S. Patnaik and N. Immerman. Dyn-FO: A parallel dynamic complexity class. In PODS'94, pages 210--221.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in a language such as recursive datalog, and the maintenance is done in relational calculus. The query that received most attention is the transitive closure. It can be easily shown that the transitive closure can be maintained under the insertion of edges [5, 3] A more interesting result of [28, 6] shows that transitive closure of undirected graphs can always be maintained, provided some auxiliary (binary) relations can be used. For directed graphs, the situation is more complex. It is known [4] that the transitive closure of acyclic graphs can be maintained in relational calculus, but the ....

....It is known [4] that the transitive closure of acyclic graphs can be maintained in relational calculus, but the question is still open for arbitrary directed graphs. In general, it is known that every query that can be incrementally maintained in relational calculus has PTIME data complexity [28, 6]. It is conjectured that the containment is strict, but, as was shown in [9] when auxiliary relations of arity 2 or higher are allowed, proving such bounds amounts to proving lower bounds for a general model of computation, and bounds of this kind are extremely hard to obtain. For auxiliary ....

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S. Patnaik and N. Immerman. Dyn-FO: A parallel dynamic complexity class. JCSS, 55 (1997), 199-209.

....auxiliary relations can hold at most O(n k ) tuples, where n is the number of constants in the input database. There are some interesting queries that can be maintained by IES(FO) with some auxiliary relations. For the transitive closure of undirected graphs, it can be maintained in IES(FO) 3 [20] and even in IES(FO) 2 [10] But it is open if there is a IES(FO) for transitive closure of general directed graphs. Also, Dong and Su [10] showed that the IES(FO) k hierarchy is strict for k 2. More recently, using a result of Cai [6] Dong and Su showed in the journal version of their paper ....

....g j X 2 e 1 g. Here X and Y denote edges ( x; y) and (y; z) respectively) whose components, x, y and z, are obtained by applying projections 1 and 2 . 3 Formal Definition of IES(L) The definition of IES(L) is very similar to the definitions of Dong Su s FOIES [10] and Immerman Patnaik s Dyn C [20]. The idea is that, in order to incrementally maintain a query Q, we do the following. At the first step, we initialize auxiliary data and compute Q assuming that the input is empty. Then we provide functions that, upon each insertion or deletion, correctly update both the answer to Q and the ....

S. Patnaik and N. Immerman. Dyn-FO: A parallel dynamic complexity class. In Proceedings of 13th ACM Symposium on Principles of Database Systems, pages 210--221, Minneapolis, Minnesota, May 1994.

....database changes, the changes must be propagated to the views as well. In the case when a view is defined in relational calculus, or at least in the same language in which update propagations are specified, the problem of incremental maintenance has been studied thoroughly. However, few papers [11, 9, 12, 27] addressed the issue of maintaining queries such as the transitive closure in first order or NRC aggr . It was shown [9] that, in the absence of auxiliary data, recursive queries such as transitive closure and same generation cannot be maintained in relational calculus or even in SQL. It was ....

S. Patnaik and N. Immerman. Dyn-FO: A parallel dynamic complexity class. In PODS'94, pages 210--221.

....database changes, the changes must be propagated to the views as well. In the case when a view is defined in relational calculus, or at least in the same language in which update propagations are specified, the problem of incremental maintenance has been studied thoroughly. However, few papers [10, 8, 11, 42] addressed the issue of maintaining queries such as the transitive closure in first order or NRC aggr . It was shown [8] that, in the absence of auxiliary data, recursive queries such as transitive closure and same generation cannot be maintained in relational calculus or even in SQL. It was ....

S. Patnaik and N. Immerman. Dyn-FO: A parallel dynamic complexity class. Journal of Computer and System Sciences, 55(2):199--209, 1997.

....cannot compute from scratch the transitive closure of such graphs. Undirected graphs are the focus of Section 3. The first known technique for maintaining the transitive closure of undirected graphs using relational calculus (or equivalently, first order logic) as the ambient language was that of [27]. A more space efficient technique using relational calculus was reported later in [14, 15] The maintenance of the transitive closure of undirected graphs using SQL is more involved and more expensive than acyclic graphs. In particular, the maintenance of the transitive closure of acyclic graphs ....

