotence, commutativity, etc. are true about program phrases is undecidable, in fact, not even recursively axiomatizable [8, 53]. A programming languages based on full fledged structural recursion on collections is therefore not an r.e. language not an easy sell As we will see however, in database programming this is not necessarily a serious inconvenience, as most programming can be done with restrictions that are ....

....a necessarily exponential implementation in complex object algebra [1] 30] show how to encode related database languages in the simply typed lambda calculus. The possibility of treating arrays as collection types is suggested in [12] Connections with parallel complexity classes are studied in [53]. 56] shows that, in the presence of suitable external functions, sr add set is strictly more expressive that sr comb set . 55] studies foundational issues concerning complex objects with queries over external functions. Acknowledgements. The authors thank Foto Afrati, Dirk Van Gucht, Leonid ....

D. Suciu, V. Tannen, A query language for NC, in "Proceedings of 13th ACM Symposium on Principles of Database Systems," 167--178, Minneapolis, Minnesota, May 1994.

....AND and OR gates, having polynomial size and constant depth (see [BIS90] The class TC 0 is defined similarly, but by allowing an additional type of gates, the threshold gates; a threshold gate with number k will output an 1 iff at least k of its inputs are 1. It is known that NRA AC 0 [SBT94]. Consider now the PTIME fragment of NRA(powerset) more formally: Definition 8.1 Let NRA(powerset) j PTIME def = ff j f 2 NRA(powerset) 9P; P is a polynomial; 8C:complex(f; C) P (size(C) g Then we can prove: Proposition 8.2 NRA(powerset) j PTIME TC 0 . Since transitive closure is ....

Dan Suciu and Val Breazu-Tannen. A query language for NC. In Proceedings of 13th ACM Symposium on Principles of Database Systems, pages 167--178, Minneapolis, Minnesota, May 1994. See also UPenn Technical Report MS-CIS-94-05.

....by: sri(e; i)hz; i def = e(z) sri(e; i)hz; fxg[ yi def = ihx; sri (e; i)hz; yii. In order for it to be defined, i has to be commutative and idempotent, see [BTS91] Again, the type r is meant to compensate for the lack of free variables. Finally the divide and conquer recursion on sets [SBT94], dcr(e; f; u) is defined by: dcr(e; f; u)hz; i def = e(z) dcr(e; f; u)hz; fxgi def = fhz; xi, and dcr (e; f; u)hz; y [ y 0 i def = uhdcr (e; f; u) y) dcr (e; f; u) y 0 )i, when y y 0 = The function dcr (e; f; u) is defined only if u is commutative, associative, and has e(z) ....

....i def = uhdcr (e; f; u) y) dcr (e; f; u) y 0 )i, when y y 0 = The function dcr (e; f; u) is defined only if u is commutative, associative, and has e(z) as identity. This form of recursion is of interest because it captures the parallel complexity class NC over flat, ordered databases [SBT94]. Proposition 4.2 All queries expressed in any of the languages below are ef domain independent: NRA( Sigma; fix ) NRA( Sigma; loop) NRA( Sigma; sri) NRA( Sigma; dcr) Proof. The proof consists in extending the induction in the proof of Proposition 4.1 to the case when f is one of the ....

Dan Suciu and Val Breazu-Tannen. A query language for NC. In Proceedings of 13th ACM Symposium on Principles of Database Systems, pages 167--178, Minneapolis, Minnesota, May 1994. See also UPenn Technical Report MS-CIS-94-05.

....it is easy to prove the following: Proposition 4.2 All queries in NRA( Sigma) fix are ef domain independent and continuous. Also, queries expressed with other forms of iterations, like loop of [18] the structural recursions sru; sri of [7, 8] and the divide and conquer recursion dcr of [23] are also ef domain independent and continuous. We take the above proposition as evidence that the notion of ef domain independence is more appropriate for queries with external functions than the notions of em domain independence or boundeddepth domain independence. 5 Computable Queries A ....

Dan Suciu and Val Breazu-Tannen. A query language for NC. In Proceedings of 13th ACM Symposium on Principles of Database Systems, pages 167--178, Minneapolis, Minnesota, May 1994. See also UPenn Technical Report MS-CIS-94-05.

....about them for example how to validate source to source optimizations. We have in mind applications to databases and this naturally brings up important complexity issues. In a previous paper we have shown a tight connection between a related data parallel language for sets and the class NC [SBT94]. This in turn has led us to the more practical questions addressed here. 2 The Target: Bounded Vector Random Access Machines To compile the higher level programming language described in section 3 only a very simple vector parallel model is needed. The Bounded Vector Random Access Machine, ....

Dan Suciu and Val Breazu-Tannen. A query language for NC. In Proceedings of 13th ACM Symposium on Principles of Database Systems, Minneapolis, Minnesota, May 1994. To appear. See also UPenn Technical Report MS-CIS-9405.

....query in NRC(sru) exists. However, given certain external functions, we can construct such a query; see Proposition 3. Still our result suggests that NRC(sri) can implement more efficient algorithms than NRC(sru) In the context of database query languages and ordered databases, it is known [22] that first order logic extended with divide and conquer recursion, which is a form of recursion closely related to sru, captures the complexity class NC, while first order logic extended with sri captures PTIME. Here NC is the class of functions computable by polynomially many processors in ....

....are allowed, but cannot be efficiently computed when functional input and output are not allowed. This result, as well as that of Colson, indicates the important effects higher order functions have on performance. We should also remark on the hom operator of Machiavelli [18] called dcr elsewhere [22]. If sru(u; f; e) is well defined, then so is hom(u; f; e) and they compute the same function. The difference is that hom(u; f; e) does not have to obey idempotence axiom. Instead, it is required to satisfy hom(u; f; e) O 1 [O 2 ) u(hom(u; f; e) O 1 ) hom(u; f; e) O 2 ) when O 1 and O 2 are ....

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D. Suciu and V. Breazu-Tannen. A query language for NC. In Proceedings of 13th ACM Symposium on Principles of Database Systems, pages 167--178, Minneapolis, Minnesota, May 1994.

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D. Suciu, V. Tannen, A query language for NC, in "Proceedings of 13th ACM Symposium on Principles of Database Systems," Minneapolis, Minnesota, May 1994.

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