....if we store additional relations which are called auxiliary relations. The auxiliary relations are used to maintain information between updates to the graph. The first IES that maintains the transitive closure of undirected graph using nothing more than pure relational calculus was given in [27] and an improved (space wise optimal) IES was subsequently developed in [14] The SQL queries sketched below are mainly derived from the former, except for the explicit maintenance and use of the total order. We again assume the following schemas: G(Start; End) for the input undirected graph and ....

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S. Patnaik and N. Immerman. Dyn-FO: A parallel dynamic complexity class. Journal of Computer and System Sciences, 55(2):199--209, October 1997.

....terms of R, R , and (x; y) Thus transitive closure can be decrementally maintained in a relational database provided the relation involved is acyclic. But this is not satisfactory because acyclicity cannot be tested in relational calculus [10] Another solution is that of Immerman and Patnaik [14]. They proved that transitive closure of undirected graphs can always be maintained, provided some auxiliary ternary relations can be used. Dong and Su [7] strengthened this result further by showing that transitive closure of undirected graphs can be maintained using only auxiliary binary ....

S. Patnaik and N. Immerman. Dyn-FO: A parallel dynamic complexity class. In Proceedings of 13th ACM Symposium on Principles of Database Systems, Minneapolis, Minnesota, pages 210--221, May 1994.

....database changes, the changes must be propagated to the views as well. In the case when a view is defined in relational calculus, or at least in the same language in which update propagations are specified, the problem of incremental maintenance has been studied thoroughly. However, few papers [10, 9, 11, 28] addressed the issue of maintaining queries such as the transitive closure in first order or NRC aggr . It was shown [9] that, in the absence of auxiliary data, recursive queries such as transitive closure and same generation cannot be maintained in relational calculus or even in SQL. It was ....

S. Patnaik and N. Immerman. Dyn-FO: A parallel dynamic complexity class. In Proceedings of 13th ACM Symposium on Principles of Database Systems, Minneapolis, Minnesota, pages 210--221, May 1994.

....auxiliary relations can hold at most O(n k ) tuples, where n is the number of constants in the input database. There are some interesting queries that can be maintained by IES(FO) with some auxiliary relations. For the transitive closure of undirected graphs, it can be maintained in IES(FO) 3 [19] and even in IES(FO) 2 [10] But it is open if there is a IES(FO) for transitive closure of general directed graphs. Also, Dong and Su [10] showed that the IES(FO) k hierarchy is strict for k 2. More recently, using a result of Cai [6] Dong and Su showed in the journal version of their paper ....

....j X 2 e 1 g. Here X and Y denote edges ( x; y) and (y; z) respectively) whose components, x, y and z, are obtained by applying projections 1 and 2 . 3 Formal Definition of IES(L) The definition of IES(L) is very similar to the definitions of Dong Su s FOIES [10] and ImmermanPatnaik s Dyn C [19]. The idea is that, in order to incrementally maintain a query Q, we do the following. At the first step, we initialize auxiliary data and compute Q assuming that the input is empty. Then we provide functions that, upon each insertion or deletion, correctly update both the answer to Q and the ....

S. Patnaik and N. Immerman. Dyn-FO: A parallel dynamic complexity class. In Proceedings of 13th ACM Symposium on Principles of Database Systems, pages 210--221, Minneapolis, Minnesota, May 1994.

....database changes, the changes must be propagated to the views as well. In the case when a view is defined in relational calculus, or at least in the same language in which update propagations are specified, the problem of incremental maintenance has been studied thoroughly. However, few papers [11, 9, 12, 34] addressed the issue of maintaining queries such as the transitive closure in first order or NRC aggr . It was shown [9] that, in the absence of auxiliary data, recursive queries such as transitive closure and same generation cannot be maintained in relational calculus or even in SQL. It was ....

S. Patnaik and N. Immerman. Dyn-FO: A parallel dynamic complexity class. In Proceedings of 13th ACM Symposium on Principles of Database Systems, Minneapolis, Minnesota, pages 210--221, May 1994.

....any auxiliary data (like the parity example above) Finally, IES(L) is the union of all IES(L) k . The most frequently considered class is IES(FO) which uses the relational calculus as its ambient language. There are several examples of queries belonging to IES(FO) that are not definable in FO [21, 7]. The most complex example is probably that of [9] which is a query that is in IES(FO) but cannot be expressed even in first order logic enhanced with counting and transitive closure operators. It is known [7] that the arity hierarchy is strict: IES(FO) k ae IES(FO) k 1 , and that IES(FO) ....

....input database, and O is the output database; and that the update functions must be expressible in the language L. For example, in the previous section we gave an incremental evaluation system for the parity query in relational calculus. That system did not use any auxiliary relations. Following [21, 7, 8, 19], we consider here only queries that operate on relational databases storing elements of the base type b. These queries are those whose inputs are of types of the form fb Theta : Theta bg. Queries whose incremental evaluation we study have to be generic, that is, invariant under ....

[Article contains additional citation context not shown here]

S. Patnaik and N. Immerman. Dyn-FO: A parallel dynamic complexity class. Journal of Computer and System Sciences 55 (1997), 199--209.

....closure of graphs of various kinds. In [DT92, DST95] some insertion only binary foies were given for generalized transitive closure of labelled graphs. For the transitive closure of acyclic directed graphs, DS93, DS95a] gave a space free foies. For undirected graphs, there is a ternary foies [PI94] which maintains an undirected spanning forest for the undirected graph, from which the reachability relation can be extracted. In [PI94] a foies is called a DynFO, meaning dynamic first order. There is also a binary foies [DS95b] which maintains a directed spanning forest of the undirected ....

....labelled graphs. For the transitive closure of acyclic directed graphs, DS93, DS95a] gave a space free foies. For undirected graphs, there is a ternary foies [PI94] which maintains an undirected spanning forest for the undirected graph, from which the reachability relation can be extracted. In [PI94] a foies is called a DynFO, meaning dynamic first order. There is also a binary foies [DS95b] which maintains a directed spanning forest of the undirected graphs, plus some approximation of a total order on the nodes in the graph; it was shown [DS95b] that there is no monadic foies for this ....

Sushant Patnaik and Neil Immerman. Dyn-FO: A parallel dynamic complexity class. In Proc. ACM Symp. on Principles of Database Systems, pages 210--221, 1994.

.... database queries such as transitive closure of (un)directed graphs and parity (whether the number of tuples in a relation is even) cannot be expressed in first order logic [1] Interestingly, many materialized database views defined by such queries Q can be maintained using first order queries [9, 10, 8, 5, 6, 7]. Roughly speaking, such maintenance for the view defined by Q is carried out through a set of first order queries (fixed for Q) which is called a first order incremental evaluation system (or foies for short) 1 . One of these queries directly maintains the answer to Q, while the others ....

....(i) Such maintenance can be implemented in all relational database systems, since first order queries are available in every relational database system, even though the views themselves cannot be defined in first order. ii) First order maintenance algorithms 1 Patnaik and Immerman s DynFO [10] is very similar, though different from our foies. 2 Permissible updates are those updates whose sizes are bounded by a constant dependent only on the query Q and which transforms the old database in the domain of Q to a new database in the domain of Q. have great potential to be adapted for ....

[Article contains additional citation context not shown here]

S. Patnaik and N. Immerman. Dyn-FO: A parallel dynamic complexity class. In Proc. ACM Symp. on Principles of Database Systems, pages 210--221, 1994.

....is a method of storing and modifying data in a data structure, so that each change and query may be performed very efficiently. There is an extensive literature in dynamic algorithms and amortized analysis; but with the exception of [MSV94] a theory of dynamic complexity had been lacking. In [PI97], Patnaik and Immerman began the development of a dynamic complexity theory from the descriptive point of view. Research supported by NSF grant CCR 9877078. In this paper we consider a simpler and more general approach to dynamic complexity than in [PI97] Rather than consider dynamic versions ....

....dynamic complexity had been lacking. In [PI97] Patnaik and Immerman began the development of a dynamic complexity theory from the descriptive point of view. Research supported by NSF grant CCR 9877078. In this paper we consider a simpler and more general approach to dynamic complexity than in [PI97]. Rather than consider dynamic versions of static problems, we consider the general problem of processing a sequence of operations. Previous work on Dyn FO considered only static functions of an input that was subject to simple updates; we consider any dynamic, continuing computation in our model. ....

[Article contains additional citation context not shown here]

S. Patnaik and N. Immerman, "Dyn-FO: A Parallel, Dynamic Complexity Class," J. Comput. Sys. Sci. 55(2) (1997), 199--209.

....is a method of storing and modifying data in a data structure, so that each change and query may be performed very efficiently. There is an extensive literature in dynamic algorithms and amortized analysis; but with the exception of [MSV94] a theory of dynamic complexity had been lacking. In [PI97], Patnaik and Immerman began the development of a dynamic complexity theory from the descriptive point of view. # Research supported by NSF grant CCR 9877078. In this paper we consider a simpler and more general approach to dynamic complexity than in [PI97] Rather than consider dynamic versions ....

....dynamic complexity had been lacking. In [PI97] Patnaik and Immerman began the development of a dynamic complexity theory from the descriptive point of view. # Research supported by NSF grant CCR 9877078. In this paper we consider a simpler and more general approach to dynamic complexity than in [PI97]. Rather than consider dynamic versions of static problems, we consider the general problem of processing a sequence of operations. Previous work on Dyn FO considered only static functions of an input that was subject to simple updates; we consider any dynamic, continuing computation in our model. ....

[Article contains additional citation context not shown here]

S. Patnaik and N. Immerman, "Dyn-FO: A Parallel, Dynamic Complexity Class," J. Comput. Sys. Sci. 55(2) (1997), 199--209.

....that this uses an ordering on the vertices. If no such ordering is given, then we can order edges by their order of insertion. In the presence of an ordering relation, we can modify the construction to always maintain the minimum spanning forest as in Theorem 4.4. This is then memoryless. 8 2 In [PI94] we asked if the proof of Theorem 4.1 could be carried out using auxiliary relations of arity two instead of three. Quite recently Dong and Su have shown that the answer is yes [DS95] They show that the arity three construction of PV can be replaced by a directed version of F and its transitive ....

S. Patnaik and N. Immerman, "Dyn-FO: A Parallel, Dynamic Complexity Class," ACM Symp. Principles Database Systems (1994), 210-221.

....paths in the forest, from x to y via z, that did not pass through a and b, are valid. Also, new paths have to be added as a result of the insertion of a new edge in the forest. PV 0 (x; y; z) j T(x; y; z) 9u; v) New(u; v) New(v; u) T(x; u; x) T(y; v; y) T(x; u; z) T(y; v; z) 2 In [PI94] we asked if the proof of Theorem 4.1 could be carried out using auxiliary relations of arity two instead of three. Quite recently Dong and Su have shown that the answer is yes [DS95] They show that the arity three construction of PV can be replaced by a directed version of F and its transitive ....

S. Patnaik and N. Immerman, "Dyn-FO: A Parallel, Dynamic Complexity Class," ACM Symp. Principles Database Systems (1994), 210-221.

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S. Patnaik and N. Immerman, "Dyn-FO: A Parallel, Dynamic Complexity Class," ACM Symp. Principles Database Systems (1994), 210-221.

....first order variables. The resulting logic, denoted FO 2 (TC) is known to have a data complexity of NSPACE[log n] I87, Var82] This means that the space requirements are manageable, the entire computation is parallelizable, and efficient incremental evaluation is possible (see, for example, [PI94, ZSS94]) Thus, it is very promising to do model checking and symbolic model checking using the language FO 2 (TC) rather than the more complex modal calculus. 2 Background on Temporal Logic and the Modal calculus Let Phi = fp 1 ; p r g be a finite set of propositional symbols. A ....

S. Patnaik and N. Immerman, "Dyn-FO: A Parallel, Dynamic Complexity Class," Proc. ACM Symp. on Principles of Database Systems (1994), 210221.

No context found.

S. Patnaik and N. Immerman. Dyn-FO: A parallel dynamic complexity class. In PODS'94, pages 210--221.

No context found.

S. Patnaik and N. Immerman. Dyn-FO: A parallel dynamic complexity class. In Proceedings of 13th ACM Symposium on Principles of Database Systems, Minneapolis, Minnesota, pages 210--221, May 1994.

No context found.

S. Patnaik and N. Immerman. Dyn-FO: A parallel dynamic complexity class. In Proceedings of 13th ACM Symposium on Principles of Database Systems, pages 210--221, Minneapolis, Minnesota, May 1994.

No context found.

S. Patnaik and N. Immerman. Dyn-FO: A parallel dynamic complexity class. Journal of Computer and System Sciences, 55(2):199--209, 1997.

No context found.

S. Patnaik and N. Immerman. Dyn-FO: A parallel dynamic complexity class. Journal of Computer and System Sciences 55 (1997), 199--209.

No context found.

S. Patnaik and N. Immerman. Dyn-FO: A parallel dynamic complexity class. JCSS, 55 (1997), 199--209.

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Sushant Patnaik and Neil Immerman. Dyn-FO: A parallel dynamic complexity class. In Proc. ACM Symp. on Principles of Database Systems, pages 210--221, 1994. 8

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